Coordinate Geometry Class 10 Previous Year Questions: 1 Mark (Chapter 7)

1-mark questions on Coordinate Geometry

This page has 45 1-mark questions on Coordinate Geometry from CBSE Class 10 board papers of 2023, 2024, 2025 and 2026. Of these, 26 appeared only in Standard papers and 19 appeared only in Basic papers. Question formats: 45 MCQs. 16 of them were asked in two or more paper sets.

0 of 45 practised

All 45 Coordinate Geometry 1-mark questions

Q1. For a point \(X(a, b)\) where (\(b \gt a \gt 0\)), the value of its [distance from \(x\)-axis – distance from \(y\)-axis] is :

Mark practised

(A) \(a - b\)
(B) \(b - a\)
(C) \(a^2 - b^2\)
(D) \(b^2 - a^2\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Basic, Set 430/1/3, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) b − a, because the distance from the x-axis is the y-coordinate b, the distance from the y-axis is the x-coordinate a, and b > a makes b − a the positive difference.
Why? For a point with positive coordinates, the perpendicular distance to the x-axis is its y-coordinate and the perpendicular distance to the y-axis is its x-coordinate.

The full 1 mark is for the correct option.

Answer:

(B) b − a

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/1/3: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Subtracting in the wrong order to get a − b; the condition b > a tells you to take the larger distance b minus the smaller distance a.
Using b² − a² or a² − b² by applying the distance formula for two points; here you only need the perpendicular distances to the two axes.
Calling the distance from the x-axis a and the distance from the y-axis b; the abscissa a measures distance from the y-axis, and the ordinate b measures distance from the x-axis.

Why this works

For X(a, b), the distance from the x-axis is b and the distance from the y-axis is a. Because b > a > 0, the required value is b − a. For example, X(2, 5): distance from the x-axis is 5, distance from the y-axis is 2, so 5 − 2 = 3 = b − a. Option A gives a negative value since a < b, and options C and D square the coordinates, which is not what 'distance from an axis' means.

Key idea to remember

Distance of a point from the coordinate axes: see Key ideas used on this page.

Explanation

The NCERT text says the distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of a point from the x-axis is called its y-coordinate, or ordinate. So for X(a, b), the distance from the x-axis is b and the distance from the y-axis is a. The expression therefore becomes b − a.

Important formulas

For P(x, y) with x > 0 and y > 0: distance from the x-axis = y, distance from the y-axis = x · Since a > 0 and b > 0, these apply directly to X(a, b).

Q2. The distance of point (– 3, 4) from y-axis is :

Mark practised

(A) – 3
(B) 3
(C) 4
(D) 5

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Basic, Set 430/3/2, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 3, because the distance from the y-axis is the absolute value of the x-coordinate, |–3| = 3.
Why? The distance of a point from the y-axis is its x-coordinate (abscissa), and distance is always non-negative, so the negative sign is dropped.

The full 1 mark is for the correct option.

Answer:

(B) 3

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/2: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing –3 because the x-coordinate is –3, forgetting that a distance cannot be negative.
Choosing 4, the y-coordinate (ordinate), instead of the x-coordinate.
Finding the distance from the origin, √((–3)² + 4²), and giving 5.

Q3. If point \((a, 2b)\) is the mid-point of the line segment joining the points \((3, 5)\) and \((-1, -1)\) , then \((a, b)\) is equal to :

Mark practised

(A) \((1, 2)\)
(B) \((2, 2)\)
(C) \((2, 1)\)
(D) \((1, 1)\)

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2025, 2026
Topic: 7.3 Section Formula
Source: (CBSE 2025 Mathematics Basic, Set 430/3/3, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) (1, 1), because the mid-point has x-coordinate (3 − 1)/2 = 1 and y-coordinate (5 − 1)/2 = 2, so a = 1 and 2b = 2, giving b = 1.
Why? The mid-point of a segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2); equating the two coordinates gives a and b.

The full 1 mark is for the correct option.

Answer:

(D) (1, 1)

✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/3: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Picking (1, 2) because that is the mid-point itself, without halving 2b to get b = 1.
Equating the second coordinate 2b with 5 or −1 instead of the average 2.
Picking (2, 1) by swapping the coordinates of the mid-point (1, 2).

Why this works

A mid-point divides the segment internally in the ratio 1 : 1, so it is the special case of the section formula. The x-coordinate of a mid-point is the average of the x-coordinates, and the y-coordinate is the average of the y-coordinates. Here the y-coordinate of the mid-point is 2, and the given point is (a, 2b), so 2b must equal 2. That makes b = 1. Another correct method also earns full marks.

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point that divides a line segment internally in the ratio m₁ : m₂. For the mid-point the ratio is 1 : 1, so the formula becomes ((x₁ + x₂)/2, (y₁ + y₂)/2). This solution uses that special case and equates the coordinates with (a, 2b).

Important formulas

Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2) · Coordinates of the mid-point of the segment joining (x₁, y₁) and (x₂, y₂).
Section formula = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) · The mid-point is this formula with m₁ = m₂ = 1.

Q4. In what ratio, does \(x\)-axis divide the line segment joining the points \(A(3, 6)\) and \(B(-12, -3)\) ?

Mark practised

(A) \(1 : 2\)
(B) \(1 : 4\)
(C) \(4 : 1\)
(D) \(2 : 1\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: 7.3 Section Formula
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 2 : 1, because the section formula gives y = (k(−3)+6)/(k+1), and on the x-axis y = 0, so k = 2.
Why? The x-axis is the line whose y-coordinate is zero, so the point of division must have y-coordinate zero; the section formula turns this condition into an equation for the ratio.

The full 1 mark is for the correct option.

Answer:

(D) 2 : 1

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Setting the x-coordinate equal to 0 instead of the y-coordinate, which gives the ratio 1 : 4; the x-axis is the line y = 0.
Reversing the ratio and writing 1 : 2 because B is nearer to the x-axis; the section formula gives AP : PB = 2 : 1.

Why this works

On the x-axis, every point has y-coordinate 0. Let the x-axis divide AB in the ratio k : 1. By the section formula, the y-coordinate of the dividing point is (k(−3)+6)/(k+1). Setting this equal to 0 gives −3k+6=0, so k=2. Hence the ratio is 2 : 1. The x-coordinate is (2(−12)+3)/3 = −7 and the y-coordinate is (2(−3)+6)/3 = 0, so the point lies on AB.

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of the point P that divides the join of A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂. Here, the x-axis is the divider, so the point of division has y-coordinate 0. Writing the ratio as k : 1 and putting the y-coordinate from the section formula equal to 0 gives the required ratio.

Important formulas

x = (m₁x₂ + m₂x₁)/(m₁ + m₂), y = (m₁y₂ + m₂y₁)/(m₁ + m₂) · Section formula for internal division in the ratio m₁ : m₂.
P = ((kx₂ + x₁)/(k + 1), (ky₂ + y₁)/(k + 1)) · Section formula when the ratio is written as k : 1.

Q5. The point of intersection of the line represented by \(3x - y = 3\) and \(y\)-axis is given by

Mark practised

(A) \((0, -3)\)
(B) \((0, 3)\)
(C) \((2, 0)\)
(D) \((-2, 0)\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2023
Topic: Mixed topics
Source: (CBSE 2023 Mathematics Standard, Set 30/6/1, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) (0, -3), because on the y-axis the x-coordinate is 0, so y = 3(0) - 3 = -3.
Why? On the y-axis, the x-coordinate is 0; substituting x = 0 into the line's equation gives the y-coordinate of the intersection.

The full 1 mark is for the correct option.

Answer:

(A) (0, -3)

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/1: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Putting y equal to zero instead of x equal to zero, because the y-axis is confused with the x-axis; that gives a point on the x-axis, not on the y-axis.
Choosing the point with x-coordinate zero and y-coordinate three, by ignoring the minus sign on y in the equation.
Selecting the options with non-zero x-coordinates, which are points on the x-axis, not on the y-axis.

Q6. If the distance between the points \((3, -5)\) and \((x, -5)\) is 15 units, then the values of \(x\) are :

Mark practised

(A) \(12, -18\)
(B) \(-12, 18\)
(C) \(18, 5\)
(D) \(-9, -12\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Standard, Set 30/1/1, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) −12, 18, because √((x − 3)²) = 15, so x − 3 = ±15, giving x = 18 or x = −12.
Why? By the distance formula, when two points have the same y-coordinate, the distance is the positive square root of (x₂ − x₁)², i.e. the absolute difference of the x-coordinates.

The full 1 mark is for the correct option.

Answer:

(B) −12, 18

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing only x = 18 and missing x = −12, because x − 3 can also be −15.
Choosing option A, 12 and −18, by taking x = 15 − 3 and x = −15 − 3 instead of x = 3 + 15 and x = 3 − 15.
Writing the horizontal distance as x + 3 instead of |x − 3|, because the y-coordinates are both −5.

Why this works

Both points have the same y-coordinate, −5, so they lie on a horizontal line. The distance between them is the horizontal distance, √((x − 3)²) = |x − 3|. Since the distance is 15, |x − 3| = 15, so x − 3 = 15 or x − 3 = −15. This gives x = 18 or x = −12. Check: the distance from (3, −5) to (18, −5) is 18 − 3 = 15, and the distance from (3, −5) to (−12, −5) is |−12 − 3| = 15.

Key idea to remember

Distance formula: see Key ideas used on this page.

Explanation

The distance formula gives PQ = √((x₂ − x₁)² + (y₂ − y₁)²). For these points, y₁ = y₂ = −5, so the vertical difference contributes nothing and the distance is √((x − 3)²) = |x − 3|. Setting this equal to 15 gives x − 3 = 15 or x − 3 = −15, hence x = 18 or x = −12.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance formula for two points.
√(a²) = |a| · The square root of a square is the absolute value.
|a| = 15 ⇒ a = 15 or a = −15 · An absolute value equation gives two values.

Q7. AD is a median of \(\triangle ABC\) with vertices \(A(5, -6)\), \(B(6, 4)\) and \(C(0, 0)\). Length AD is equal to :

Mark practised

(A) \(\sqrt{68}\) units
(B) \(2\sqrt{15}\) units
(C) \(\sqrt{101}\) units
(D) 10 units

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Standard, Set 30/1/1, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) √68 units, because AD is a median, so D is the midpoint of BC; its x-coordinate is (6+0)/2 = 3, its y-coordinate is (4+0)/2 = 2, and AD = √((5 − 3)² + (−6 − 2)²) = √68.
Why? By the definition of a median, D is the midpoint of BC, so its coordinates are found by the mid-point formula; the distance formula then gives the length AD.

The full 1 mark is for the correct option.

Answer:

(A) √68 units

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/1/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using A(5, −6) and B(6, 4) in the distance formula, because D is not B; it is the midpoint of BC.
Taking D as the midpoint of AC, since a median from A must end on the side opposite A, which is BC.
Writing the coordinates of B unchanged, (6, 4), as D instead of halving the sum of the coordinates of B and C.

Why this works

In triangle ABC, the median AD ends at D, the midpoint of the side opposite A, which is BC. So the mid-point formula gives the x-coordinate of D as (6 + 0)/2 = 3 and the y-coordinate of D as (4 + 0)/2 = 2. Then the distance formula gives AD = √((5 − 3)² + (−6 − 2)²) = √68. This is option A.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula gives the length of a line segment from the coordinates of its end points. To apply it to the median AD, first find D, the midpoint of BC, using the mid-point formula, because a median from A ends at the midpoint of the opposite side.

Important formulas

\(PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) · Used to find AD from the coordinates of A and D.
\(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) · Used to find D, the midpoint of BC.

Q8. Point P divides the line segment joining the points A(4, -5) and B(1, 2) in the ratio 5:2. Co-ordinates of point P are

Mark practised

(A) \(\left(\frac{5}{2}, \frac{-3}{2}\right)\)
(B) \(\left(\frac{11}{7}, 0\right)\)
(C) \(\left(\frac{13}{7}, 0\right)\)
(D) \(\left(0, \frac{13}{7}\right)\)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2024
Topic: 7.3 Section Formula
Source: (CBSE 2024 Mathematics Standard, Set 30/2/2, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) (13/7, 0), because by the section formula, x = (5(1) + 2(4))/7 = 13/7 and y = (5(2) + 2(−5))/7 = 0.
Why? The section formula gives the coordinates of a point dividing a line segment internally in a given ratio, so substituting the given coordinates and ratio gives the required point.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/2: option (C).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Putting the ratio as 2 : 5 gives x = (2(1) + 5(4))/7 = 22/7, which is not one of the options.
Interchanging A and B gives x = (5(4) + 2(1))/7 = 22/7 and y = (5(−5) + 2(2))/7 = −3, which is not listed.

Why this works

The section formula gives the coordinates of P(x, y) dividing A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ as P = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)). Here the given ratio is 5 : 2, and the coordinates are A(4, −5) and B(1, 2). Putting these into the formula gives x = (5×1 + 2×4)/7 = 13/7 and y = (5×2 + 2×(−5))/7 = 0. So P = (13/7, 0), a point on the x-axis. Another correct method also earns full marks.

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point which divides the line segment joining two given points internally in the ratio m₁ : m₂. This solution substitutes the given ratio and coordinates directly into the formula.

Important formulas

P(x, y) = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)) · Coordinates of the point dividing A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂.

Q9. Two of the vertices of Δ PQR are P(-1, 5) and Q(5, 2). The coordinates of a point which divides PQ in the ratio 2 : 1 are :

Mark practised

(A) (3, -3)
(B) (5, 5)
(C) (3, 3)
(D) (5, 1)

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2025
Topic: 7.3 Section Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/3/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) (3, 3), because the section formula gives x = (2×5 + 1×(−1))/(2+1) = 3 and y = (2×2 + 1×5)/(2+1) = 3.
Why? By the section formula, a point dividing a segment internally in the ratio m₁ : m₂ has coordinates ((m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂)); substituting the given points and ratio 2 : 1 gives the required point.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the ratio in the reverse order (1 : 2 instead of 2 : 1) puts the point at (1, 4), which is not one of the given options.
Taking P as the second endpoint and Q as the first in the section formula also gives (1, 4).
Using the midpoint formula treats the ratio as 1 : 1, not 2 : 1.

Why this works

Here P(-1, 5) is the first endpoint and Q(5, 2) is the second. A point dividing PQ in the ratio 2 : 1 means its distance from P is twice its distance from Q, so it lies closer to Q. Using the section formula with m₁ = 2 and m₂ = 1, the x-coordinate is (2×5 + 1×(−1))/(2+1) = 9/3 = 3 and the y-coordinate is (2×2 + 1×5)/(2+1) = 9/3 = 3. So the point is (3, 3). Quick check: the midpoint of P(-1, 5) and Q(5, 2) is (2, 3.5), and a point dividing the segment in the ratio 2 : 1 lies between the midpoint and Q, which (3, 3) does.

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point P that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ as ((m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂)). In this question, A is P(-1, 5), B is Q(5, 2), and m₁ : m₂ = 2 : 1.

Important formulas

((m₁x₂ + m₂x₁)/(m₁+m₂), (m₁y₂ + m₂y₁)/(m₁+m₂)) · Coordinates of the point dividing A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂.

Q10. The mid-point of the line segment joining the points \((5, -4)\) and \((6, 4)\) lies on :

Mark practised

(A) \(x\)-axis
(B) \(y\)-axis
(C) origin
(D) neither \(x\)-axis nor \(y\)-axis

Marks: 1
Level: Standard
Asked in: 3 paper sets
Years: 2026
Topic: 7.3 Section Formula
Source: (CBSE 2026 Mathematics Standard, Set 30/1/1, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) x-axis, because the mid-point has y-coordinate (-4+4)/2 = 0, and every point with y-coordinate 0 lies on the x-axis.
Why? By the mid-point formula, the y-coordinate of the mid-point is the average of the y-coordinates of the two end points. A point on the x-axis has y-coordinate 0.

The full 1 mark is for the correct option.

Answer:

(A) x-axis

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing the y-axis because -4 and 4 add to 0, without checking that the x-coordinate is not 0.
Thinking (11/2, 0) lies on neither axis because 11/2 is not 0; a point lies on the x-axis when its y-coordinate is 0.
Swapping the averages and writing the mid-point as (0, 11/2).

Why this works

The mid-point divides the line segment in the ratio 1 : 1, so each coordinate of the mid-point is the average of the two corresponding coordinates. The y-coordinates are -4 and 4, whose average is 0. Therefore the mid-point is on the x-axis. The x-coordinate is (5+6)/2 = 5.5, which is not 0, so the mid-point is not on the y-axis and not the origin.

Key idea to remember

Mid-point formula: see Key ideas used on this page.

Explanation

The mid-point of the line segment joining two points is found by taking the average of the x-coordinates and the average of the y-coordinates. On the x-axis, every point has y-coordinate 0.

Important formulas

((x₁ + x₂)/2, (y₁ + y₂)/2) · Coordinates of the mid-point of the segment joining (x₁, y₁) and (x₂, y₂).
(x, 0) · Coordinates of a point on the x-axis.

Q11. In what ratio does x-axis divide the line segment joining the points A(2, -3) and B(5, 6) ?

Mark practised

(A) 2 : 3
(B) 2 : 1
(C) 3 : 4
(D) 1 : 2

Marks: 1
Level: Basic
Asked in: 2 paper sets
Years: 2023
Topic: 7.3 Section Formula
Source: (CBSE 2023 Mathematics Basic, Set 430/2/1, Q2, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 1 : 2, because on the x-axis y = 0, so the section formula gives (6m − 3n)/(m + n) = 0, hence m : n = 1 : 2.
Why? By the section formula, the y-coordinate of the point that divides AB in the ratio m : n is (m y₂ + n y₁)/(m + n); a point on the x-axis has y-coordinate 0, and equating it to 0 gives m : n.

The full 1 mark is for the correct option.

Answer:

(D) 1 : 2

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/2/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Putting the condition y = 0 into the x-coordinate instead of the y-coordinate; the x-axis condition applies only to the y-coordinate.
Reversing the order and writing 2 : 1 because the y-distance from B to the x-axis is twice the y-distance from A to the x-axis; the ratio must be AP : PB.
Confusing the x-axis with the y-axis and taking the point of division as (0, y) instead of (x, 0).

Why this works

Let the x-axis cut AB at P, and let P divide AB in the ratio m : n. The section formula gives the y-coordinate of P as (6m − 3n)/(m + n). Since P lies on the x-axis, its y-coordinate is 0, so 6m = 3n, giving n = 2m, i.e. m : n = 1 : 2. Quick check: with m = 1 and n = 2, P = ((5 + 4)/3, (6 − 6)/3) = (3, 0), which lies on the x-axis.

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of the point P that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ as ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)). Here the dividing line is the x-axis, so P has y-coordinate 0; putting this condition into the section formula gives the required ratio.

Important formulas

\(P(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right)\) · Section formula for a point dividing AB internally in the ratio m₁ : m₂.
A point on the x-axis has coordinates (x, 0). · Its y-coordinate is 0.

Q12. The distance between the points \((0, 2\sqrt{5})\) and \((-2\sqrt{5}, 0)\) is

Mark practised

(A) \(2\sqrt{10}\) units
(B) \(4\sqrt{10}\) units
(C) \(2\sqrt{20}\) units
(D) 0

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Standard, Set 30/1/1, Q13, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 2√10 units, because √((−2√5 − 0)² + (0 − 2√5)²) = √(20 + 20) = √40 = 2√10.
Why? The distance formula comes from the Pythagoras theorem: square the difference of x-coordinates, square the difference of y-coordinates, add them, and take the positive square root.

The full 1 mark is for the correct option.

Answer:

(A) 2√10 units

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/1/1: option (A).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 0 because the two points look close, without applying the distance formula; the points are different, so the distance is not 0.
Writing √(20 + 20) as 2√20; the correct simplification is √40 = 2√10, since 40 = 4 × 10.
Forgetting to square the negative sign: (−2√5)² is 20, not −20.

Why this works

Take the first point as (x₁, y₁) = (0, 2√5) and the second as (x₂, y₂) = (−2√5, 0). The differences are x₂ − x₁ = −2√5 and y₂ − y₁ = −2√5. The distance formula gives √((−2√5)² + (−2√5)²) = √(20 + 20) = √40. Since 40 = 4 × 10, √40 = 2√10. A quick check: (2√10)² = 4 × 10 = 40, which matches the sum of the squares.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It is derived from the Pythagoras theorem: the difference of the x-coordinates and the difference of the y-coordinates are the two legs of a right triangle, and the distance is the hypotenuse.

Important formulas

√((x₂ − x₁)² + (y₂ − y₁)²) · Distance formula for the distance between two points.
√40 = √(4 × 10) = 2√10 · Simplifying a square root by taking out the square factor.

Q13. The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0, 0), C(3, 0) and D(0, 4) are :

Mark practised

(A) (4, 0)
(B) (0, 3)
(C) (3, 4)
(D) (4, 3)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2023
Topic: Mixed topics
Source: (CBSE 2023 Mathematics Standard, Set 30/2/1, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) (3, 4), because A shares the x-coordinate of C, which is 3, and the y-coordinate of D, which is 4.
Why? In a rectangle with sides parallel to the axes, the vertex opposite B lies on the same vertical line as C and on the same horizontal line as D.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/2/1: option (C).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing (4, 3) by taking the x-coordinate of D and the y-coordinate of C; the fourth vertex must use the x-coordinate of C and the y-coordinate of D.
Choosing (0, 3) by reading 3 as a y-coordinate from C; the y-coordinate of A must match D, which is 4.

Why this works

Plot B(0,0), C(3,0) and D(0,4). The side BC lies along the x-axis, so it is horizontal. The side BD lies along the y-axis, so it is vertical. Therefore the rectangle has one side along the x-axis and the other along the y-axis. The fourth vertex A must be directly above C and directly to the right of D, so it is (3,4). The complete rectangle is B(0,0), C(3,0), A(3,4), D(0,4). Another correct method also earns full marks.

Key idea to remember

Coordinates of a point: see Key ideas used on this page.

Explanation

A point is named by its x-coordinate and its y-coordinate. In a rectangle whose sides are parallel to the axes, two vertices on the same vertical side have the same x-coordinate, and two vertices on the same horizontal side have the same y-coordinate. Here A is on the same vertical line as C and on the same horizontal line as D.

Q14. XOYZ is a rectangle with vertices X(-3, 0), O(0, 0), Y(0, 4) and Z(x, y). The length of its each diagonal is

Mark practised

(A) 5 units
(B) \(\sqrt{5}\) units
(C) \(x^2 + y^2\) units
(D) 4 units

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Standard, Set 30/2/1, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 5 units, because the diagonals of a rectangle are equal, and XY = √((0 − (−3))² + (4 − 0)²) = √(9 + 16) = 5 units.
Why? In a rectangle both diagonals are equal; the distance formula gives the length of a diagonal as the positive square root, so XY = √(3² + 4²) = 5.

The full 1 mark is for the correct option.

Answer:

(A) 5 units

✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/1: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing (C), because x² + y² is the square of the diagonal (for Z(−3, 4), x² + y² = 25), not its length.
Choosing (D), because OY = 4 is a side of the rectangle, not a diagonal.

Why this works

The diagonals of rectangle XOYZ are XY and OZ. Since the diagonals of a rectangle are equal, finding XY is enough. XY = √((0−(−3))² + (4−0)²) = √(9+16) = 5. Quick check: the fourth vertex is Z(−3, 4), so OZ = √((−3)² + 4²) = 5. Choice (C), x² + y², is the square of the diagonal, not its length.

Key idea to remember

Distance formula: see Key ideas used on this page.

Explanation

The distance formula gives the length of a segment joining two points P(x₁, y₁) and Q(x₂, y₂) as √((x₂ − x₁)² + (y₂ − y₁)²). In a rectangle the diagonals are equal, so the length of one diagonal, XY, gives the length of each diagonal.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between two points P(x₁, y₁) and Q(x₂, y₂).
OP = √(x² + y²) · Distance of a point P(x, y) from the origin O(0, 0).

Q15. The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is :

Mark practised

(A) \(x = 3\)
(B) \(x = -3\)
(C) \(y = -3\)
(D) \(y = 3\)

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/2/1, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) y = -3, because a line parallel to the x-axis has equation y = constant, and 3 units below the x-axis means y = -3.
Why? The y-coordinate of a point is its distance from the x-axis, and below the x-axis this value is negative.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/1: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Picking the line above the x-axis because the distance is 3, without reading the word 'below'.
Picking a vertical line, because a vertical line is parallel to the y-axis, not the x-axis.
Thinking the line contains only the single point (0, -3), instead of all points whose y-coordinate is -3.

Q16. AOBC is a rectangle whose three vertices are A(0, 2), O(0, 0) and B(4, 0). The square of the length of its diagonal is equal to :

Mark practised

(A) 36
(B) 20
(C) 16
(D) 4

Marks: 1
Level: Standard
Asked in: 2 paper sets
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/3/1, Q16, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 20, because the diagonal AB joins A(0, 2) and B(4, 0), so AB² = (4 − 0)² + (0 − 2)² = 20.
Why? By the distance formula, the square of the distance between two points is the sum of the squares of the differences of their x-coordinates and y-coordinates.

The full 1 mark is for the correct option.

Answer:

(B) 20

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/3/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 16 because only the x-difference 4 is squared, and the y-difference 2 is left out.
Choosing 4 because only the y-difference 2 is squared, and the x-difference 4 is left out.
Choosing 36 because the coordinates 4 and 2 are added to get 6 and then 6 is squared, instead of subtracting coordinates in the distance formula.
Taking AO or OB as a diagonal; they are sides of the rectangle, not diagonals.

Why this works

In rectangle AOBC, the vertices are A(0, 2), O(0, 0), B(4, 0) and C(4, 2). The diagonal AB joins A and B. Draw the right triangle AOB with the right angle at O. Then OA = 2 − 0 = 2 and OB = 4 − 0 = 4. By Pythagoras theorem, AB² = OA² + OB² = 2² + 4² = 4 + 16 = 20. The distance formula gives the same square: AB² = (4 − 0)² + (0 − 2)² = 20. The other diagonal OC also has the same square: (4 − 0)² + (2 − 0)² = 20. A quick check: 2² = 4, 4² = 16, and 4 + 16 = 20.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula gives the distance between two points P(x₁, y₁) and Q(x₂, y₂) as PQ = √((x₂ − x₁)² + (y₂ − y₁)²). The square of the distance is PQ² = (x₂ − x₁)² + (y₂ − y₁)². Here the diagonal is the segment joining A(0, 2) and B(4, 0).

Important formulas

PQ² = (x₂ − x₁)² + (y₂ − y₁)² · Square of the distance between two points.
PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance formula.

Q17. The coordinates of the point A, where AB is the diameter of the circle whose centre is \((3, -2)\) and B \((7, 4)\) is :

Mark practised

(A) \((-1, -8)\)
(B) \((-1, 8)\)
(C) \((1, 8)\)
(D) \((1, -8)\)

Marks: 1
Level: Standard
Compartment: 2023
Topic: 7.3 Section Formula
Source: (CBSE 2023 Compartment Mathematics Standard, Set 30/C/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) (-1, -8), because the centre is the midpoint of AB, so (x+7)/2 = 3 and (y+4)/2 = -2, giving x = -1 and y = -8.
Why? The centre of a circle is the midpoint of every diameter, so the midpoint formula gives the unknown endpoint.

The full 1 mark is for the correct option.

Answer:

(A) (-1, -8)

✓ Answer matches the CBSE answer key, 2023 Compartment, Paper 30/C/1: option (A).

Open CBSE’s official marking scheme for the 2023 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the centre as one endpoint and averaging the centre with B, instead of letting the centre be the midpoint of A and B.
Finding only the x-coordinate and choosing (-1, 8) without checking that the y-coordinate must make (y+4)/2 equal to -2.
Subtracting the centre from B instead of applying the midpoint formula.

Why this works

Let the x-coordinate of A be x and the y-coordinate be y. Since AB is a diameter, the centre is the midpoint of A and B. By the midpoint formula, (x+7)/2 = 3 and (y+4)/2 = -2. From (x+7)/2 = 3, we get x = 2 × 3 − 7 = −1. From (y+4)/2 = −2, we get y = 2 × (−2) − 4 = −8. So A has x-coordinate −1 and y-coordinate −8. Check: the midpoint of A and B has x-coordinate (−1+7)/2 = 3 and y-coordinate (−8+4)/2 = −2, which is the given centre.

Key idea to remember

Section Formula (special case: mid-point): see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point dividing a line segment in a given ratio. When the ratio is 1 : 1, the point is the midpoint, whose coordinates are the averages of the corresponding coordinates of the endpoints. Here the centre is the midpoint of the diameter AB, so we equate the midpoint of A and B to the centre.

Important formulas

Midpoint of A(x₁, y₁) and B(x₂, y₂) = ((x₁+x₂)/2, (y₁+y₂)/2) · Special case of the section formula when the ratio is 1 : 1; the centre of a circle is the midpoint of any diameter.

Q18. The distance between the points \(\left(\frac{-5}{2}, 7\right)\) and \(\left(\frac{-1}{2}, 7\right)\) is :

Mark practised

(A) 3
(B) 2
(C) 4
(D) 9

Marks: 1
Level: Basic
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Basic, Set 430/1/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 2, because the y-coordinates are equal, so the distance is the difference of the x-coordinates: \(|-1/2 - (-5/2)| = 2\).
Why? For two points with the same y-coordinate, the distance formula reduces to the absolute difference of their x-coordinates.

The full 1 mark is for the correct option.

Answer:

(B) 2

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/1/2: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Adding the x-coordinates and taking \(|-5/2| + |-1/2|\) gives 3, which is option A; the distance needs subtraction, not addition.
Using the distance formula but forgetting the square root: \((-1/2 - (-5/2))^2 + (7-7)^2 = 4\), and choosing option C instead of taking the square root to get 2.

Why this works

The two points are \((-5/2, 7)\) and \((-1/2, 7)\). Their y-coordinates are both 7, so the segment joining them is horizontal. The distance is the absolute difference of the x-coordinates: \(|-1/2 - (-5/2)| = 2\). This agrees with the distance formula, because \((7-7)^2 = 0\) leaves only the x-difference. So the correct option is B.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

NCERT's distance formula is \(PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Here the y-coordinates are equal, so the vertical part is zero and the distance becomes the absolute difference of the x-coordinates.

Important formulas

\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) · Distance between two points \((x_1, y_1)\) and \((x_2, y_2)\).
\(|x_2 - x_1|\) · Distance between two points with the same y-coordinate.

Q19. The distance between the points \((6, 2)\) and \((-6, 2)\) is :

Mark practised

(A) \(6\sqrt{2}\) units
(B) 12 units
(C) \(2\sqrt{6}\) units
(D) 6 units

Marks: 1
Level: Basic
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Basic, Set 430/5/2, Q15, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 12 units, because the distance formula gives √((−6−6)² + (2−2)²) = √(144 + 0) = 12 units.
Why? By the distance formula, the distance between two points is the positive square root of the sum of the squares of the differences of their x-coordinates and y-coordinates.

The full 1 mark is for the correct option.

Answer:

(B) 12 units

✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/5/2: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 6√2 by using √(6² + (−6)²) instead of the distance formula √((−6−6)² + (2−2)²).
Choosing 6 units by taking the x-coordinate 6 of the first point as the whole distance and ignoring that the second point is (−6, 2).

Q20. The distance between the points \(P\left(-\frac{11}{3}, 5\right)\) and \(Q\left(-\frac{2}{3}, 5\right)\) is :

Mark practised

(A) 6 units
(B) 4 units
(C) 2 units
(D) 3 units

Marks: 1
Level: Standard
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Standard, Set 30/5/1, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 3 units, because the y-coordinates are equal, so PQ = |−2/3 − (−11/3)| = 3.
Why? Two points with the same y-coordinate lie on a line parallel to the x-axis, so the distance between them is the absolute difference of their x-coordinates.

The full 1 mark is for the correct option.

Answer:

(D) 3 units

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/5/1: option (D).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the difference as −2/3 − 11/3, forgetting that subtracting −11/3 means adding 11/3; the absolute value of this wrong difference is not an option.
Finding the y-difference 5 − 5 and thinking the distance is 0, forgetting that the x-coordinates are different.
Subtracting the x-coordinates in the wrong order and getting a negative number, without taking the absolute value; the distance must be positive.

Q21. The distance of the point \((-4, 3)\) from y-axis is

Mark practised

(A) \(-4\)
(B) 4
(C) 3
(D) 5

Marks: 1
Level: Standard
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Standard, Set 30/6/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 4, because the distance of a point from the y-axis is the absolute value of its x-coordinate, so |-4| = 4.
Why? By the definition in Coordinate Geometry, the distance of a point from the y-axis is its abscissa; distance is always non-negative, so the abscissa is taken in absolute value.

The full 1 mark is for the correct option.

Answer:

(B) 4

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/2: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing -4 because the x-coordinate is -4, without remembering that distance is never negative.
Choosing 3 because it is the y-coordinate, but that is the distance from the x-axis, not from the y-axis.
Choosing 5 by using √((-4)² + 3²), which gives the distance from the origin, not from the y-axis.

Why this works

The y-axis is the vertical line through the origin. For (-4, 3), the x-coordinate is -4, so the point lies 4 units to the left of the y-axis. Distance is measured as a positive number, so the answer is 4. Option C is the distance from the x-axis, option D is the distance from the origin, and option A is the x-coordinate itself, which cannot be a distance. Quick check: 0 - (-4) = 4.

Key idea to remember

Abscissa (x-coordinate): see Key ideas used on this page.

Explanation

In Coordinate Geometry, the distance of a point from the y-axis is called its x-coordinate, or abscissa. For the point (-4, 3), the abscissa is -4; since distance is non-negative, the distance from the y-axis is the absolute value of the abscissa.

Important formulas

Distance of a point P(x, y) from the y-axis = |x| · This distance is the x-coordinate, or abscissa; use the absolute value because distance is non-negative.

Q22. The distance between the points (0, 5) and (-3, 1) is :-

Mark practised

(A) 8 units
(B) 5 units
(C) 3 units
(D) 25 units

Marks: 1
Level: Standard
Years: 2023
Topic: 7.2 Distance Formula
Source: (CBSE 2023 Mathematics Standard, Set 30/6/3, Q1, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 5 units, because √((−3−0)² + (1−5)²) = √(9 + 16) = √25 = 5.
Why? By the distance formula, the distance between two points is the positive square root of the sum of the squares of the differences of their x-coordinates and y-coordinates.

The full 1 mark is for the correct option.

Answer:

(B) 5 units

✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/3: option (B).

Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 25 (option D), which is 9 + 16, without taking the square root; the distance is √25 = 5.
Choosing 3 (option C) by using only the difference of the x-coordinates, −3, and ignoring the y-coordinates.
Writing the distance as 25, the value of the sum of the squares, and forgetting that the formula asks for the square root of that sum.

Why this works

The points are (0, 5) and (−3, 1). The change in x is −3 and the change in y is −4. These are the two perpendicular sides of a right triangle, so the distance is the hypotenuse: √((−3)² + (−4)²) = √(9 + 16) = √25 = 5 units. This is the 3-4-5 Pythagorean triplet, so 5 is a quick check. Option D, 25, is the square of the distance, not the distance itself.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula is derived from Pythagoras theorem. For two points P(x₁, y₁) and Q(x₂, y₂), the horizontal difference x₂ − x₁ and the vertical difference y₂ − y₁ form the legs of a right triangle, and PQ is the hypotenuse. Substituting the given coordinates into the formula gives the distance.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between the points P(x₁, y₁) and Q(x₂, y₂).

Q23. If the coordinates of the mid-points of the line joining the points \((3a, 4)\) and \((-2, 2b)\) are \((5, a)\) , then

Mark practised

(A) \(a = 2, b = 5\)
(B) \(a = 4, b = 2\)
(C) \(a = 3, b = 4\)
(D) \(a = 5, b = 8\)

Marks: 1
Level: Basic
Compartment: 2024
Topic: 7.3 Section Formula
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430(B), Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) a = 4, b = 2, because by the mid-point formula (3a − 2)/2 = 5 gives a = (10 + 2)/3 = 4, and (4 + 2b)/2 = a = 4 gives b = (8 − 4)/2 = 2.
Why? The mid-point of a segment is the average of the x-coordinates and the average of the y-coordinates, so each coordinate gives a separate equation.

The full 1 mark is for the correct option.

Answer:

(B) a = 4, b = 2

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430(B): option (B).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing (3a + 2)/2 instead of (3a − 2)/2, because the second x-coordinate is −2, not +2.
Putting 5 in place of a in the y-coordinate equation instead of (4 + 2b)/2 = a; this leads to the wrong b-value 3, which is not in the options.
Choosing an option after checking only one coordinate; both the x-coordinate and the y-coordinate equations must be satisfied.

Why this works

The mid-point of the join of (3a, 4) and (−2, 2b) is ((3a − 2)/2, (4 + 2b)/2). It is given to be (5, a). Equating x-coordinates gives (3a − 2)/2 = 5, so 3a − 2 = 10 and a = 4. Equating y-coordinates gives (4 + 2b)/2 = a. With a = 4, this becomes (4 + 2b)/2 = 4, so 4 + 2b = 8, giving b = 2. Substitution gives the mid-point (5, 4), matching (5, a).

Key idea to remember

Section Formula (Special Case: mid-point): see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point that divides a line segment in a given ratio. A mid-point divides the segment in the ratio 1 : 1, so its coordinates are the averages of the corresponding coordinates of the end points. Equating these averages to the given coordinates gives linear equations in a and b.

Important formulas

Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2) · Coordinates of the mid-point of the segment joining (x₁, y₁) and (x₂, y₂).

Q24. The coordinates of a point on the x-axis, which is equidistant from \((-2, 5)\) and \((2, -3)\) are :

Mark practised

(A) \((-5, 0)\)
(B) \((-3, 0)\)
(C) \((-4, 0)\)
(D) \((-2, 0)\)

Marks: 1
Level: Basic
Compartment: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430(B), Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) (-2, 0), because for P(x, 0) on the x-axis, (x + 2)² + 25 = (x − 2)² + 9 gives x = −2.
Why? A point on the x-axis has coordinates (x, 0). Equidistant means the two distances are equal; comparing their squares removes the square roots.

The full 1 mark is for the correct option.

Answer:

(D) -2

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430(B): option (D).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Using (x − 2)² + 25 = (x + 2)² + 9 (x-terms reversed), which leads to x = 2, not among the options.
Using (x + 2)² + 5 = (x − 2)² + 3 without squaring the y-differences 5 and 3.
Choosing (−2, 5) or (2, −3) directly, forgetting that the required point must lie on the x-axis, so its y-coordinate must be 0.

Why this works

On the x-axis every point is (x, 0). Write the distance formula for both given points and square both sides: (x + 2)² + (0 − 5)² = (x − 2)² + (0 + 3)². Expanding gives x² + 4x + 4 + 25 = x² − 4x + 4 + 9, so 4x + 29 = −4x + 13. Then 8x = −16, so x = −2. Checking: the distance from (−2, 0) to (−2, 5) is √(0² + 5²) = 5, and the distance from (−2, 0) to (2, −3) is √(4² + 3²) = 5. So the point (−2, 0) is equidistant. Another correct method also earns full marks.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). A point on the x-axis is (x, 0). Equidistant means the two distances are equal, and comparing their squares makes the working simpler.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance formula, used to compare the two distances from the required point.

Q25. The coordinates of the mid-point of a line segment joining the points A(3, -1) and B(7, 5) are (5a, 2b). The value of (a + b) is :

Mark practised

(A) 7
(B) 3
(C) 2
(D) 5

Marks: 1
Level: Basic
Compartment: 2024
Topic: 7.3 Section Formula
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430/S/1, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2, because the mid-point of AB is ((3+7)/2, (-1+5)/2) = (5, 2), so 5a = 5 and 2b = 2, giving a = 1 and b = 1.
Why? By the mid-point formula, each coordinate of the mid-point is the average of the corresponding coordinates of the endpoints. Equating the coordinates of the same point gives two equations in a and b.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430/S/1: option (C).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Reading the mid-point (5, 2) as the values of a and b, and giving 7, instead of solving 5a = 5 and 2b = 2.
Swapping the coordinates and thinking 5a matches 2 while 2b matches 5, which does not match the mid-point of A and B.

Why this works

The mid-point of a segment is the point halfway between its endpoints, so each coordinate of the mid-point is the average of the two corresponding endpoint coordinates. For A(3, -1) and B(7, 5), the mid-point is ((3+7)/2, (-1+5)/2) = (5, 2). The question gives the same mid-point as (5a, 2b). Two ordered pairs are equal only when their first coordinates are equal and their second coordinates are equal, so 5a = 5 and 2b = 2. Solving gives a = 1 and b = 1, and therefore a + b = 2.

Key idea to remember

Mid-point formula: see Key ideas used on this page.

Explanation

The mid-point of a line segment divides the segment in the ratio 1 : 1, so its coordinates are the averages of the corresponding coordinates of the endpoints. In this question, we first find the mid-point of AB and then compare it with the given point (5a, 2b) to get two simple equations for a and b.

Important formulas

Mid-point of A(x₁, y₁) and B(x₂, y₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2) · Use it to find the mid-point, then equate corresponding coordinates.

Q26. If origin is the mid-point of the line segment joining the points P(a, b) and Q(3, 3), then the value of (a + b) is :

Mark practised

(A) 0
(B) 3
(C) 6
(D) -6

Marks: 1
Level: Basic
Compartment: 2024
Topic: 7.3 Section Formula
Source: (CBSE 2024 Compartment Mathematics Basic, Set 430/S/2, Q12, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) −6, because the mid-point ((a+3)/2, (b+3)/2) equals the origin, so a = b = −3 and a + b = −6.
Why? The mid-point of a segment is the average of the coordinates of its ends; when it is the origin, both averages are 0.

The full 1 mark is for the correct option.

Answer:

(D) -6

✓ Answer matches the CBSE answer key, 2024 Compartment, Paper 430/S/2: option (D).

Open CBSE’s official marking scheme for the 2024 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Forgetting to subtract 3 after equating (a+3)/2 to 0, and taking a as 3, which gives the wrong answer 6.
Reversing the order inside the formula, for example using (a−3)/2, which leads to the wrong sign and gives 6.
Choosing 0 just because the origin is (0, 0), without actually solving for a and b.

Why this works

The origin is the point (0, 0). The mid-point of P(a, b) and Q(3, 3) is ((a+3)/2, (b+3)/2). Since this is the origin, the x-coordinate (a+3)/2 is 0 and the y-coordinate (b+3)/2 is 0. Solving gives a as −3 and b as −3, so a + b is −6. Quick check: (−3+3)/2 equals 0, so the mid-point of (−3, −3) and (3, 3) is the origin.

Key idea to remember

Mid-point formula: see Key ideas used on this page.

Explanation

The mid-point of a line segment has coordinates that are the averages of the corresponding coordinates of its end points: ((x₁+x₂)/2, (y₁+y₂)/2). Here the mid-point is given to be the origin (0, 0), so each average is set equal to 0 and solved for a and b.

Important formulas

Mid-point of P(x₁, y₁) and Q(x₂, y₂) = ((x₁+x₂)/2, (y₁+y₂)/2) · Substitute a for x₁, b for y₁, and Q as (3, 3); set the result equal to the origin.

Q27. The distance of the point \((3, 4)\) from the origin is

Mark practised

(A) \(25\)
(B) \(5\)
(C) \(7\)
(D) \(1\)

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430/2/2, Q17, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 5, because the distance of (3, 4) from the origin is √(3² + 4²) = 5.
Why? The distance formula gives OP = √(x² + y²); since distance is always non-negative, take the positive square root.

The full 1 mark is for the correct option.

Answer:

(B) 5

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/2/2: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 25, the square of the distance, without taking the positive square root.
Choosing 7, the sum of the two coordinates, instead of applying the distance formula.

Why this works

Plot the point (3, 4). The x-coordinate 3 is the distance from the y-axis, and the y-coordinate 4 is the distance from the x-axis. These are the two legs of a right triangle, and the distance from the origin is the hypotenuse. By Pythagoras theorem, distance² = 3² + 4² = 9 + 16 = 25, so distance = √25 = 5. This is the 3-4-5 right triangle. Option A is 25, the square of the distance; option C is 7, the sum of the coordinates; option D is 1.

Key idea to remember

Distance formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). When Q is the origin (0, 0), this simplifies to √(x² + y²). For (3, 4), the distance is √(3² + 4²) = 5.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between two points P(x₁, y₁) and Q(x₂, y₂).
OP = √(x² + y²) · Distance of a point P(x, y) from the origin O(0, 0); this is the formula used here.

Q28. The distance between the points A(5, -4) and B(4, -5) is

Mark practised

(A) \(\sqrt{2}\) units
(B) 2 units
(C) 1 unit
(D) \(9\sqrt{2}\) units

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430/3/2, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) √2 units, because AB = √((4−5)² + (−5+4)²) = √(1+1) = √2.
Why? The distance formula is the Pythagoras theorem in coordinate form: the two coordinate differences are the legs of a right triangle, and the required distance is the hypotenuse.

The full 1 mark is for the correct option.

Answer:

(A) √2 units

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/3/2: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 2 units because the squared differences add to 2, but forgetting the final square root.
Choosing 9√2 units by taking B as (−4, 5) instead of (4, −5).

Why this works

Take A(5, −4) as (x₁, y₁) and B(4, −5) as (x₂, y₂). Then x₂ − x₁ = 4 − 5 = −1 and y₂ − y₁ = −5 − (−4) = −1. Squaring removes the negative signs: (−1)² = 1 and (−1)² = 1, and 1 + 1 = 2. So AB = √2 units, which is option A. Option B is the value before the square root; option D comes from misreading B as (−4, 5).

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula gives the length of the line segment joining two points. It comes from the Pythagoras theorem: the horizontal separation x₂ − x₁ and the vertical separation y₂ − y₁ form the two legs of a right triangle, and the required distance is the hypotenuse. Here both separations are −1, so the squared sum is 2 and the distance is √2.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between P(x₁, y₁) and Q(x₂, y₂); distance is always non-negative.
PQ = √((x₁ − x₂)² + (y₁ − y₂)²) · Equivalent form; the order of subtraction does not change the result.

Q29. If \(C(1, -1)\) is the mid-point of the line segment AB joining points \(A(4, x)\) and \(B(-2, 4)\), then value of \(x\) is :

Mark practised

(A) 5
(B) -5
(C) 6
(D) -6

Marks: 1
Level: Basic
Years: 2024
Topic: 7.3 Section Formula
Source: (CBSE 2024 Mathematics Basic, Set 430/4/2, Q11, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) -6, because the mid-point formula gives (x + 4)/2 = -1, so x = -6.
Why? The mid-point formula states that the y-coordinate of the mid-point is the average of the y-coordinates of the endpoints, so this equation is solved for x.

The full 1 mark is for the correct option.

Answer:

(D) -6

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/2: option (D).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Forgetting the division by 2 in the mid-point formula, which changes the answer.
Writing (x - 4)/2 for the y-coordinate of the mid-point instead of (x + 4)/2, which changes the answer.
Using only the x-coordinates of the endpoints, (4 + (-2))/2 = 1, which does not contain the variable x and so gives no equation for x.

Why this works

The mid-point of a line segment divides it in the ratio 1 : 1, so the mid-point formula is the section formula in that special case. For A(4, x) and B(-2, 4), the y-coordinate of the mid-point is (x + 4)/2. Since C(1, -1) is the mid-point, equate the y-coordinates: (x + 4)/2 = -1. Multiplying by 2 gives x + 4 = -2, so x = -6. Check: with A(4, -6), the y-coordinate of the mid-point is (-6 + 4)/2 = -1, and the x-coordinate is (4 + (-2))/2 = 1, so the mid-point is C(1, -1).

Key idea to remember

Mid-point Formula: see Key ideas used on this page.

Explanation

The mid-point of a line segment divides it in the ratio 1 : 1, so its coordinates are the averages of the corresponding coordinates of the endpoints. Here we put the given y-coordinate -1 equal to (x + 4)/2 to find x.

Important formulas

Mid-point of A(x₁, y₁) and B(x₂, y₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2) · This is the section formula with the ratio 1 : 1.

Q30. The distance of the point (4, 5) from x-axis is :

Mark practised

(A) 5
(B) 4
(C) 9
(D) 1

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430/4/2, Q14, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 5, because the y-coordinate of the point (4, 5) is 5, which is its distance from the x-axis.
Why? The perpendicular distance from a point to the x-axis is measured along the y-direction, and the y-coordinate of the point is exactly that perpendicular distance.

The full 1 mark is for the correct option.

Answer:

(A) 5

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/2: option (A).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 4 because it is the first number in (4, 5), but 4 is the x-coordinate, which is the distance from the y-axis.
Choosing 9 by adding 4 and 5, but the distance from an axis is a coordinate, not the sum of the coordinates.
Choosing 1 by subtracting 5 minus 4, but the distance from the x-axis is the y-coordinate itself, not the difference.

Q31. The distance between the points \((2, -1)\) and \((-1, -5)\) is :

Mark practised

(A) 15 units
(B) 5 units
(C) 25 units
(D) 41 units

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430/4/3, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 5 units, because the distance formula gives √((−1−2)² + (−5−(−1))²) = √(9+16) = √25 = 5.
Why? The distance between two points (x₁,y₁) and (x₂,y₂) is the positive square root of (x₂−x₁)² + (y₂−y₁)².

The full 1 mark is for the correct option.

Answer:

(B) 5

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430/4/3: option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 25, the sum 9 + 16, without taking the square root; the distance is 5 units.
Thinking the square of a negative number is negative, so the squared differences would not add to 25.

Why this works

Take the first point as (2, −1) and the second as (−1, −5). The difference in x-coordinates is −1 − 2 = −3, and the difference in y-coordinates is −5 − (−1) = −4. Their squares are 9 and 16, whose sum is 25. So the distance is √25 = 5. The numbers 3, 4, 5 form a right triangle, which is a quick check. Another correct method also earns full marks.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula comes from applying the Pythagoras theorem to the right triangle formed by the two points: the distance is the hypotenuse, and the differences of the x-coordinates and y-coordinates are the two legs. This solution substitutes the given coordinates into PQ = √((x₂−x₁)² + (y₂−y₁)²).

Important formulas

PQ = √((x₂−x₁)² + (y₂−y₁)²) · Distance between P(x₁,y₁) and Q(x₂,y₂).

Q32. The distance of the point P(3, -5) from x-axis is :

Mark practised

(A) 3 units
(B) 5 units
(C) -5 units
(D) \(\sqrt{34}\) units

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430(B), Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 5 units, because the distance from the x-axis is the absolute value of the y-coordinate, |-5| = 5.
Why? By definition, the distance of a point from the x-axis is its y-coordinate (ordinate), and distance is always non-negative, so we take the absolute value.

The full 1 mark is for the correct option.

Answer:

(B) 5 units

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430(B): option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 3 units, which is the distance from the y-axis (the x-coordinate), not from the x-axis.
Choosing -5 units, forgetting that distance can never be negative.
Choosing √34 units, which is the distance of P(3, -5) from the origin, not from the x-axis.

Why this works

The x-axis is the horizontal line. To find how far P(3, -5) is from it, measure the vertical distance, which is the absolute value of the y-coordinate: |-5| = 5. Option (A) 3 units is the distance from the y-axis, not the x-axis. Option (C) -5 units cannot be correct because distance is never negative. Option (D) √34 units is the distance of P from the origin, not from the x-axis.

Key idea to remember

Distance of a point from the x-axis: see Key ideas used on this page.

Explanation

In coordinate geometry, the distance of a point from the x-axis is its y-coordinate, or ordinate. Since distance cannot be negative, we take the absolute value. For P(3, -5), the y-coordinate is -5, so the distance is |-5| = 5 units.

Important formulas

Distance of a point P(x, y) from the x-axis = |y| · Use the y-coordinate (ordinate), not the x-coordinate, and take the absolute value because distance is non-negative.

Q33. The distance between the points A(0, -1) and B(0, -9) is :

Mark practised

(A) 6
(B) 8
(C) 4
(D) 2

Marks: 1
Level: Basic
Years: 2024
Topic: 7.2 Distance Formula
Source: (CBSE 2024 Mathematics Basic, Set 430(B), Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 8, because the distance between two points on the y-axis is the absolute difference of their y-coordinates: |−1 − (−9)| = 8.
Why? The distance formula gives PQ = √((x₂ − x₁)² + (y₂ − y₁)²); when x₁ = x₂, this becomes |y₂ − y₁|, and distance is always non-negative.

The full 1 mark is for the correct option.

Answer:

(B) 8

✓ Answer matches the CBSE answer key, 2024 Main, Paper 430(B): option (B).

Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Taking √64 as 4 instead of 8, which gives option C.
Counting the units from −1 to −9 as 9, forgetting that the starting point −1 is not counted as a unit.
Forgetting that distance is non-negative when the difference of the y-coordinates comes out negative.

Why this works

Both points have x-coordinate 0, so they lie on the y-axis. On the y-axis, the distance between two points is the absolute difference of their y-coordinates, which is 8. The same result comes from the distance formula: √((0 − 0)² + (−1 − (−9))²) = √(0 + 64) = √64 = 8. Quick check: on a number line, from −9 to −1 you move 8 units upwards.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is the positive square root of (x₂ − x₁)² + (y₂ − y₁)². Here both points have x-coordinate 0, so they lie on the y-axis and the distance is simply the absolute difference of the y-coordinates.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance formula for points P(x₁, y₁) and Q(x₂, y₂).
When x₁ = x₂, PQ = |y₂ − y₁| · Special case when both points lie on the y-axis.

Q34. If \((\frac{1}{2}, 6)\) is the mid-point of the line segment joining \((-5, k)\) and \((6, 3)\), then the value of 'k' is :

Mark practised

(A) 0
(B) 3
(C) \(\frac{9}{2}\)
(D) 9

Marks: 1
Level: Basic
Compartment: 2025
Topic: 7.3 Section Formula
Source: (CBSE 2025 Compartment Mathematics Basic, Set 430/S/3, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) 9, because the y-coordinate of the mid-point is (k + 3)/2 = 6, so k = 9.
Why? The y-coordinate of the mid-point is the average of the y-coordinates of the two end points, so it gives a linear equation in k.

The full 1 mark is for the correct option.

Answer:

(D) 9

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/3: option (D).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Forgetting the division by 2 and treating k + 3 as 6, which leads to option B.
Reading the y-coordinate of the given mid-point as the value of k, without forming the mid-point equation.

Why this works

The mid-point of a line segment joining (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). Here the end points are (-5, k) and (6, 3), and the mid-point is (1/2, 6). The y-coordinate gives (k + 3)/2 = 6. Multiplying by 2 gives k + 3 = 12, so k = 9. Check: with k = 9, (9 + 3)/2 = 6 and (-5 + 6)/2 = 1/2, so both coordinates match.

Key idea to remember

Section Formula (Special Case: Mid-point): see Key ideas used on this page.

Explanation

The section formula gives the coordinates of a point that divides a line segment in a given ratio. When the ratio is 1 : 1, the point is the mid-point, so its coordinates are the averages of the corresponding coordinates of the end points. This solution uses the average of the y-coordinates to form and solve an equation for k.

Important formulas

Mid-point of A(x1, y1) and B(x2, y2) = ((x1 + x2)/2, (y1 + y2)/2) · The mid-point divides the line segment in the ratio 1 : 1.

Q35. M is a point on y-axis at a distance of 4 units from x-axis and it lies below the x-axis. The distance of point M from point Q (5, 1) is :

Mark practised

(A) \(\sqrt{2}\) units
(B) \(\sqrt{34}\) units
(C) \(\sqrt{50}\) units
(D) \(\sqrt{90}\) units

Marks: 1
Level: Standard
Compartment: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/1, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) √50 units, because M has x-coordinate 0 and y-coordinate −4, so MQ = √((5 − 0)² + (1 + 4)²) = √(25 + 25) = √50.
Why? By the distance formula, PQ = √((x₂ − x₁)² + (y₂ − y₁)²); a point on the y-axis has x-coordinate 0, and a point below the x-axis has a negative y-coordinate.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/1: option (C).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Writing the y-coordinate as 4 instead of −4, because the point lies below the x-axis.
Thinking the 4 units give the x-coordinate instead of the y-coordinate.
Subtracting the y-coordinates as 1 − 4 instead of 1 + 4.

Why this works

M is on the y-axis, so its x-coordinate is 0. A distance of 4 units from the x-axis means the y-coordinate is 4 or −4; since M lies below the x-axis, M has y-coordinate −4. The horizontal gap from M to Q(5, 1) is 5 − 0 = 5, and the vertical gap is 1 + 4 = 5. So MQ is the hypotenuse of a right triangle with legs 5 and 5. By the distance formula, MQ = √(5² + 5²) = √(25 + 25) = √50, which is option C. As a check, (√50)² = 50 = 5² + 5².

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). This formula comes from applying the Pythagoras theorem to the right triangle whose legs are the horizontal gap and the vertical gap between the two points. In this question, first find M, whose x-coordinate is 0 and y-coordinate is −4, from the description, then substitute M and Q(5, 1) into the formula.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between P(x₁, y₁) and Q(x₂, y₂).
A point on the y-axis has x-coordinate 0 · Used to find M, whose y-coordinate is −4 because it lies below the x-axis.

Q36. ABC is a triangle. B lies on x-axis at a distance of 4 units from y-axis at its right. C is on y-axis and it is 3 units away from the origin. If the coordinates of A are \((0, 0)\) , the perimeter of \(\Delta ABC\) is :

Mark practised

(A) 7 units
(B) 12 units
(C) 6 units
(D) 15 units

Marks: 1
Level: Standard
Compartment: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Compartment Mathematics Standard, Set 30/S/2, Q8, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 12 units, because AB = 4, AC = 3, BC = 5, so the perimeter = 4 + 3 + 5 = 12 units.
Why? The distance formula gives the length of each side from the coordinates, and the perimeter is the sum of the three side lengths.

The full 1 mark is for the correct option.

Answer:

(B) 12 units

✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 30/S/2: option (B).

Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.

Common mistakes to avoid

Adding only the two given distances 4 and 3, and forgetting the third side BC.
Finding BC by subtracting 3 from 4, because B and C are not on the same axis.
Thinking the perimeter changes when C is below the origin; the side lengths are the same for both positions.

Why this works

B lies on the x-axis 4 units to the right of the y-axis, so its coordinates are (4, 0). C lies on the y-axis 3 units from the origin, so its coordinates are (0, 3) or (0, −3). A is (0, 0). So AB is 4 and AC is 3. By the distance formula, BC = √((4−0)² + (0−3)²) = √(16 + 9) = 5. The perimeter is 4 + 3 + 5 = 12. A quick check: 3² + 4² = 9 + 16 = 25, so BC = 5. If C is below the origin, the same side lengths occur, so the answer does not change.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance formula gives the length of a segment from the coordinates of its endpoints. For points P(x₁, y₁) and Q(x₂, y₂), the distance is √((x₂ − x₁)² + (y₂ − y₁)²). Here, the axes give the coordinates of B and C directly, and the formula finds BC. The perimeter is the sum of the three sides.

Important formulas

PQ = √((x₂ − x₁)² + (y₂ − y₁)²) · Distance between two points P(x₁, y₁) and Q(x₂, y₂).
Perimeter of a triangle = AB + BC + CA · Sum of the three side lengths.

Q37. The mid-point of the line segment joining the points \(P(-4, 5)\) and \(Q(4, 6)\) lies on :

Mark practised

(A) x-axis
(B) y-axis
(C) origin
(D) neither x-axis nor y-axis

Marks: 1
Level: Standard
Years: 2025
Topic: 7.3 Section Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/1/1, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) y-axis, because the mid-point is (0, 5.5), whose x-coordinate is 0.
Why? The mid-point formula averages the x-coordinates and y-coordinates of the two endpoints, and a point whose x-coordinate is 0 lies on the y-axis.

The full 1 mark is for the correct option.

Answer:

(B) y-axis

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing the x-axis because the y-coordinate 5.5 is positive, although a point on the x-axis must have y = 0.
Calling (0, 5.5) the origin because its x-coordinate is 0; the origin is (0, 0), so y must also be 0.
Swapping the averages, giving a point with x-coordinate 5.5 and y-coordinate 0, which would wrongly lie on the x-axis.

Why this works

The mid-point divides the segment in the ratio 1 : 1, so each coordinate of the mid-point is the average of the two corresponding coordinates. For P(-4, 5) and Q(4, 6), the x-average is (-4 + 4)/2 = 0 and the y-average is (5 + 6)/2 = 5.5. Hence the mid-point is (0, 5.5). A point lies on the y-axis when its x-coordinate is 0, and on the x-axis when its y-coordinate is 0. Here x = 0 but y = 5.5, so the point is on the y-axis only; it is not the origin, because the origin is (0, 0).

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of the point that divides a line segment internally in a given ratio. The mid-point is the point that divides the segment in the ratio 1 : 1, so the section formula becomes ((x1 + x2)/2, (y1 + y2)/2).

Important formulas

Mid-point = ((x1 + x2)/2, (y1 + y2)/2) · Coordinates of the point halfway between (x1, y1) and (x2, y2).
A point on the y-axis has coordinates (0, y) · Its x-coordinate, or abscissa, is 0.

Q38. The perimeter of the triangle formed by the vertices \((0, 0)\), \((2, 0)\) and \((0, 2)\) is :

Mark practised

(A) 4 units
(B) 6 units
(C) \(6\sqrt{2}\) units
(D) \((4 + 2\sqrt{2})\) units

Marks: 1
Level: Standard
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/2/2, Q3, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) (4 + 2√2) units, because the distance formula gives the sides as 2, 2 and 2√2, and the perimeter is their sum.
Why? The distance formula gives the length of each side; the perimeter of a triangle is the sum of the lengths of its three sides.

The full 1 mark is for the correct option.

Answer:

(D) (4 + 2√2) units

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/2: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 4 units by adding only 2 and 2, and leaving out the side between (2, 0) and (0, 2).
Taking the side between (2, 0) and (0, 2) as 2 units, instead of using √(2² + 2²), which is 2√2.
Choosing 6 units by thinking all three sides are 2 units.

Q39. The equation of a line parallel to y-axis and at a distance of 5 units to the right of y-axis is :

Mark practised

(A) x = 5
(B) x = -5
(C) y = 5
(D) y = -5

Marks: 1
Level: Standard
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/2/3, Q6, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) x = 5, because every point on this line is 5 units to the right of the y-axis, so its x-coordinate is 5.
Why? A line parallel to the y-axis keeps the same x-coordinate for all its points; being to the right fixes that x-coordinate as 5.

The full 1 mark is for the correct option.

Answer:

(A) x = 5

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/3: option (A).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing y = 5 because it is a line at distance 5, but a line parallel to the y-axis has equation x = constant, not y = constant.
Choosing x = -5, forgetting that 'to the right' means the positive direction from the y-axis.
Thinking both x = 5 and x = -5 are correct because both are 5 units from the y-axis; the question fixes the right side.

Why this works

The y-axis is the vertical line through the origin. A line parallel to the y-axis is also vertical, so its x-coordinate never changes. To the right of the y-axis means the positive x-direction. Therefore every point on the required line has x-coordinate 5, for any y-value, so its equation is x = 5. Option (B) x = -5 is 5 units to the left. Options (C) y = 5 and (D) y = -5 are horizontal lines parallel to the x-axis, not parallel to the y-axis.

Key idea to remember

Line parallel to the y-axis: see Key ideas used on this page.

Explanation

A vertical line parallel to the y-axis has the same x-coordinate for every point on it. If it is 5 units to the right of the y-axis, that fixed x-coordinate is 5, so the equation is x = 5.

Important formulas

x = a · Equation of a line parallel to the y-axis at distance a from it; a is positive to the right.
y = b · Equation of a line parallel to the x-axis at distance b from it.

Q40. The distance of a point A from x-axis is 3 units. Which of the following cannot be coordinates of the point A ?

Mark practised

(A) (1, 3)
(B) (-3, -3)
(C) (-3, 3)
(D) (3, 1)

Marks: 1
Level: Standard
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/4/1, Q4, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(D) (3, 1), because the distance of a point from the x-axis is the size of its y-coordinate, and for (3, 1) that size is 1, not 3.
Why? The y-coordinate (ordinate) measures how far a point is above or below the x-axis; since distance is never negative, the distance equals |y|.

The full 1 mark is for the correct option.

Answer:

(D) (3, 1)

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/4/1: option (D).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Using the x-coordinate as the distance from the x-axis, so taking (3, 1) as distance 3 instead of 1.
Thinking (-3, -3) is at distance -3 from the x-axis; the distance is |-3| = 3.
Choosing (1, 3) because its y-coordinate is 3 without checking that (3, 1) has y-coordinate 1.

Q41. The distance of the point A(-3, -4) from x-axis is

Mark practised

(A) 3
(B) 4
(C) 5
(D) 7

Marks: 1
Level: Standard
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/6/1, Q5, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(B) 4, because the distance of a point from the x-axis is the absolute value of its y-coordinate, so |-4| = 4.
Why? The perpendicular from a point to the x-axis is vertical, so its length is the vertical gap between the point and the x-axis.

The full 1 mark is for the correct option.

Answer:

(B) 4

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/1: option (B).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 3 because the x-coordinate of A is -3, but the distance from the x-axis is decided by the y-coordinate.
Choosing 5 by using √((-3)² + (-4)²), which gives the distance from the origin (0, 0), not from the x-axis.
Choosing 7 by adding 3 and 4, because the distance from the x-axis is only the vertical distance, not the sum of the two distances.

Why this works

The x-axis is horizontal, and the perpendicular from a point to the x-axis is vertical. For A(-3, -4), the vertical gap between A and the x-axis is |-4| = 4. The x-coordinate -3 only tells the distance from the y-axis, so it does not affect this distance. A quick check with the distance formula gives √((−3 − (−3))² + (0 − (−4))²) = √(0 + 16) = 4.

Key idea to remember

Distance of a point from the x-axis (ordinate): see Key ideas used on this page.

Explanation

The chapter defines the distance of a point from the x-axis as its y-coordinate, or ordinate. For A(-3, -4), the ordinate is negative, and distance cannot be negative, so the distance is |-4| = 4. The x-coordinate, or abscissa, -3 measures the distance from the y-axis, not from the x-axis.

Important formulas

Distance of P(x, y) from x-axis = |y| · The y-coordinate is also called the ordinate.
Distance of P(x, y) from y-axis = |x| · The x-coordinate is also called the abscissa.

Q42. The distance of point (a, -b) from x-axis is

Mark practised

(A) a
(B) -a
(C) b
(D) -b

Marks: 1
Level: Standard
Years: 2025
Topic: 7.2 Distance Formula
Source: (CBSE 2025 Mathematics Standard, Set 30/6/2, Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) b, because the distance from the x-axis is the absolute value of the y-coordinate, and |−b| = b.
Why? The distance of a point from the x-axis is its ordinate, and distance is always non-negative, so the negative sign is dropped.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/2: option (C).

Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing D (−b), because the ordinate of (a, −b) is −b, without remembering that distance cannot be negative.
Choosing A (a), because a is the abscissa, which measures distance from the y-axis, not from the x-axis.

Q43. The ratio in which the \(x\)-axis divides the line segment joining the points \(A(-8, 4)\) and \(B(-6, -2)\) is :

Mark practised

(A) \(5 : 1\)
(B) \(3 : 1\)
(C) \(2 : 1\)
(D) \(1 : 2\)

Marks: 1
Level: Basic
Years: 2026
Topic: 7.3 Section Formula
Source: (CBSE 2026 Mathematics Basic, Set 430(B), Q7, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(C) 2 : 1, because on the x-axis y = 0, so by the section formula \(\frac{k(-2)+1(4)}{k+1}=0\), giving \(k=\frac{4}{2}=2\).
Why? A point on the x-axis has y-coordinate 0; the section formula gives the y-coordinate of the point that divides AB in the ratio k : 1.

The full 1 mark is for the correct option.

Answer:

✓ Answer matches the CBSE answer key, 2026 Main, Paper 430(B): option (C).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Putting the x-coordinate equal to 0 because the point is on the x-axis; a point on the x-axis has y = 0, not x = 0.
Writing the ratio as 1 : 2 by directly comparing the y-values 4 and -2, without solving the section formula.
Solving \(-2k+4=0\) as k = -2 and then choosing the wrong option.

Why this works

The point where the x-axis cuts the segment lies on the x-axis, so its y-coordinate is 0. Take the ratio as k : 1. By the section formula, the y-coordinate of the dividing point is \(\frac{k(-2)+1(4)}{k+1}\). Setting this equal to 0 gives \(-2k+4=0\), so k = 2 and the ratio is 2 : 1. With k = 2, the dividing point is \(\left(\frac{2(-6)+1(-8)}{2+1}, 0\right)=\left(-\frac{20}{3},0\right)\), which lies between A(-8,4) and B(-6,-2).

Key idea to remember

Section Formula: see Key ideas used on this page.

Explanation

The section formula gives the coordinates of the point P that divides the line segment joining A(x₁,y₁) and B(x₂,y₂) internally in the ratio k : 1. Here we use the y-coordinate of this formula and the fact that a point on the x-axis has y-coordinate 0.

Important formulas

\(\left(\frac{kx_2+x_1}{k+1}, \frac{ky_2+y_1}{k+1}\right)\) · Section formula when the point divides AB internally in the ratio k : 1.
\((x, 0)\) · Coordinates of any point on the x-axis; hence its y-coordinate is 0.

Q44. The distance of the point A(4a, 3a) from x-axis is :

Mark practised

(A) 3a
(B) -3a
(C) 4a
(D) -4a

Marks: 1
Level: Standard
Years: 2026
Topic: 7.2 Distance Formula
Source: (CBSE 2026 Mathematics Standard, Set 30/2/1, Q9, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) 3a, because the distance of a point from the x-axis is the absolute value of its y-coordinate, and |3a| = 3a.
Why? For any point (x, y), the distance from the x-axis (the horizontal axis) is |y|, and distance is never negative.

The full 1 mark is for the correct option.

Answer:

(A) 3a

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/2/1: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing 4a because it is the x-coordinate, which gives the distance from the y-axis, not from the x-axis.
Choosing -3a, forgetting that distance can never be negative.

Why this works

The x-axis is the horizontal coordinate axis. A point's distance from it is its vertical distance, which is given by the y-coordinate. For A(4a, 3a), the vertical coordinate is 3a, so the point lies 3a units away from the x-axis when a is positive. The x-coordinate 4a measures the distance from the y-axis, not from the x-axis. Also, distance cannot be negative, so -3a cannot be correct. As a quick check, think of the point (4, 3): it is 3 units above the x-axis, and this matches 3a when a is positive.

Key idea to remember

Y-coordinate (ordinate): see Key ideas used on this page.

Explanation

For any point (x, y), the y-coordinate, also called the ordinate, is the distance of the point from the x-axis. Since distance is always non-negative, we take the absolute value of the y-coordinate. Here the y-coordinate is 3a, so the distance from the x-axis is |3a| = 3a.

Important formulas

Distance of a point (x, y) from the x-axis = |y| · The y-coordinate gives this distance, and distance is taken as non-negative.
Distance of a point (x, y) from the y-axis = |x| · The x-coordinate gives the distance from the y-axis.

Q45. If the distance between the points (4, p) and (1, 0) is 5, then p is equal to :

Mark practised

(A) \(\pm 4\)
(B) 4
(C) -4
(D) 0

Marks: 1
Level: Standard
Years: 2026
Topic: 7.2 Distance Formula
Source: (CBSE 2026 Mathematics Standard, Set 30/2/2, Q10, 1 mark)
Status: Verified solution

Step-by-step solution

Step 1: Choose the option
1 mark
(A) ±4, because the distance formula gives √((4−1)²+(p−0)²)=5, so 9+p²=25 and p²=16.
Why? The distance formula turns the given distance into an equation in p. Squaring both sides removes the square root; p²=16 has two solutions, p=4 and p=−4.

The full 1 mark is for the correct option.

Answer:

(A) ±4

✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/2/2: option (A).

Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.

Common mistakes to avoid

Choosing only p=4 because 4²=16, and forgetting that (−4)²=16 also satisfies p²=16.
Writing 9+p²=5 without the square root, and then finding no real value for p.

Why this works

Put the two points into the distance formula: distance = √((4−1)²+(p−0)²) = √(9+p²). The question says this distance is 5, so √(9+p²)=5. Squaring both sides gives 9+p²=25, so p²=16. Both p=4 and p=−4 satisfy the equation, because 4²=16 and (−4)²=16. Quick check: with p=4, the distance is √(3²+4²)=√25=5; with p=−4, the distance is √(3²+(−4)²)=√25=5. So both values are correct.

Key idea to remember

Distance Formula: see Key ideas used on this page.

Explanation

The distance between two points P(x₁,y₁) and Q(x₂,y₂) is √((x₂−x₁)²+(y₂−y₁)²). In this question, substituting the given distance 5 into the formula gives an equation in p, and solving that equation gives two possible values because squaring a number and squaring its negative give the same result.

Important formulas

PQ = √((x₂−x₁)² + (y₂−y₁)²) · Distance between points P(x₁,y₁) and Q(x₂,y₂).

1 / 45

Key ideas used on this page

Distance formula
Used in questions 6 7 12 14 16 18 19 20 22 24 27 28 31 33 35 36 38 45
Section Formula
Used in questions 3 4 8 9 11 37 43
Mid-point formula
Used in questions 10 25 26 29
Section Formula (special case: mid-point)
Used in questions 17 23 34
Ordinate
Used in questions 15 42
Y-coordinate (ordinate)
Used in questions 30 44
Distance of a point from the x-axis
Used in questions 32 40
Distance of a point from the coordinate axes
Used in question 1
Distance of a point from the y-axis (x-coordinate, or abscissa)
Used in question 2
Coordinates of a point on the y-axis
Used in question 5
Coordinates of a point
Used in question 13
Abscissa (x-coordinate)
Used in question 21
Line parallel to the y-axis
Used in question 39
Distance of a point from the x-axis (ordinate)
Used in question 41

How to use these questions

Try each question on paper first. Then read the steps: the mark beside each step shows what the examiner looks for. If a step is new to you, revise Section 7.2 and Section 7.3 of the NCERT chapter.

To revise the lessons behind these questions, read the NCERT chapter, exactly as NCERT publishes it on its own website.

Official papers behind this page

Every question on this page comes from an official CBSE Class 10 Mathematics board paper. The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.

Year	Exam	Course	Question paper	Marking scheme
2026	Main	Standard (041)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given.
2026	Main	Basic (241)	Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper.	Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 29.1 MB, Class 10 Maths Basic (241)) to see how marks are given.
2025	Main	Standard (041)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Main	Basic (241)	Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given.
2025	Compartment	Standard (041)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given.
2025	Compartment	Basic (241)	Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Main	Standard (041)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2024	Main	Basic (241)	Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given.
2024	Compartment	Basic (241)	Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Main	Standard (041)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given.
2023	Main	Basic (241)	Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper.	Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given.
2023	Compartment	Standard (041)	Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper.	Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given.

Sources and session

Each question is reproduced as printed in its CBSE paper, with its year, paper code and question number; only the layout is tidied. The same question printed in several paper sets counts once, and the card says how often it was printed in 2023–2026. These counts describe past papers; they are not predictions.

Every solution is original and follows NCERT Mathematics Class X (reprint 2026-27). The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.

Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.

1-mark questions on Coordinate Geometry

All 45 Coordinate Geometry 1-mark questions

Q1. For a point \(X(a, b)\) where (\(b \gt a \gt 0\)), the value of its [distance from \(x\)-axis – distance from \(y\)-axis] is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q2. The distance of point (– 3, 4) from y-axis is :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q3. If point \((a, 2b)\) is the mid-point of the line segment joining the points \((3, 5)\) and \((-1, -1)\) , then \((a, b)\) is equal to :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q4. In what ratio, does \(x\)-axis divide the line segment joining the points \(A(3, 6)\) and \(B(-12, -3)\) ?

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q5. The point of intersection of the line represented by \(3x - y = 3\) and \(y\)-axis is given by

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q6. If the distance between the points \((3, -5)\) and \((x, -5)\) is 15 units, then the values of \(x\) are :

Step-by-step solution

Step 1: Choose the option

Answer:

Common mistakes to avoid

Q7. AD is a median of \(\triangle ABC\) with vertices \(A(5, -6)\), \(B(6, 4)\) and \(C(0, 0)\). Length AD is equal to :

Step-by-step solution

Step 1: Choose the option

Answer:

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Q8. Point P divides the line segment joining the points A(4, -5) and B(1, 2) in the ratio 5:2. Co-ordinates of point P are

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Q9. Two of the vertices of Δ PQR are P(-1, 5) and Q(5, 2). The coordinates of a point which divides PQ in the ratio 2 : 1 are :

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Q10. The mid-point of the line segment joining the points \((5, -4)\) and \((6, 4)\) lies on :

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Q11. In what ratio does x-axis divide the line segment joining the points A(2, -3) and B(5, 6) ?

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Q12. The distance between the points \((0, 2\sqrt{5})\) and \((-2\sqrt{5}, 0)\) is

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Q13. The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0, 0), C(3, 0) and D(0, 4) are :

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Q14. XOYZ is a rectangle with vertices X(-3, 0), O(0, 0), Y(0, 4) and Z(x, y). The length of its each diagonal is

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Q15. The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is :

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Q16. AOBC is a rectangle whose three vertices are A(0, 2), O(0, 0) and B(4, 0). The square of the length of its diagonal is equal to :

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