Mathematics Board Sample Paper - Basic 2025-2026 English Medium Class 10 Question Paper Solution

The exponent of 3 in the prime factorization of 2025 is ______.

1

2

3

4

If 2024x + 2025y = 1, 2025x + 2024y = –1, then x – y = ______.

0

–2

2

–1

The number of polynomials having –2 and 5 as its zeroes is ______.

one

two

three

Infinitely many

Which of the following is not a quadratic equation?

(x + 2)² = 2(x + 3)

x² + 3x = (–1)(1 – 3x²)

(x + 2)(x – 1) = x² – 2x – 3

x³ – x² + 2x + 1 = (x + 1)³

The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an AP is ______.

6

–6

–2

2

If 1 + 2 + 3 + 4 + .... + 50 = 25 k, then k = ______.

50

51

49

26

The distance between the points (cos 30°, sin 30°) and (cos 60°, –sin 60°) is ______.

0 unit

`sqrt(3)` units

1 unit

`sqrt(2)` units

The coordinates of the point which is the mirror image of the point (–3, 5) about the x-axis are ______.

(3, 5)

(3, –5)

(–3, –5)

(–3, 5)

If in ∆ABC and ∆DEF, `(AB)/(EF) = (AC)/(DE)` then they will be similar when ______.

∠A = ∠D

∠A = ∠E

∠C = ∠F

∠B = ∠E

If ∆ABC ~ ∆PQR, then the perimeter of the triangle PQR (in cm) is ______.

12

24

18

20

In the figure given below, radius r of the circle which touches the sides of the triangle is:

3 cm

6 cm

7 cm

4 cm

Which one of the following is not equal to unity?

sin²x + cos²x

cot²x – cosec²x

sec²x – tan²x

tan x ⋅ cot x

Consider the following frequency distribution:

The upper limit of the median class is ______.

10

13

15

20

Let empirical relationship between the three measures of central tendency be a(Median) = Mode + b(Mean), then (2b + 3a) = ______.

11

12

13

14

From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is ______.

10

5

12

7

In the given figure, PA is a tangent from an external point P to a circle with centre O and diameter AB. If ∠POB = 115°, then the measure of ∠APO is:

25°

30°

20°

65°

The circumferences of two circles are in the ratio 3 : 4. The ratio of their areas is ______.

3 : 4

4 : 3

9 : 16

16 : 9

An event is most unlikely to happen. Its probability is ______.

0.0001

0.001

0.01

0.1

Assertion (A): Line joining the midpoints of two sides of triangle is parallel to the third side.

Reason (R): If a line divides two sides of a triangle in the same ratio then it is parallel to the third side.

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Assertion (A) is true but reason (R) is false.

Assertion (A) is false but reason (R) is true.

Assertion (A): Two coins are tossed simultaneously. Possible outcomes are two heads, one head and one tail, two tails. Hence, the probability of getting two heads is `1/3`.

Reason (R): Probabilities of ‘equally likely’ outcomes of an experiment are always equal.

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Assertion (A) is true but reason (R) is false.

Assertion (A) is false but reason (R) is true.

Show that the number 2 × 5 × 7 × 11 + 11 × 13 is a composite number.

Find the smallest number which is divisible by both 306 and 657.

Find the radius of the circle with centre at the origin if line (l) given by x + y = 5 is tangent to the circle at point P.

If the zeroes of the quadratic polynomial x² + (a + 1)x + b are 2 and –3, then find the values of a and b.

Find the nature of roots of the quadratic equation `x^2 + 4x - 3sqrt(2) = 0`.

Evaluate: 

2 sin 30° tan 60° – 3 cos² 60° sec² 30°

If sin x = `7/25`, where x is an acute angle, then find the value of sin x ⋅ cos x (tan x + cot x).

Show that `sqrt(2) - sqrt(5)` is an irrational number.

The frequency distribution for agriculture holdings in a village is given below:

If the mean of the following distribution is 54, find the value of p.

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the given figure. Show that `(AB + CD)/(AD + BC)` = 1.

On a particular day, 50000 people attended a Cricket Test Match between India and Australia in Sydney Cricket Ground. Let x be the number of adults attended the cricket match and y be the number of children attended the cricket match. Cost of an adult ticket was ₹ 1000 while cost of a child ticket was ₹ 200. On that day Revenue earned by selling all 50,000 tickets, was ₹ 4,20,00,000. Find how many adults and how many children attended the cricket match?

Solve for x and y graphically:

2x + y = 6; x + y = 5

Prove that (sin x – cos x + 1) ⋅ (sec x – tan x) = (sin x + cos x – 1).

The sum of the first n terms of an AP is 5n² − n. Find the n^th term of the AP.

Prove that a line drawn parallel to one side of a triangle intersecting the other two sides in distinct points divides the other two sides in the same ratio.

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original fraction is `29/20`. Find the original fraction.

A train covers a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km/hour, it takes 2 hours less in the journey. Find the original speed of the train.

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 meters, find the height of the chimney. Also, find the length of the wire tied from the top of the chimney to the top of the tower.

The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. (Use `sqrt(3)` = 1.73)

A solid toy is in the form of a hemisphere surmounted by a right circular cone of height 2 cm and diameter of base 4 cm. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. [Use π = 3.14]

Carpooling is the sharing of car journeys so that more than one person travels in a car, and prevents the need for others to have to drive to a location themselves. By having more people using one vehicle, carpooling reduces each person’s travel costs such as: fuel costs, tolls, and the stress of driving. Carpooling is also a more environmentally friendly and sustainable way to travel as sharing journeys reduces air pollution, carbon emissions, traffic congestion on the roads, and the need for parking spaces. Three friends Amar, Bhavin and Chetanya live in societies represented by the points A(4, 5), B(6, 2) and C(2, 6) respectively. They all work in offices located in a same building represented by the point O(0, 0). Since they all go to same building every day, they decided to do carpooling to save money on petrol.

Based on the above information, answer the following questions.

1. What is the distance between B and C?   (1)
2. If Bhavin and Chetanya planned to meet at a club situated at the mid-point of the line joining the points B and C, find the coordinates of this point.   (1)
3. 1. Which society is farthest from the office? Also find its distance from the office.   (2)
      OR
   2. Out of B and C which society is nearer to A? Also find their distances.

1. What is the area of sector in terms of arc length?  (1)
2. What is the area of the watered region (in terms of π)?   (1)
3. 1. If the radius (r) changes to 28 m, find the angle θ so that the area of the watered region remains the same.  (2)
      OR
   2. If the radius (r) is increased from 21 m to 28 m and the angle remains the same, what is the increase in the area of the watered region?

One of four main blood types can be found in a human body. They are known as A, B, AB and O. Each blood type can be further classified as either a Rhesus positive (+) or Rhesus negative (–). For example, a possible combination is blood type O and Rhesus negative which is written as O–. The data below shows the distribution of the blood types and Rhesus types of given blood type for a Blood Donation Center recorded (in percentages) for the year 2023. Blood Group Rhesus factor Number of person (in %) O O− x O+ 30 A A− 8 A+ 24 B B− 6 B+ 18 AB AB− 1 AB+ 3

1.  Find the value of x.   (1)
2. Find the probability that a randomly selected person has a Rhesus negative blood type.   (1)
3. 1. What is the probability that the person selected from the record is Rhesus positive but neither blood type A nor B?   (2)
      OR
   2. People with blood type AB positive (AB⁺) are known as the universal recipient and with blood type O negative (O^–) are known as universal donor. Find the probability of a selected person to be neither universal recipient nor universal donor.