Mathematics Standard Board Sample Paper 2024-2025 English Medium Class 10 Question Paper Solution

The graph of a quadratic polynomial p(x) passes through the points (−6, 0), (0, −30), (4, −20) and (6, 0). The zeroes of the polynomial are __________.

−6, 0

4, 6

−30, −20

−6, 6

The value of k for which the system of equations 3x − ky = 7 and 6x + 10y = 3 is inconsistent, is ______.

−10

−5

5

7

Which of the following statements is not true?

A number of secants can be drawn at any point on the circle.

Only one tangent can be drawn at any point on a circle.

A chord is a line segment joining two points on the circle.

From a point inside a circle only two tangents can be drawn.

If the nth term of an A.P. is 7n − 4, then the common difference of the A.P. is ______.

7

7n

−4

4

The radius of the base of a right circular cone and the radius of a sphere are each 5 cm in length. If the volume of the cone is equal to the volume of the sphere then the height of the cone is ________.

5 cm

20 cm

10 cm

4 cm

If tanθ = `5/2` then `(4 sinθ + cosθ)/(4sinθ − cosθ)` is equal to ________.

`11/9`

`3/2`

`9/11`

4

In the given figure, a tangent has been drawn at a point P on the circle centred at O.

If ∠TPQ = 110°, then ∠POQ is equal to:

110°

70°

140°

55°

A quadratic polynomial having zeroes `-sqrt(5/2)` and `sqrt(5/2)` is _________.

`x^2 − 5sqrt2 x + 1`

8x² − 20

15x² − 6

`x^2 - 2sqrt5 x - 1`

Consider the frequency distribution of 45 observations.

The upper limit of the median class is:

20

10

30

40

O is the point of intersection of two chords AB and CD of a circle.

If ∠BOC = 80° and OA = OD then ΔODA and ΔOBC are ____________.

equilateral and similar

isosceles and similar

isosceles but not similar

not similar

The roots of the quadratic equation x² + x − 1 = 0 are __________.

Irrational and distinct

not real

rational and distinct

real and equal

If θ = 30° then the value of 3tanθ is ________.

1

`1/sqrt3`

`3/sqrt3`

not defined

The volume of a solid hemisphere is `396/7` cm³. The total surface area of the solid hemisphere (in sq. cm) is ______.

`396/7`

`594/7`

`549/7`

`604/7`

In a bag containing 24 balls, 4 are blue, 11 are green and the rest are white. One ball is drawn at random. The probability that drawn ball is white in colour is ________.

`1/6`

`3/8`

`11/24`

`5/8`

The point on the x-axis nearest to the point (-4, -5) is __________.

(0, 0)

(−4, 0)

(−5, 0)

`(sqrt41, 0)`

Which of the following gives the middle most observation of the data?

Median

Mean

Range

Mode

A point on the x-axis divides the line segment joining the points A(2, −3) and B(5, 6) in the ratio 1 : 2. The point is __________.

(4, 0)

`(7/2, 3/2)`

(3, 0)

(0, 3)

A card is drawn from a well shuffled deck of playing cards. The probability of getting red face card is _________.

`3/13`

`1/2`

`3/52`

`3/26`

Assertion (A): HCF of any two consecutive even natural numbers is always 2.

Reason (R): Even natural numbers are divisible by 2.

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 and 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): If the radius of sector of a circle is reduced to its half and angle is doubled then the perimeter of the sector remains the same.

Reason (R): The length of the arc subtending angle θ at the centre of a circle of radius r = `(pirθ)/180`.

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 and 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.

Find the H.C.F and L.C.M of 480 and 720 using the prime factorisation method.

The H.C.F of 85 and 238 is expressible in the form 85m -238. Find the value of m.

Two dice are rolled together bearing numbers 4, 6, 7, 9, 11, 12. Find the probability that the product of numbers obtained is an odd number.

How many positive three-digit integers have the hundredths digit 8 and unit’s digit 5? Find the probability of selecting one such number out of all three-digit numbers.

Evaluate: `(2sin^2 60° − tan^2 30°)/(sec^2 45°)`

Find the point(s) on the x-axis which is at a distance of `sqrt41` units from the point (8, −5).

Show that the points A(−5, 6), B(3, 0) and C(9, 8) are the vertices of an isosceles triangle.

In ΔABC, D, E and F are midpoints of BC, CA and AB respectively. Prove that ΔFBD ∼ ΔDEF and ΔDEF ∼ ΔABC.

In ΔABC, P and Q are points on AB and AC, respectively, such that PQ is parallel to BC. Prove that the median AD drawn from A on BC bisects PQ.

The sum of two numbers is 18 and the sum of their reciprocals is `9/40`. Find the numbers.

If α and β are zeroes of a polynomial 6x² − 5x + 1 then form a quadratic polynomial whose zeroes are α² and β².

If cosθ + sinθ = 1, then prove that cosθ − sinθ = ±1.

The minute hand of a wall clock is 18 cm long. Find the area of the face of the clock described by the minute hand in 35 minutes.

AB is a chord of a circle centred at O such that ∠AOB = 60°. If OA = 14 cm, then find the area of the minor segment. `("take" sqrt3 = 1.73)`

Prove that `sqrt3` is an irrational number.

Solve the following system of linear equations graphically:

x + 2y = 3, 2x − 3y + 8 = 0

Places A and B are 180 km apart on a highway. One car starts from A and another from B at the same time. If the car travels in the same direction at different speeds, they meet in 9 hours. If they travel towards each other with the same speeds as before, they meet in an hour. What are the speeds of the two cars?

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Using the above result, find the length BC of ΔABC. Given that, a circle is inscribed in ΔABC touching the sides AB, BC and CA at R, P and Q respectively and AB = 10 cm, AQ = 7 cm, CQ = 5 cm.

A boy whose eye level is 1.35 m from the ground, spots a balloon moving with the wind in a horizontal line at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is 60°. After 12 seconds, the angle of elevation reduces to 30°. If the speed of the wind is 3m/s then find the height of the balloon from the ground. `("Use"  sqrt3= 1.73)`

Find the mean and median of the following data:

The monthly expenditure on milk in 200 families of a Housing Society is given below:

Find the value of x and also, find the median and mean expenditure on milk.

On the basis of the above situation, answer the following questions:

1. Write an A.P. whose terms represent the number of jars in different layers starting from top. Also, find the common difference.
2. Is it possible to arrange 34 jars in a layer if this pattern is continued? Justify your answer.
3. 1. If there are ‘n’ number of rows in a layer, then find the expression for finding the total number of jars in terms of n. Hence, find S₈.
                                       OR
   2. The shopkeeper added 3 jars in each layer. How many jars are there in the 5^th layer from the top?

Triangle is a very popular shape used in interior designing. The picture given above shows a cabinet designed by a famous interior designer.

Here the largest triangle is represented by ΔABC and smallest one with shelf is represented by ΔDEF. PQ is parallel to EF.

1. Show that ΔDPQ ∼ ΔDEF.
2. If DP = 50 cm and PE = 70 cm then find `(PQ)/(EF)`.
3. 1. If 2AB = 5DE and ΔABC ∼ ΔDEF then show that `"perimeter of ΔABC"/"perimeter of ΔDEF"` is constant.
                                            OR
   2. If AM and DN are medians of triangles ABC and DEF respectively then prove that ΔABM ∼ ΔDEN.

On the basis of the above information, answer the following questions:.

1. Calculate the slant height of the conical part of one silo.
2. Find the curved surface area of the conical part of one silo.
3. 1. Find the cost of metal sheet used to make the curved cylindrical part of 1 silo at the rate of ₹ 2000 per m².
                                         OR
   2. Find the total capacity of one silo to store grains.