Mathematics Standard - 30/5/3 2025-2026 English Medium Class 10 Question Paper Solution

A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (Use π = 3.14)

`314 sqrt(2)  cm^2`

314 cm²

`3140/3 cm^2`

`3140 sqrt(2)  cm^2`

If a_n represents n^th term of the A.P. `- 15/4, - 10/4, - 5/4,` ...... then value of a₁₆ – a₁₂ is ______.

4

`5/4`

5

`25/4`

Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is ______.

64

640

100

10

A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 m². Outer surface area of the tent is ______.

11.78 m²

12.32 m²

11.82 m²

12.86 m²

PQ is tangent to a circle with centre O. If OQ = a, OP = a + 2 and PQ = 2b, then relation between a and b is

a² + (a + 2)² = (2b)²

b² = a + 4

2a² + 1 = b²

b² = a + 1

Simplest form of `(sec A)/sqrt(sec^2 A - 1)` is ______.

sin A

tan A

cosec A

cos A

The line segment joining the points P(–4. –2) and Q(10, 4) is divided by y-axis in the ratio

2 : 5

1 : 2

2 : 1

5 : 2

A wire is attached from a point A on the ground to the top of a pole BC. making an angle of elevation as 60°. If AB = `5sqrt(3)` m. then length of the wire is

10 m

`10 sqrt(3)  m`

15 m

`5/2 sqrt(3)  m`

In the given figure, AB || EF. If AB = 24 cm, EF = 36 cm and DA = 7 cm, then AE equals

2.5 cm

10.5 cm

3.5 cm

`14/3` cm

Devansh proved that ΔABC ∼ ΔPQR using SAS similarity criteria. If he found ∠C = ∠R, then which of the following was proved true?

`(AC)/(AB) = (PR)/(PQ)`

`(BC)/(AC) = (PR)/(QR)`

`(AC)/(BC) = (PR)/(PQ)`

`(AC)/(BC) = (PR)/(QR)`

While calculating mean of a grouped frequency distribution, step deviation method was used `((x - a)/h = u)`. It was found that `barx = 64`, h = 5 and a = 62.5. The value of `baru` is ______.

0.5

1.5

0.3

7.5

For an acute angle θ, if `sin θ = 1/9`, then value of `(9  "cosec"  θ + 1)/(9  "cosec"  θ - 1)` is ______.

0

`80/81`

1

`82/80`

Which of the following cannot be the probability of an event?

`39/100`

`0.001/20`

`10/0.2`

10%

The value of m for which the quadratic equation 3x² – 7x + m = 0 has real and equal roots, is ______.

7

`49/12`

`49/3`

4

If the zeroes of a polynomial p(x) are –3 and 8, then p(x) equals

x² + 5x – 4

(x + 3)(–x + 8)

a(x² + 5x – 24)

x² – 24

The value of p for which roots of the quadratic equation x² – px + 6 = 0 are rational, is ______.

1

–5

25

`sqrt(5)`

An arc of length 2.2 cm subtends an angle θ at the centre of the circle with radius 2.8 cm. The value of θ is ______.

50°

60°

45°

30°

Two dice are rolled together. The probability of getting an outcome (x, y) where x > y, is ______.

`5/12`

`5/6`

1

0

Assertion (A): H.C.F. (36 m², 18 m) = 18 m, where m is a prime number.

Reason (R): H.C.F. of two numbers is always less than or equal to the smaller number.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

Assertion (A): The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent.

Reason (R): When two linear equations don’t have unique solution, they always represent parallel lines.

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

Assertion (A) is true, but Reason (R) is false.

Assertion (A) is false, but Reason (R) is true.

In the given figure, point D divides the side BC of ΔABC in the ratio 1 : 2. Find the length AD.

Evaluate: `(sin^3 60^circ - tan 30^circ)/(cos^2 45^circ)`

For acute angles A and B and A + 2B and 2A + B are acute if `tan (A + 2B) = sqrt(3)` and `sin (2A + B) = 1/sqrt(2)`, then find the measures of angles A and B.

A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is `3/5`, then find the number of yellow balls.

In the given figure, AB || DE and AC || DF. Show that ΔABC ~ ΔDEF. If BC = 10 cm, EB = CF = 5 cm and AB = 7 cm, then find the length DE.

Prove that `14 - 2sqrt(3)` is an irrational number, given that `sqrt(3)` is irrational.

A circle centered at (2, 1) passes through the points A(5, 6) and B(–3, K). Find the value(s) of K. Hence find length of chord AB.

Prove that the point P dividing the line segment joining the points A(–1, 7) and B(4, –3) in the ratio 3 : 2, lies on the line x – 3y = –1. Also find length of PA and PB.

Use graphical method to solve the system of linear equations: x = –3 and 5x – 2y = –5.

In an A.P., 15^th term exceeds the 8^th term by 21. If sum of first 10 terms is 55, then form the A.Р.

The sum of first n terms of an A.P. is 2n² + 13n. Find its n^th term and hence 10^th term.

The dimensions of a window are 156 cm × 216 cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

Prove that: `(tan θ)/(1 - cot θ) + (cot θ)/(1 - tan θ) = 1 + tan θ + cot θ`.

A chord of a circle, of radius 14 cm, subtends an angle of 60° at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R.

Prove that

1. AQ = QR
2. AP = 2PQ
3. PR = 2AP

The mean of the following distribution is 53. Find the missing frequency p.

Hence, find mode of the distribution.

Compute median of the following data:

PQ and PR are two tangents to a circle with centre O and radius 5 cm. AВ is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

Two water taps together can fill a tank in `8 8/9` hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank.  From a window (W) of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°.

(i) Write a relation between d (the height of window) and y.   [1]

(ii) Determine the value of h.   [1]

(iii) (a) Determine height of the water tank.   [2]

OR

(iii) (b) Find the value of x and height of the window above ground level.   [2]

An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial p(x) = –0.0025x2 – 0.025x + 136.

Observe the diagram and based on above information, answer the following questions:

(i) Write the co-ordinates of point A.   [1]

(ii) Find the span of the arch.   [1]

(iii) (a) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.   [2]

OR

(iii) (b) Find the values of p(x) at x = 100 and x = –100. Are they same?   [2]

A wall mounted lamp, made of fabric, is shown below. Lamp has cuboidal shape, open from top and bottom. A spherical bulb of diameter 7 cm is latched with a very thin rod. (Ignore the rod while making calculations.) Dimensions of the cuboid are 24 cm × 12 cm × 17 cm.

(i) Find the surface area of the bulb.   [1]

(ii) What could be the maximum diameter of the bulb if at least 1 cm space is left from each side?   [1]

(iii) (a) Find the area of the fabric used if there is a fold of 2 cm on top and bottom edges.   [2]

OR

(iii) (b) Find the space available inside the lamp.   [2]