Mathematics Basic Official Outside Delhi set 3 2024-2025 English Medium Class 10 Question Paper Solution

In two concentric circles centred at O, a chord AB of the larger circle touches the smaller circle at C. If OA = 3.5 cm, OC = 2.1 cm, then AB is equal to:

5.6 cm

2.8 cm

3.5 cm

4.2 cm

Three coins are tossed together. The probability that at least one head comes up is ______.

`3/8`

`7/8`

`1/8`

`3/4`

The volume of air in a hollow cylinder is 450 cm³. A cone of the same height and radius as that of the cylinder is kept inside it. The volume of empty space in the cylinder is:

225 cm³

150 cm³

250 cm³

300 cm³

If the length of the shadow of a tower is `sqrt(3)` times its height, then the angle of elevation of the sun is ______.

45°

30°

60°

0°

22^nd term of the A.P. : `3/2, 1/2, (-1)/2, (-3)/2, ...` is:

`45/2`

–9

`(-39)/2`

–21

In the given graph, the polynomial p(x) is shown. Number of zeroes of p(x) is:

3

2

1

4

If the probability of happening of an event is 57%, then the probability of non-happening of the event is ______.

0.43

0.57

53%

`1/57`

OAB is a sector of a circle with centre O and radius 7 cm. If the length of arc `bar(AB) = 22/3` cm, then ∠AOB is equal to ______.

`(120/7)^°`

45°

60°

30°

If the sum of the first n terms of an A.P. is given by `S_n = n/2 (3n + 1)`, then the first term of the A.P. is ______.

2

`3/2`

4

`5/2`

To calculate the mean of grouped data, Rahul used the assumed mean method. He used d = (x – А), where A is the assumed mean. Then `bar(x)` is equal to ______.

`A + bar(d)`

`A + hbar(d)`

`h(A + bard)`

`A - hbar(d)`

The point (3, –5) lies on the line mx – y = 11. The value of m is ______.

3

–2

8

2

If `sqrt(3) sin θ = cos θ`, then the value of θ is ______.

`sqrt(3)`

60°

`1/sqrt(3)`

30°

ABCD is a rectangle with its vertices at (2, –2), (8, 4), (4, 8) and (–2, 2) taken in order. Length of its diagonal is ______.

`4sqrt(2)`

`6sqrt(2)`

`4sqrt(26)`

`2sqrt(26)`

Two dice are rolled together. The probability of getting a sum more than 9 is ______.

`5/6`

`5/18`

`1/6`

`1/2`

In ΔABC, DE || BC. If AE = (2x + 1) cm, EC = 4 cm, AD = (x + 1) cm and DB = 3 cm, then value of x is:

1

`1/2`

–1

`1/3`

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

−10

−5

5

7

In the given figure, PA is tangent to a circle with centre O. If ∠APO = 30° and OA = 2.5 cm, then OP is equal to:

2.5 cm

5 cm

`5/sqrt(3)` cm

2 cm

Two identical cones are joined as shown in the figure. If the radius of the base is 4 cm and the slant height of the cone is 6 cm, then the height of the solid is:

8 cm

`4sqrt(5)` cm

`2sqrt(5)` cm

12 cm

Assertion (A): `(a + sqrt(b)) · (a - sqrt(b))` is a rational number, where a and b are positive integers.

Reason (R): Product of two irrationals is always rational.

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

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

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

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

Assertion (A): ΔАВС ~ ΔРQR such that ∠A = 65°, ∠C = 60°. Hence ∠Q = 55°. 

Reason (R): Sum of all angles of a triangle is 180°.

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

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

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

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

A box contains 120 discs, which are numbered from 1 to 120. If one disc is drawn at random from the box, find the probability that:

1. it bears a 2-digit number. 
2. the number is a perfect square.

Evaluate: 

`(cos 45°)/(tan 30° + sin 60°)`

Verify that sin 2A = `(2 tan A)/(1 + tan^2A)`, for A = 30°.

Solve the quadratic equation `sqrt(3)x^2 + 10x + 7sqrt(3) = 0` using quadratic formula.

Find the nature of the roots of the equation 4x² – 4a²x + a⁴ – b⁴ = 0, b ≠ 0.

Using prime factorisation, find the HCF of 144, 180 and 192.

The perimeters of two similar triangles are 22 cm and 33 cm, respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.

Given that `sqrt(5)` is an irrational number, prove that `2 + 3sqrt(5)` is an irrational number.

Find the A.P. whose third term is 16 and seventh term exceeds the fifth term by 12. Also, find the sum of first 29 terms of the A.P.

Find the sum of the first 20 terms of an A.P. whose n^th term is given by a_n = 5 + 2n. Can 52 be a term of this A.P.?

Prove that `sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2  "cosec"  θ`.

Find length and breadth of a rectangular park whose perimeter is 100 m and area is 600 m².

AB and CD are diameters of a circle with centre O and radius 7 cm. If ∠BOD = 30°, then find the area and perimeter of the shaded region.

α, β are zeroes of the polynomial 3x² – 8x + k. Find the value of k, if `α^2 + β^2 = 40/9`.

Find the zeroes of the polynomial 2x² + 7x + 5 and verify the relationship between its zeroes and coefficients.

Find the ‘mean’ and ‘mode’ marks of the following data: 

Solve the following pair of linear equations by the graphical method:

2x + y = 9 and x – 2y = 2

Nidhi received a simple interest rate of 1,200 when she invested ₹ x at 6% p.a. and ₹ y at 5% p.a. for 1 year. Had she invested ₹ x at 3% p.a. and ₹ y at 8% p.a. for that year, she would have received simple interest of ₹ 1,260. Find the values of x and y.

The given figure shows a circle with centre O and radius 4 cm circumscribed by ΔABC. BС touches the circle at D such that BD = 6 cm, DC = 10 cm. Find the length of AE.

PA and PB are tangents drawn to a circle with centre O. If ∠AOB = 120° and OA = 10 cm, then 

1. Find ∠OPA.  (1)
2. Find the perimeter of ΔOAP.  (3)
3. Find the length of chord AB.  (1)

A drone is flying at a height of h metres. At an instant it observes the angle of elevation of the top of an industrial turbine as 60° and the angle of depression of the foot of the turbine as 30°. If the height of the turbine is 200 metres, find the value of h and the distance of the drone from the turbine. (Use `sqrt(3)` = 1.73)

A triangular window of a building is shown above. Its diagram represents a ΔABC with ∠A = 90° and AB = AC. Points P and R trisect AB and PQ || RS || AС.

Based on the above, answer the following questions:

1. Show that ΔBPQ ~ ΔBAC.  (1)
2. Prove that PQ = `1/3` AC.  (1)
3. 1. If AB = 3 m, find the lengths of BQ and BS. Verify that BQ = `1/2` BS.  (2)
      OR
   2. Prove that `BR^2 + RS^2 = 4/9 BC^2`.  (2)

Based on the above, answer the following questions:

Based on the above, answer the following questions:

1. Find the dimensions of the cuboidal box.  (1)
2. Find the total outer surface area of the box.  (1)
3. 1. Find the difference between the capacity of the bowl and the volume of the box. (Use π = 3.14).   (2)
      OR
   2. The inner surface of the bowl and the thickness is to be painted. Find the area to be painted.  (2)