Mathematics Basic Official Outside Delhi set 2 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 exactly one coin shows heads is ______.

`1/8`

`1/4`

1

`3/8`

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

In ΔABC, ∠B = 90°. If `(AB)/(AC) = 1/2`, then cos C is equal to ______.

`3/2`

`1/2`

`sqrt(3)/2`

`1/sqrt(3)`

15^th term of the A.P. `13/3, 9/3, 5/3,` ........ is:

23

`(-53)/3`

–11

`(-43)/3`

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`

A quadratic polynomial having zeroes 0 and –2, is ______.

x(x – 2)

4x(x + 2)

x² + 2

2x² + 2x

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°

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)`

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`

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)`

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 dice are rolled together. The probability of getting an outcome (a, b) such that b = 2a is ______.

`1/6`

`1/12`

`1/36`

`1/9`

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

If sin θ = `1/9`, then tan θ is equal to ______.

`1/(4sqrt(5))`

`(4sqrt(5))/9` 

`1/8`

`4sqrt(5)`

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

The line 2x – 3y = 6 intersects the x-axis at ______.

(0, –2)

(0, 3)

(–2, 0)

(3, 0)

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.

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.

Evaluate: 

`(cos 45°)/(tan 30° + sin 60°)`

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

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.

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

Solve the equation 4x² – 9x + 3 = 0, using the quadratic formula.

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

In the given figure, AB || DE and BD || EF. Prove that DC² = CF × AC.

Three friends plan to go for a morning walk. They step off together and their steps measures 48 cm, 52 cm and 56 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ten times?

Prove that `(1 + 1/(tan^2θ))(1 + 1/(cos^2θ)) = 1/(sin^2θ - sin^4θ)`.

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.

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.?

If α, β are zeroes of the polynomial 8x² – 5x – 1, then form a quadratic polynomial in x whose zeroes are `2/α` and `2/β`.

Find the zeroes of the polynomial p(x) = 3x² + x – 10 and verify the relationship between the zeroes and its coefficients.

The sum of a number and its reciprocal is `13/6`. Find the number.

Two poles of equal heights are standing opposite each other on either side of the road, which is 85 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles. (Use `sqrt(3)` = 1.73)

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.

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

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)

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)

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: