Mathematics Basic - Outside Delhi Set 2 2022-2023 English Medium Class 10 Question Paper Solution

The HCF of the smallest 2-digit number and the smallest composite number is ______.

4

20

2

10

40

The value of ‘p’ if (– 2, p) lies on the line represented by the equation 2x – 3y + 7 = 0, is ______.

`-13/2`

`13/2`

– 1

1

Distance of the point (6, 5) from the y-axis is ______.

6 units

5 units

`sqrt(61)` units

0 unit

Area of a quadrant of a circle of radius 7 cm is ______.

154 cm²

77 cm²

`77/2`cm²

`77/4`cm²

If HCF (72, 120) = 24, then LCM (72, 120) is ______.

72

120

360

9640

One card is drawn at random from a well-shuffled deck of 52 playing cards. What is the probability of getting a black king?

`1/26`

`1/13`

`1/52`

`1/2`

The graph of y = f(x) is shown in the figure for some polynomial f(x).

The number of zeroes of f(x) is ______.

0

2

3

4

The value of k, if (6, k) lies on the line represented by x – 3y + 6 = 0, is ______.

– 4

12

– 12

4

The prime factorisation of the number 2304 is ______.

2⁸ × 3² 

2⁷ × 3³

2⁸ × 3¹

2⁷ × 3²

If n is a natural number, then 8ⁿ cannot end with digit

0

2

4

6

The median of first seven prime numbers is ______.

5

7

11

13

If (2, 4) is the mid-point of the line segment joining (6, 3) and (a, 5), then the value of a is ______.

2

4

– 4

– 2

The 20^th term of an A.P, whose first term is – 2 and the common difference is 4, is ______.

78

74

– 36

– 34

The zeroes of the polynomial p(x) = 25x² – 49 are ______.

`49/25, 49/25`

`-49/25, 49/25`

`7/5, -7/5`

`7/5, 7/5`

The mean of first ten natural numbers is ______.

5.5

55

45

4.5

If –5, x, 3 are three consecutive terms of an A.P., then the value of x is ______.

– 2

2

1

– 1

An unbiased die is thrown. The probability of getting an odd prime number is ______.

`1/6`

`1/2`

`2/3`

`1/3`

If the mean of 6, 7, x, 8, y, 14 is 9, then ______.

 x + y = 21

x + y = 19

x − y = 19

x − y = 21

Assertion (A): The probability that a leap year has 53 Sundays is `2/7`.

Reason (R): The probability that a non-leap year has 53 Sundays is `1/2`.

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, 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): For 0 < 0 ≤ 90°, cosec θ – cot θ and cosec θ + cot θ are reciprocal of each other.

Reason (R): cot² θ – cosec² θ = 1.

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.

Evaluate: 5 cosec² 45° – 3 sin² 90° + 5 cos 0°.

Find a quadratic polynomial whose zeroes are 6 and – 3.

Find the zeroes of the polynomial x² + 4x – 12.

Find the value of k for which the roots of the quadratic equation 5x² – 10x + k = 0 are real and equal.

If one root of the quadratic equation 3x² – 8x – (2k + 1) = 0 is seven times the other, then find the value of k.

A box contains 20 discs which are numbered from 1 to 20. If one disc is drawn at random from the box, then find the probability that the number the drawn disc is a 2-digit number.

A box contains 20 discs which are numbered from 1 to 20. If one disc is drawn at random from the box, then find the probability that the number the drawn disc is a number less than 10.

Evaluate: `(5  "cosec"^2  30^circ - cos 90^circ)/(4 tan^2 60^circ)`

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

If A and B are (– 2, – 2) and (2, – 4) respectively; then find the co-ordinates of the point P such that `(AB)/(AB) = 3/7`.

A solid is in the shape of a cone standing on a hemisphere with both their diameters being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid. [Use π = 3.14]

Find the co-ordinates of the points of trisection of the line segment joining the points (5, 3) and (4, 5).

Prove that `3 - 2sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.

Prove that `(cot A - cos A)/(cot A + cos A) = (cos^2 A)/(1 + sin A)^2`

Prove that (sec θ + tan θ) (1 – sin θ) = cos θ

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. (Use `sqrt(3)` = 1.73)

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower. (Use `sqrt3` = 1.73)

The first term of an A.P. is 22, the last term is –6 and the sum of all the terms is 64. Find the number of terms of the A.P. Also, find the common difference.

An ice-cream filled cone having radius 5 cm and height 10 cm is as shown in the figure. Find the volume of the ice-cream in 7 such cones.

Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.

In the following figure, `("QR")/("QS") = ("QT")/("PR")` and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.

Read the following passage:

Based on the above information, answer the following questions:

1. What is the area of square ABCD?
2. What is the length of diagonal AC of square ABCD?
3. Find the area of sector OPRQO.
   OR
   Find the area of remaining part of square ABCD when area of circle is excluded.

Read the following passage:

Lokesh, a production manager in Mumbai, hires a taxi everyday to go to his office. The taxi charges in Mumbai consists of a fixed charges together with the charges for the distance covered. His office is at a distance of 10 km from his home. For a distance of 10 km to his office, Lokesh paid ₹ 105. While coming back home, he took another roµte. He covered a distance of 15 km and the charges paid by him were ₹ 155.

Based on the above information, answer the following questions:

1. What are the fixed charges?
2. What are the charges per km?
3. If fixed charges are ₹ 20 and charges per km are ₹ 10, then how much Lokesh have to pay for travelling a distance of 10 km? 
   OR
   Find the total amount paid by Lokesh for travelling 10 km from home to office and 25 km from office to home. [Fixed charges and charges per km are as in (i) and (ii).

Read the following passage:

People of a circular village Dharamkot want to construct a road nearest to it. The road cannot pass through the village. But the people want the road at a shortest distance from the centre of the village. Suppose the road starts from A which is outside the circular village (as shown in the figure) and touch the boundary of the circular village at B such that AB = 20 m. Also the distance of the point A from the centre O of the village is 25 m.

Based on the above information, answer the following questions:

1. If B is the mid-point of AC, then find the distance AC.
2. Find the shortest distance of the road from the centre of the village.
3. Find the circumference of the village.
   OR
   Find the area of the village.