Mathematics Basic - Outside Delhi Set 3 2022-2023 English Medium Class 10 Question Paper Solution

The prime factorisation of the number 5488 is ______.

2³ × 7³

2⁴ × 7³

2⁴ × 7⁴

2³ × 7⁴

The Empirical relation between the three measures of central tendency is ______.

Mode = 3 Mean – 2 Median

Mode = 2 Median – 3 Mean

Mode = 2 Mean – 3 Median

Mode = 3 Median – 2 Mean

In the given figure, ΔPQR is a right triangle right angled at Q. If PQ = 4 cm and PR = 8 cm, then P is ______.

60°

45°

30°

15°

The median of first 10 natural numbers is ______.

5

6

5.5

6.5

The zeroes of the polynomial p(x) = 2x² – x – 3 are ______.

`-3/2, 1`

`3/2, 1`

`-3/2, -1`

`3/2, -1`

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

4

3

2

1

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 value of ‘k’ for which the system of equations kx + 2y = 5 and 3x + 4y = 1 have no solution, is ______.

k = `3/2`

k ≠ `3/2`

k ≠ `2/3`

k = 15

In the given figure, PQ and PR are tangents drawn from P to the circle with centre O such that ∠QPR = 65°. The measure of ∠QOR is ______.

65°

125°

115°

90°

The zeroes of the quadratic polynomial 16x² – 9 are ______.

`3/4, 3/4`

`-3/4, 3/4`

`9/16, 9/16`

`-3/4, -3/4`

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.

A bag contains 30 discs numbered from 1 to 30. One disc is drawn at random from the bag. Find the probability that it bears a number divisible by 6.

A bag contains 30 discs numbered from 1 to 30. One disc is drawn at random from the bag. Find the probability that it bears a number greater than 25.

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.

From a point P, the length of the tangent to a circle is 24 cm and the distance of P from the centre of the circle is 25 cm. Find the radius of the circle.

Prove that `7 + 4sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.

Solve for x:

`1/x - 1/(x-2)=3`, x ≠ 0, 2

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB.

In the given figure, CM and RN are respectively the medians of ΔABC and ΔPQR. If ΔABC ∼ ΔPQR, then prove that ΔAMC ∼ ΔPNR.

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.

The sum of the 4^th and 8^th term of an A.P. is 24 and the sum of the 6^th and 10^th term of the A.P. is 44. Find the A.P. Also, find the sum of first 25 terms of the A.P.

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

 [Use `pi = 22/7`]

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.