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

The value of k for which the equations 3x – y + 8 = 0 and 6x – ky + 16 = 0 represent coincident lines is ______.

`1/2`

`-1/2`

2

– 2

A circle of radius 5.2 cm has two tangents AB and CD parallel to each other. What is the distance between the two tangents?

5.2 cm

10.4 cm

20.8 cm

can't find

The number of polynomials having zeroes – 3 and 4 is ______.

1

2

3

more than 3

Tick the correct answer in the following and justify your choice: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is:

2 units

π units

4 units

2π units

7 units

If p(x) = x² + 5x + 6, then p(– 2) is ______.

20

0

– 8

8

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

0.1

`5/3`

3%

`1/3`

The pair of linear equations x + 2y + 5 = 0 and – 3x – 6y + 1 = 0 has ______.

a unique solution

exactly two solutions

infinitely many solutions

no solution

If ΔABC ~ ΔDEF and ∠A = 47°, ∠E = 83°, then ∠C is equal ______.

47°

50°

83°

130°

If the pair of linear equations x – y = 1, x + ky = 5 has a unique solution x = 2, y = 1, then the value of k is ______.

– 2

– 3

3

4

The value of 5 sin² 90° – 2 cos² 0° is ______.

– 2

5

3

– 3

The length of the arc of a circle of radius 14 cm which subtends an angle of 60° at the centre of the circle is ______.

`44/3` cm

`88/3` cm

`308/3` cm

`616/3` cm

The angle of elevation of the top of a 30 m high tower at a point 30 m away from the base of the tower is ______.

30°

45°

60°

90°

The mode of the numbers 2, 3, 3, 4, 5, 4, 4, 5, 3, 4, 2, 6, 7 is ______.

2

3

4

5

From a well-shuffled deck of 52 playing cards, a card is drawn at random. What is the probability of getting a red queen?

`1/52`

`1/26`

`1/13`

`12/13`

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is ______.

x² + 4 = 0

x² − 4 = 0

4x² − 1 = 0

x² − 2 = 0

Which of the following is not a quadratic equation?

2(x – 1)² = 4x² – 2x + 1

2x – x² = x² + 5

`(sqrt(2)x + sqrt(3))^2 + x^2 = 3x^2 - 5x`

(x² + 2x)² = x⁴ + 3 + 4x³

What is the length of arc of a circle of radius 7 cm which subtends an angle of 90° at the centre of the circle?

22 cm

11 cm

`77/2` cm

`11/2` cm

(3 sin² 30° – 4 cos² 60°) is equal to ______.

`5/4`

`-3/4`

`-1/4`

`-9/4`

Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact.

Reason (R): The lengths of tangents drawn from an external point to a circle are equal.

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

Both Assertion (A) and Reason (R) are true but Reason (R) does not give 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): If one root of the quadratic equation 4x² – 10x + (k – 4) = 0 is reciprocal of the other, then value of k is 8.

Reason (R): Roots of the quadratic equation x² – x + 1 = 0 are real.

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

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

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

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

If sin α = `1/2`, then find the value of (3 cos α – 4 cos³ α).

Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.

If the points A(2, 3), B(–5, 6), C(6, 7) and D(p, 4) are the vertices of a parallelogram ABCD, find the value of p.

Find the discriminant of the quadratic equation `3x^2 - 2x + 1/3` = 0 and hence find the nature of its roots.

Find the roots of the quadratic equation x² – x – 2 = 0.

In the adjoining figure, PT is a tangent at T to the circle with centre O. If ∠TPO = 30°, find the value of x.

In a right triangle PQR, right angled at Q. If tan P = `sqrt(3)`, then evaluate 2 sin P cos P.

Prove the following trigonometric identities.

`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`

An unbiased coin is tossed twice. Find the probability of getting at least one head.

An unbiased coin is tossed twice. Find the probability of getting exactly one tail.

An unbiased coin is tossed twice. Find the probability of getting at most one head.

A lending library has a fixed charge for first three days and an additional charge for each day thereafter. Rittik paid 27 for a book kept for 7 days and Manmohan paid ₹ 21 for a book kept for 5 days. Find the fixed charges and the charge for each extra day.

Find the values of 'a' and 'b' for which the system of linear equations 3x + 4y = 12, (a + b)x + 2(a – b)y = 24 has infinite number of solutions.

A die is thrown. Find the probability of getting:

an even prime number

A die is thrown, find the probability of getting:

a number greater than 4

A die is thrown once. Find the probability of getting an odd number.

Find the area of the sector of a circle of radius 7 cm and of central angle 90°. Also, find the area of corresponding major sector.

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Two concentric circles with centre O are of radii 3 cm and 5 cm. Find the length of chord AB of the larger circle which touches the smaller circle at P.

The shadow of a tower standing on a level ground is found to be 40 m longer when Sun’s altitude is 30° than when it was 60°. Find the height of the tower.

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

Find the sum of first 25 terms of the A.P. whose nth term is given by a_n = 5 + 6n. Also, find the ratio of 20^th term to 45^th term.

In an A.P., if S_n = 3n² + 5n and a_k = 164, find the value of k.

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

From a point P on the ground the angle of elevation of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag-staff from P is 45°. Find the length of the flag-staff and the distance of the building from the point P. (Take `sqrt3` = 1.732)

Read the following passage:

Based on the above information, answer the following questions:

1. Find the coordinates of the point of intersection of diagonals AC and BD.
2. Find the length of the diagonal AC.
3. 1. Find the area of the campaign Board ABCD.
      OR
   2. Find the ratio of the length of side AB to the length of the diagonal AC.

Read the following passage:

Based on the above information, answer the following questions:

1. How many guests Khushi can invite at the most?
2. How many apples and bananas will each guest get?
3. 1. If Khushi decides to add 42 mangoes, how many guests Khushi can invite at the most?
      OR
   2. If the cost of 1 dozen of bananas is ₹ 60, the cost of 1 apple is ₹ 15 and cost of 1 mango is ₹ 20, find the total amount spent on 60 bananas, 36 apples and 42 mangoes.
      

Observe the figures given below carefully and answer the questions:

Figure A
Figure B
Figure C

1. Name the figure(s) where in two figures are similar.
2. Name the figure(s) where in the figures are congruent.
3. 1. Prove that congruent triangles are also similar but not the converse.
      OR
   2. What more is least needed for two similar triangles to be congruent?