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Board Paper of Class 10 2020 Maths (Standard) Delhi(Set 2) - Solutions

General Instructions :
(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory:
(ii) Section A : Q. No. 1 to 20 comprises of 20 questions of one mark each.
(iii) Section B : Q. No. 21 to 26 comprises of 6 questions of two marks each.
(iv) Section C: Q. No. 27 to 34 comprises of 8 questions of three marks each.
(v) Section D : Q. No. 35 to 40 comprises of 6 questions of four marks each.
(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 2 questions of one mark each, 2 questions of two marks each, 3 questions of three marks each and 3 questions of four marks each. You have to attempt only one of the choices in such questions.
(vii) In addition to this, separate instructions are given with each section and question, wherever necessary.
(viii) Use of calculators is not permitted.

• Question 1
The HCF and the LCM of 12, 21, 15 respectively are
(a) 3, 140
(b) 12, 420
(c) 3, 420
(d) 420, 3 VIEW SOLUTION

• Question 2
The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an AP, is
(a) 6
(b) $-$6
(c) 18
(d) $-$18 VIEW SOLUTION

• Question 3
The value of k for which the system of equations x + y – 4 = 0 and 2x + ky = 3, has no solution, is
(a) $-2$
(b) $\ne$2
(c) 3
(d) 2 VIEW SOLUTION

• Question 4
The first term of an AP is p and the common difference is q, then its 10th term is
(a) q + 9p
(b) p – 9p
(c) p + 9q
(d) 2p + 9q VIEW SOLUTION

• Question 5
The quadratic polynomial, the sum of whose zeroes is –5 and their product is 6, is
(a) x2 + 5x + 6
(b) x2 – 5x + 6
(c) x2 – 5x – 6
(d) –x2 + 5x + 6 VIEW SOLUTION

• Question 6
The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ), is
(a) ${a}^{2}+{b}^{2}$
(b) ${a}^{2}-{b}^{2}$
(c) $\sqrt{{a}^{2}+{b}^{2}}$
(d) $\sqrt{{a}^{2}-{b}^{2}}$ VIEW SOLUTION

• Question 7
The total number of factors of a prime number is
(a) 1
(b) 0
(c) 2
(d) 3 VIEW SOLUTION

• Question 8
If the point P(k, 0) divides the line segment joining the points A(2, –2) and B(–7, 4) in the ratio 1 : 2, then the value of k is
(a) 1
(b) 2
(c) –2
(d) –1 VIEW SOLUTION

• Question 9
The value of p, for which the points A(3, 1), B(5, p) and C(7, –5) are collinear, is
(a) –2
(b) 2
(c) –1
(d) 1 VIEW SOLUTION

• Question 10
If one of the zeroes of the quadratic polynomial x2 + 3x + k is 2, then the value of k is
(a) 10
(b) –10
(c) –7
(d) –2 VIEW SOLUTION

• Question 11
Fill in the blanks.
ABC is an equilateral triangle of side 2a, then length of one of its altitude is ____________. VIEW SOLUTION

• Question 12
Fill in the blank.
In the given figure ∆ABC is circumscribing a circle, the length of BC is _____ cm.
VIEW SOLUTION

• Question 13
Fill in the blank.
The value of = ____________.

OR

Fill in the blank.
The value of (1 + tan2 θ) (1 – sin θ) (1 + sin θ) = _____________. VIEW SOLUTION

• Question 14

• Question 15
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is _________. VIEW SOLUTION

• Question 16
A die is thrown once. What is the probability of getting a number less than 3?

OR

If the probability of winning a game is 0.07, what is the probability of losing it? VIEW SOLUTION

• Question 17
If the mean of the first n natural number is 15, then find n. VIEW SOLUTION

• Question 18
Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. What is the ratio of their volumes? VIEW SOLUTION

• Question 19
The ratio of the length of a vertical rod and the length of its shadow is $1:\sqrt{3}.$ Find the angle of elevation of the sun at that moment? VIEW SOLUTION

• Question 20
A die is thrown once. What is the probability of getting an even prime number? VIEW SOLUTION

• Question 21
In the given Figure, DE || AC and DC || AP. Prove that $\frac{\mathrm{BE}}{\mathrm{EC}}=\frac{\mathrm{BC}}{\mathrm{CP}}$

OR

In the given Figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ.
VIEW SOLUTION

• Question 22
The rod AC of a TV disc antenna is fixed at right angles to the wall AB and a rod CD is supporting the disc as shown in the given figure. If AC = 1.5 m long and CD = 3 m, find (i) tanθ (ii) secθ + cosecθ

VIEW SOLUTION

• Question 23
If a number x is chosen at random from the numbers –3, –2, –1, 0, 1, 2, 3. What is probability that x2 ≤ 4? VIEW SOLUTION

• Question 24
Find the mean of the following distribution:
 Class: 3 – 5 5 – 7 7 – 9 9 – 11 11 – 13 Frequency: 5 10 10 7 8

OR

Find the mode of the following data :
 Class: 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120 120 – 140 Frequency: 6 8 10 12 6 5 3
VIEW SOLUTION

• Question 25
Find the sum of first 20 terms of the following AP:
1, 4, 7, 10, _________ VIEW SOLUTION

• Question 26
The perimeter of a sector a circle of radius 5.2 cm is 16.4 cm. Find the area of the sector. VIEW SOLUTION

• Question 27
A cone of base radius 4 cm is divided into two parts by drawing a plane through the mid-points of its height and parallel to its base. Compare the volume of the two parts. VIEW SOLUTION

• Question 28
In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle. VIEW SOLUTION

• Question 29
Find the area of triangle PQR formed by the points P(–5, 7), Q(–4, –5) and R (4, 5).

OR

If the point C(–1, 2) divides internally the line segment joining A(2, 5) and B(x, y) in the ratio 3 : 4, find the coordinates of B. VIEW SOLUTION

• Question 30
Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial f(x) = ax2 + bx + c, a ≠ 0, c ≠ 0.

OR

Divide the polynomial f(x) = 3x2x3 – 3x + 5 by the polynomial g(x) = x – 1 – x2 and verify the division algorithm. VIEW SOLUTION

• Question 31
Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are given by 2y – x = 8, 5yx = 14 and y – 2x = 1.

OR

If 4 is a zero of the cubic polynomial x3 – 3x2 – 10x + 24, find its other two zeroes. VIEW SOLUTION

• Question 32
A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the original speed of the train. VIEW SOLUTION

• Question 33
Prove that the parallelogram circumscribing a circle is a rhombus. VIEW SOLUTION

• Question 34
Prove that: 2(sin6θ + cos6θ) – 3 (sin4θ + cos4θ) + 1 = 0. VIEW SOLUTION

• Question 35
The following table gives production yield per hectare (in quintals) of wheat of 100 farms of a village :
 Production yield/hect. 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 No. of farms 4 6 16 20 30 24

Change the distribution to 'a more than' type distribution and draw its ogive.

OR

The median of the following data is 525. Find the values of x and y, if total frequency is 100:
 Class : Frequency: 0 – 100 2 100 – 200 5 200 – 300 x 300 – 400 12 400 – 500 17 500 – 600 20 600 – 700 y 700 – 800 9 800 – 900 7 900 – 1000 4
VIEW SOLUTION

• Question 36
A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height 6 m. At a point on the plane, the angle of elevation of the bottom and top of the flag-staff are 30° and 45° respectively. Find the height of the tower. VIEW SOLUTION

• Question 37
Show that the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.

OR

Prove that one of every three consecutive positive integers is divisible by 3. VIEW SOLUTION

• Question 38
The sum of four consecutive numbers in AP is 32 and the ratio of the product of  the first and last terms to the product of two middle terms is 7 : 15. Find the numbers.

OR

Solve : 1 + 4 + 7 + 10 + ... + x = 287 VIEW SOLUTION

• Question 39
A bucket is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper circular ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket, at the rate of  40 per litre. (Use π = 3.14) VIEW SOLUTION

• Question 40
Construct a triangle with sides 4 cm, 5 cm and 6 cm. Then construct another triangle whose sides are $\frac{2}{3}$ times the corresponding sides of the first triangle. VIEW SOLUTION
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