Board Paper of Class 10 2019 Maths Delhi(Set 1) - Solutions

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.

(iii) Section A comprises 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.

(v) Use of calculators is not permitted.

Question 1
Find the coordinates of a point A, where AB is diameter of a circle whose centre is (2, –3) and B is the point (1, 4). VIEW SOLUTION

Question 2
For what value of k, the roots of the equation x² + 4x + k = 0 are real?
OR
Find the value of k for which the roots of the equation 3x² – 10x + k = 0 are reciprocal of each other. VIEW SOLUTION

Question 3
Find A if tan 2A = cot (A – 24°)
OR
Find the value of (sin²33° + sin² 57°) VIEW SOLUTION

Question 4
How many two digits numbers are divisible by 3? VIEW SOLUTION

Question 5
In Fig. 1, DE || BC, AD = 1 cm and BD = 2 cm. What is the ratio of the ar (Δ ABC) to the ar (Δ ADE)?

VIEW SOLUTION

Question 6
Find a rational number between $\sqrt{2} and \sqrt{3}$ . VIEW SOLUTION

Question 7
Find the HCF of 1260 and 7344 using Euclid's algorithm.
OR
Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer. VIEW SOLUTION

Question 8
Which term of the AP 3, 15, 27, 39, .... will be 120 more than its 21st term?
OR
If S_n, the sum of first n terms of an AP is given by S_n = 3n² – 4n, find the nth term. VIEW SOLUTION

Question 9
Find the ratio in which the segment joining the points (1, –3) and (4, 5) is divided by x-axis? Also find the coordinates of this point on x-axis. VIEW SOLUTION

Question 10
A game consists of tossing a coin 3 times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game. VIEW SOLUTION

Question 11
A die is thrown once. Find the probability of getting a number which (i) is a prime number (ii) lies between 2 and 6. VIEW SOLUTION

Question 12
Find c if the system of equations cx + 3y + (3 – c) = 0; 12x + cy – c = 0 has infinitely many solutions? VIEW SOLUTION

Question 13
Prove that $\sqrt{2}$ is an irrational number. VIEW SOLUTION

Question 14
Find the value of k such that the polynomial x² − (k + 6)x + 2(2k −1) has sum of its zeros equal to half of their product. VIEW SOLUTION

Question 15
A father's age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
Or
A fraction becomes $\frac{1}{3}$ when 2 is subtracted from the numerator and it becomes $\frac{1}{2}$ when 1 is subtracted from the denominator. Find the fraction. VIEW SOLUTION

Question 16
Find the point on y-axis which is equidistant from the points (5, −2) and (−3, 2).
OR
The line segment joining the points A(2, 1) and B(5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x − y + k = 0, find the value of k. VIEW SOLUTION

Question 17
Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ.
Or
Prove that (1 + cot A − cosec A) (1 + tan A + sec A) = 2 VIEW SOLUTION

Question 18
In Fig. 2, PQ is a chord of length 8 cm of a circle of radius 5 cm and centre O. The tangents at P and Q intersect at point T. Find the length of TP.

VIEW SOLUTION

Question 19
In Fig. 3, ∠ACB = 90° and CD ⊥ AB, prove that CD² = BD × AD.

OR
If P and Q are the points on side CA and CB respectively of Δ ABC, right angled at C, prove that (AQ² + BP²) = (AB² + PQ²) VIEW SOLUTION

Question 20
Find the area of the shaded region in Fig. 4, if ABCD is a rectangle with sides 8 cm and 6 cm and O is the centre of circle. (Take π = 3.14)
VIEW SOLUTION

Question 21
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/hour. How much area will it irrigate in 30 minutes; if 8 cm standing water is needed? VIEW SOLUTION

Question 22
Find the mode of the following frequency distribution.

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Frequency 8 10 10 16 12 6 7

VIEW SOLUTION

Question 23
Two water taps together can fill a tank in $1 \frac{7}{8}$ hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.
OR
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. VIEW SOLUTION

Question 24
If the sum of first four terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms. VIEW SOLUTION

Question 25
Prove that $\frac{\sin A - \cos A + 1}{\sin A + \cos A - 1} = \frac{1}{\sec A - \tan A}$ VIEW SOLUTION

Question 26
A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60° to 30°.
Find the speed of the boat in metres per minute. $[Use \sqrt{3} = 1.732]$
OR
Two poles of equal heights are standing opposite each other on either side of the road, which is 80 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. VIEW SOLUTION

Question 27
Construct a ΔABC in which CA = 6 cm, AB = 5 cm and ∠BAC = 45°. Then construct a triangle whose sides are $\frac{3}{5}$ of the corresponding sides of ΔABC. VIEW SOLUTION

Question 28
A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm³. The radii of the top and bottom of circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of the metal sheet used in making it. (Use π = 3.14) VIEW SOLUTION

Question 29
Prove that in a right angle triangle, the square of the hypotenuse is equal the sum of squares of the other two sides. VIEW SOLUTION

Question 30

If the median of the following frequency distribution is 32.5. Find the values of f₁ and f₂.

Class	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	Total
Frequency	f₁	5	9	12	f₂	3	2	40

The marks obtained by 100 students of a class in an examination are given below.

Mark	No. of Students
0 – 5	2
5 – 10	5
10 – 15	6
15 – 20	8
20 – 25	10
25 – 30	25
30 – 35	20
35 – 40	18
40 – 45	4
45 – 50	2

Draw 'a less than' type cumulative frequency curves (ogive). Hence find median. VIEW SOLUTION

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	8	10	10	16	12	6	7