Board Paper of Class 10 2019 Maths All India(Set 2) - 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
Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. VIEW SOLUTION

Question 2

In Figure 1, PS = 3 cm, QS = 4 cm, ∠PRQ = θ, ∠PSQ = 90°, PQ ⊥ RQ and RQ = 9 cm. Evaluate tan θ.

OR

If tan $α = \frac{5}{12}$ , find the value of sec α. VIEW SOLUTION

Question 3
Write the discriminant of the quadratic equation (x + 5)² = 2 (5x − 3). VIEW SOLUTION

Question 4
Find after how many places of decimal the decimal form of the number $\frac{27}{2^{3} \cdot 5^{4} \cdot 3^{2}}$ will terminate.

OR

Express 429 as a product of its prime factors. VIEW SOLUTION

Question 5
Find the sum of first 10 multiples of 6. VIEW SOLUTION

Question 6
Find the positive value of m for which the distance between the points A(5, −3) and B(13, m) is 10 units. VIEW SOLUTION

Question 7
A die is thrown once. Find the probability of getting (i) a composite number, (ii) a prime number. VIEW SOLUTION

Question 8
Cards numbered 7 to 40 were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of 7 ? VIEW SOLUTION

Question 9
Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

OR

Points P and Q trisect the line segment joining the points A(−2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q. VIEW SOLUTION

Question 10
Solve the following pair of linear equations:

3x − 5y = 4
2y + 7 = 9x VIEW SOLUTION

Question 11
If HCF of 65 and 117 is expressible in the form 65n − 117, then find the value of n.

OR

On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ? VIEW SOLUTION

Question 12
In the quadratic equation kx² − 6x − 1 = 0, determine the values of k for which the equation does not have any real root. VIEW SOLUTION

Question 13
A, B and C are interior angles of a triangle ABC. Show that
(i) $\sin (\frac{B + C}{2}) = \cos \frac{A}{2}$
(ii) If ∠A = 90°, then find the value of $\tan (\frac{B + C}{2})$ .
OR
If tan (A + B) = 1 and $\tan (A - B) = \frac{1}{\sqrt{3}}$ , 0° < A + B < 90°, A > B, then find the values of A and B. VIEW SOLUTION

Question 14
In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.

OR
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. VIEW SOLUTION

Question 15
A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

Number of days: 0-6 6-12 12-18 18-24 24-30 30-36 36-42

Number of students: 10 11 7 4 4 3 1

VIEW SOLUTION

Question 16
A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle 120°. Find the total area cleaned at each sweep of the blades. $(Take π = \frac{22}{7})$ VIEW SOLUTION

Question 17
The perpendicular from A on side BC of a Δ ABC meets BC at D such that DB = 3CD. Prove that 2AB² = 2AC²+ BC².
OR
AD and PM are medians of triangles ABC and PQR respectively where Δ ABC ∼ Δ PQR. Prove that $\frac{AB}{PQ} = \frac{AD}{PM}$ . VIEW SOLUTION

Question 18
Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x⁵ − 4x³ + x² + 3x + 1, g(x) = x³ − 3x + 1 VIEW SOLUTION

Question 19
Prove that $\sqrt{3}$ is an irrational number.
OR
Find the largest number which on dividing 1251, 9377 and 15628 leaves remainders 1, 2 and 3 respectively. VIEW SOLUTION

Question 20
Find the area of the triangle ABC with the coordinates of A as (1, −4) and the coordinates of the mid-points of sides AB and AC respectively are (2, −1) and (0, −1). VIEW SOLUTION

Question 21
Two numbers are in the ratio of 5 : 6. If 7 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers. VIEW SOLUTION

Question 22
Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe. VIEW SOLUTION

Question 23

In Figure 3, a decorative block is shown which is made of two solids, a cube and a hemisphere. The base of the block is a cube with edge 6 cm and the hemisphere fixed on the top has a diameter of 4⋅2 cm. Find

(a) the total surface area of the block.
(b) the volume of the block formed. (Take $π = \frac{22}{7}$ )
OR
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 circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (Use π = 3.14) VIEW SOLUTION

Question 24
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio

OR

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

Question 25
Change the following distribution to a 'more than type' distribution. Hence draw the 'more than type' ogive for this distribution.

Class interval : 20−30 30−40 40−50 50−60 60−70 70−80 80−90

Frequency : 10 8 12 24 6 25 15

VIEW SOLUTION

Question 26
The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30° than when it was 60°. Find the height of the tower. (Given $\sqrt{3} = 1 \cdot 732$ ) VIEW SOLUTION

Question 27
If m times the m^th term of an Arithmetic Progression is equal to n times its n^th term and m ≠ n, show that the (m + n)^th term of the A.P. is zero.

OR

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers. VIEW SOLUTION

Question 28
A shopkeeper buys a number of books for ₹ 80. If he had bought 4 more books for the same amount, each book would have cost ₹ 1 less. How many books did he buy? VIEW SOLUTION

Question 29
Construct a pair of tangents to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre. VIEW SOLUTION

Question 30
Prove the following:
$\frac{1}{1 + \sin^{2} θ} + \frac{1}{1 + \cos^{2} θ} + \frac{1}{1 + \sec^{2} θ} + \frac{1}{1 + {cosec}^{2} θ} = 2$ VIEW SOLUTION

Number of days:	0-6	6-12	12-18	18-24	24-30	30-36	36-42
Number of students:	10	11	7	4	4	3	1

Class interval :	20−30	30−40	40−50	50−60	60−70	70−80	80−90
Frequency :	10	8	12	24	6	25	15