Board Paper of Class 10 2017 Maths (SET 1) - Solutions

General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.

Question 1
The ratio of the height of a tower and the length of its shadow on the ground is $\sqrt{3} : 1$ . What is the angle of elevation of the sun? VIEW SOLUTION

Question 2
Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere? VIEW SOLUTION

Question 3
A number is chosen at random from the number –3, –2, –1, 0, 1, 2, 3. What will be the probability that square of this number is less then or equal to 1? VIEW SOLUTION

Question 4
If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k? VIEW SOLUTION

Question 5
Find the roots of the quadratic equation $\sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0$ . VIEW SOLUTION

Question 6
Find how many integers between 200 and 500 are divisible by 8. VIEW SOLUTION

Question 7
Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other. VIEW SOLUTION

Question 8
Find the value of k for which the equation x² + k(2x + k − 1) + 2 = 0 has real and equal roots. VIEW SOLUTION

Question 9
Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5. VIEW SOLUTION

Question 10
In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.
VIEW SOLUTION

Question 11
If the m^th term of an A. P. is $\frac{1}{n}$ and n^thterm is $\frac{1}{m}$ then show that its (mn)^th term is 1. VIEW SOLUTION

Question 12
Find the sum of n terms of the series $(4 - \frac{1}{n}) + (4 - \frac{2}{n}) + (4 - \frac{3}{n}) + . . . . . . . . . .$ VIEW SOLUTION

Question 13
If the equation (1 + m²) x² + 2mcx + c² – a² = 0 has equal roots then show that c² = a² (1 + m²). VIEW SOLUTION

Question 14
The $\frac{3}{4}$ th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel. VIEW SOLUTION

Question 15
In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.

VIEW SOLUTION

Question 16
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 17
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles. VIEW SOLUTION

Question 18
The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is $(\frac{7}{2}, y),$ find the value of y. VIEW SOLUTION

Question 19
Two different dice are thrown together. Find the probability that the numbers obtained

(i) have a sum less than 7

(ii) have a product less than 16

(iii) is a doublet of odd numbers. VIEW SOLUTION

Question 20
A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/h. VIEW SOLUTION

Question 21
Construct an isosceles triangle with base 8 cm and altitude 4 cm. Construct another triangle whose sides are $\frac{2}{3}$ times the corresponding sides of the isosceles triangle. VIEW SOLUTION

Question 22
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION

Question 23
The ratio of the sums of first m and first n terms of an A. P. is m² : n². Show that the ratio of its m^th and n^th terms is (2m−1):(2n−1). VIEW SOLUTION

Question 24
Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. VIEW SOLUTION

Question 25
If $a \neq b \neq 0$ , prove that the points (a, a²), (b, b²) (0, 0) will not be collinear. VIEW SOLUTION

Question 26
The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts. VIEW SOLUTION

Question 27
Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25. VIEW SOLUTION

Question 28
A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle. VIEW SOLUTION

Question 29
The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of water. VIEW SOLUTION

Question 30
In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.

VIEW SOLUTION

Question 31
In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water. VIEW SOLUTION