# Board Paper of Class 10 2014 Maths - Solutions

General Instructions:Answers to this paper must be written on the paper provided separately.You will not be allowed to write during the first 15 minutes.This time is to be spent in reading the question paper.The time given at the head of this paper is the time allowed for writing the answers.

- Question 1
**Solve any five sub-questions:****[5]**(i) In the following figure RP: PK= 3:2, then find the value of A(ΔTRP):A(ΔTPK).

(ii) If two circles with radii 8 cm and 3 cm, respectively, touch internally, then find the distance between their centers. (iii) If the angle *θ*= – 60°, find the value of sin*θ*.(iv) Find the slope of the line passing through the points A(2, 3) and B(4,7). (v) The radius of a circle is 7 cm. find the circumference of the circle. (vi) If the sides of a triangle are 6 cm, 8 cm and 10 cm, respectively, then determine whether the triangle is a right angle triangle or not.

- Question 2
**Solve any four sub-questions:****[8]**(i) In the figure given below, Ray PT is bisector of ∠QPR. If PQ = 5.6 cm, QT = 4 cm and TR = 5 cm, find the value of *x*.

(ii) In the following figure, Q is the center of the circle. PM and PN are tangents to the circle. If ∠MPN = 40°, find ∠MQN.

(iii) Write the equation 2 *x*– 3*y*– 4 = 0 in the slope intercept form. Hence, write the slope and y-intercept of the line.(iv) If $\mathrm{cos}\theta =\frac{1}{\sqrt{2}}$, where *θ*is an acute angle, then find the value of sin*θ.*(v) If (4, –3) is a point on the line AB and slope of the line is (–2), write the equation of the line AB. (vi) Draw a tangent at any point ‘P’ on the circle of radius 3.5 cm and centre O.

- Question 3
**Solve any three sub-questions:****[9]**(i) In a triangle ABC, line *l*|| Side BC and line*l*intersects side AB and AC in points P and Q, respectively. Prove that: $\frac{\mathrm{AP}}{\mathrm{BP}}=\frac{\mathrm{AQ}}{\mathrm{QC}}$ .

(ii) In figure, ΔABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and side AC are congruent. A circle touches the three sides as shown in the figure below. Find the length of the tangent segment from A to the circle.

(iii) Draw tangents to the circle with center ‘C’ and radius 3.6 cm, from a point B at a distance of 7.2 cm from the center of the circle. (iv) Prove that: ${\mathrm{sec}}^{2}\theta +{\mathrm{cosec}}^{2}\theta ={\mathrm{sec}}^{2}\theta \times {\mathrm{cosec}}^{2}\theta $ (v) Write the equation of each of the following lines: 1. The x-axis and the y-axis.

2. The line passing through the origin and the point (–3, 5).

3. The line passing through the point (–3, 4) and parallel to X-axis.

- Question 4
**Solve any two sub-questions :****[8]**(i) From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be 60°. If the height of the lighthouse is 90 meters, then find how far is that ship from the lighthouse? $\left(\sqrt{3}=1.73\right)$ (ii) Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”. (iii) The sum of length, breadth and height of a cuboid is 38 cm and the length of its diagonal is 22 cm. Find the total surface area of the cuboid.

- Question 5
**Solve any two sub-questions :****[10]**(i) In triangle ABC, ∠C = 90°. Let BC = *a*, CA =*b*, AB =*c*and let ‘*p*’ be the length of the perpendicular from ‘C’ on AB, prove that:1.*cp*=*ab*

2. $\frac{1}{{p}^{2}}=\frac{1}{{a}^{2}}+\frac{1}{{b}^{2}}$(ii) Construct the circumcircle and incircle of an equilateral triangle ABC with side 6 cm and centre O. Find the ratio of radii of circumcircle and incircle. (iii) There are three stair-steps as shown in the figure below. Each stair step has width 25 cm, height 12 cm and length 50 cm. How many bricks have been used in it, if each brick is 12.5 cm × 6.25 cm × 4 cm?