BOARD PAPER - MARCH 2014

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1 BOARD PAPER - MARCH 2014 Time : 2 Hours Marks : 40 Notes : (i) Solve all questions. Draw diagrams wherever necessary. Use of calculator is not allowed. Figures to the right indicate full marks. Marks of constructions should be distinct. They should not be rubbed off. (v) Diagram is essential for the proof of the theorem. Q.1.Solve any five sub-questions : 5 (i) In the following figure RP : PK = 3 : 2, then find the value of A( TRP) : A( TPK) If two circles with radii 8 cm and 3 cm respectively touch internally, then find the distance between their centres. If the angle = 60º, find the value of sin 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, determine whether the triangle is right angled triangle or not. Q.2. Solve any four sub-questions : 8 (i) In the figure given below, ray RS is a bisector of PRQ. If PS = 6 cm, SQ = 8 cm and PR = 15 cm, find the value of x. In the following figure, C is the centre of the circle. PA and PB are tangents to the circle. If APB = 50º, find ACB. Write the equation 2x 3y + 5 = 0 in the slope-intercept form. Hence write the slope and y-intercept of the line.

Unique Solutions If sin θ = 1, where is an acute angle, then find the value of cos. 2 (v) If ( 4, 3) is a point on the line PQ and slope of the line PQ is ( 2), write the equation of the line PQ.

4 cm and centre P. Q.3. Solve any three sub-questions : 9 (i) In a triangle PQR, line l side QR and line l intersects side PQ and side PR in points M and N respectively.

A circle touches the three sides as shown in the figure below. Find the length of the tangent segments from point P to the circle. Draw tangents to the circle with centre 'M' and radius 3 cm.

2 Unique Solutions If sin θ = 1, where is an acute angle, then find the value of cos. 2 (v) If ( 4, 3) is a point on the line PQ and slope of the line PQ is ( 2), write the equation of the line PQ. (vi) Draw a tangent at any point 'B' on the circle of radius 3.4 cm and centre P. Q.3. Solve any three sub-questions : 9 (i) In a triangle PQR, line l side QR and line l intersects side PQ and side PR in points M and N respectively. Prove that : PM PN = MQ NR In the following figure PQR is an isosceles triangle with perimeter 52 cm. The base QR is of length 16 cm. Side PQ and side PR are congruent. A circle touches the three sides as shown in the figure below. Find the length of the tangent segments from point P to the circle. Draw tangents to the circle with centre 'M' and radius 3 cm. From a point P at a distance of 6.8 cm from the centre of the circle. Prove that : tan + cot = cosec sec. (v) Write the equation of each of the following lines : 1) The line passing through the origin and the point ( 2, 3). 2) The line passing through the point ( 3, 5) and parallel to Y axis. 3) The X-axis and the Y-axis. Q.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 60º. If the height of the lighthouse is 90 metres, then find how far is that ship from the lighthouse? ( ) Prove that "the opposite angles of the cyclic quadrilateral are supplementary". 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. Q.5. Solve any two sub-questions : 10 (i) In triangle ABC, C = 90º. 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 p a b 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. There are three stair-steps as shown in the figure below. Each stairstep 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?

3 BOARD PAPER - MARCH 2013 Time : 2½ Hours Marks : 60 Q.1. Solve any six sub-questions : 6 (i) In the following figure, line l side BC, AP = 3, PB = 6, AY = 5, YC = x, then find x. In the following figure, m(arc APC) = 60º, find ABC. In the following figure, point A is the centre of the circle, seg AM line MN, AN = 10 cm, line MN is tangent at M Determine radius of the circle if MN = 5 cm. If sin = 1, then find the value of acute angle. 2 (v) Find the slope and y-intercept of the line y = 3x 5 (vi) Find the area of sector whose are-length and radius are 10 cm and 5 cm respectively. (vii) Find the volume of cube with side 6 cm. Q.2. Solve any five sub-questions : 10 (i) A ladder, 10 m long, reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. In the following circle, m(arc XAZ) = m(arc YBW), prove that XY ZW. If tan = 2, where is an acute angle, find sec using the identity. Show that : 1 + cos 1 cos = 1 + cos sin (v) Write the equation of line 2y 3x + 5 = 0 in slope intercept form and then find the slope and y- intercept of the line. (vi) The radius of the base of a right circular cylinder is 3 cm and height is 7 cm. Find its volume. 22 7

4 Unique Solutions Q.3. Solve any four of the following sub-questions : 12 (i) In the following figure, PQRV is a trapezium in which seg PQ seg VR, SR = 4, PQ = 6, then find VR. Draw incircle of an equilateral XYZ with side 6.3 cm. Using slope concept check whether the points A(7, 8), B( 5, 2) and C(3, 6) are collinear. The diameter of a circle is 10 cm. Find the length of the arc, if the corresponding central angle is 144º. Q.4. Solve any three of the following sub-questions : 12 (i) Prove that "If a line divides any two sides of a triangle in the same ratio, then that line is parallel to the third side." Prove that "An exterior angle of cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle." Construct GPR that GR = 5.3 cm, GPR = 70º, PM GR and PM = 3.5 cm. Two electric poles of equal heights are opposite to each other on either side of a road. Width of the road is 100 m. The angles of elevation observed from a point between two poles on the road to the tops of the poles are 30º and 60º. Find the height of the pole and the distance of that point from each pole. Q.5. Solve any four of the following sub-questions : 20 (i) Prove that "In a right-angled triangle, the square of hypotenuse is equal to the sum of the squares of the remaining two sides." In the following figure, ABC in point P. Prove that : CQ = CA. A funnel of tin sheet consists of a cylindrical portion. 10 cm long, attached to a frustum of a cone. If diameter of the top and bottom of the frustum of cone is 13 cm, find the surface area of the tin required to make the funnel. (Express your answer in terms of ) If the points A(1, 2), B(4, 6), C(3, 5) are the vertices of a ABC, find the equation of the line passing through the midpoints of AB and BC. Draw a triangle ABC, right-angled at B, such that AB = 3 cm, BC = 4 cm. Now construct a triangle PBQ similar to ABC, each of whose sides is 7 times the corresponding side of ABC. 4

5 BOARD PAPER - MARCH 2012 Time : 2½ Hours Marks : 60 Q.1. Solve any six sub-questions : 6 (i) If the angle = 60º, find the value of sin Find the side fo a square whose diagonal is 16 2 cm. In the adjoining figure, O is the centre of the circle. PA and PB are the tangents to the circle at points A and B respectively. If l(pa) = 7 cm, then find l(pb). State the slope and y-intercept fo the line y = 3x 5 (v) Find the total surface area of a cube with side 1 metre. (vi) Two circles with radii 4 cm and 3 cm touch each other externally. Find the distance between their centres. (vii) If F = 6, V= 8. Using Euler's formula find the value of E. Q.2. Solve any five sub-questions : 10 (i) In the adjoining figure, in PQR, seg RS is the angle bisector of PRQ. If PS = 8, SQ = 12, PR = 20, Find QR. In the given figure, Q is the centre of the circle. Line PM and line PN are tangents to the circle. If MPN = 80º then find MQN. If sin = 7 25, where is acute angle, find the value of cos If a sector of a circle with radius 10 cm has central angle 72º, find the area of the sector. x y (v) Convert the following equation into y = mx + c form and find the slope: + =1 3 4 (vi) Eliminate, if : x = a sec, y = b tan. Q.3. Solve any four of the following sub-questions : 12 (i) The curved surface area of a cone with base radius 16 cm is 320 sq. cm. Find the height of the cone. Construct the circumcircle of an equilateral PQR with side 5.9 cm. Find the value of K if A(3, 9), B(1,3), C(5, K) are collinear points.

6 Unique Solutions (v) A boy is at a distance of 50 metres from a tree and makes an angle of elevation of 60º with the top of the tree. What is the height of the tree? In the adjacent figure, in ABC, AP is the median. If AP = 9, AB 2 + AC 2 = 260, then find BC. Q.4. Solve any three of the following sub-questions : 12 (i) Seg AN and seg CM are the medians fo ABC in which B = 90º Prove that 4(AN 2 + CM 2 ) = 5AC 2 Prove that the opposite angles of a cyclic quadrilateral are supplementary. Construct LMN such that LM = 6.6 cm, LNM = 65º and ND is median and ND = 5 cm. From the top of a lighthouse 120m high two ships on the same side of the lighthouse are observed. The angles fo depression of the ships as seen from the lightouse are found to be 30º and 60º. Find the distance between the two ships. (Assume that the two ships and bottom of the lighthouse are in a line.) Q.5. Solve any four of the following sub-questions : 20 (i) A (5,4), B( 3, 2) and C(1, 8) are the vertices of a triangle ABC. Find the equation of median AD and equation of line parallel to AC passing through point B. AMT AHE. In AMT, AM = 6.3 cm, MAT = 120º, AT = 4.9 cm and MA HA = 7 5 Construct AHE. Write l (AH) and l (AE). Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.] In the adjacent figure, the inscribed circle of ABC with centre P touches the sides AB, BC and AC at points L,M,N respectively. Show that A ( ABC) = 1 (perimeter of ABC) (radius of inscribed circle.) 2 (v) A cuboidal shape vessel with dimensions 44 cm 35 cm 20 cm is filled with water upto the height of 17 cm. A spherical solid metal ball is placed into the vessel; due to this 231 cm 3 water overflows. Find the radius of the ball. [ = 22 7 ]

BOARD ANSWER PAPER : MARCH 2014

BOARD ANSWER PAPER : MARCH 2014 OARD ANSWER PAPER : MARH 04 GEOMETRY. Solve any five sub-questions: i. RP : PK 3 : ----[Given] A( TRP) RP ---- [Ratio of the areas of two triangles having equal heights A( TPK) PK is equal to the ratio