04 00 Seat No. MT - MTHEMTIS (7) GEOMETRY - PRELIM II - (E) Time : Hours (Pages 3) Max. Marks : 40 Note : ll questions are compulsory. Use of calculator is not allowed. Q.. Solve NY FIVE of the following : 5 Line is a tangent and line D is a secant. If 6 units, 4 units, find D. D (iv) (v) If the radius is cm and length of corresponding arc is 3.4 cm, find the area of a sector. What is the directed angle, whose terminal arm lies along the coordinate axes, called? line has the equation y 3x. State its y-intercept. Find the area of a circle with radius 7 cm. (vi) If sec, find the value of? 3 Q.. Solve NY FOUR of the following : 8 Find the side of square whose diagonal is 6 cm. In PQR, seg PM is the median. If PM 9 and PQ + PR 90. Find QR.
/ MT (iv) In the adjoining figure, point P is centre of the circle and line is the tangent to the circle at T. The radius of the circle is 6 cm. Find P iftp 60º. Draw a circle of radius 3.6 cm, take a point M on it. Draw a tangent to the circle at M without using centre of the circle. P T (v) Prove : sec + cosec sec. cosec (vi) If the angle 60º, find the value of sin, cos, sec and tan. Q.3. Solve NY THREE of the following : 9 (iv) (v) D is a trapezium in which D and its diagonals intersect each other at the point O. Show that O O O DO. Two circles which are not congruent touch externally. The sum of their areas is 30cm and the distance between their centers is 4 cm. Find radii of circles. onstruct the incircle of RST in which RS 6 cm, ST 7 cm and RT 6.5 cm. Find the equation of the line which passes through (, 7) and whose y-intercept is 3. The curved surface area of the frustum of a cone is 80 sq. cm and the perimeters of its circular bases are 8 cm and 6 cm respectively. Find the slant height of the frustum of a cone. Q.4. Solve NY TWO of the following : 8 In a right angled triangle, 90º a circle is inscribed in the triangle with radius r. a, b, c are the lengths of the sides, and respectively. Prove that r a + b c. P O Q R
3 / MT Show that (, ), (0, 3), (, ) and (0, ) are the vertices of a parallelogram. Two poles of height 8 metres and 7 metres are erected on the ground. wire of length metres tied to the top of the poles. Find the angle made by the wire with the horizontal. Q.5. Solve NY TWO of the following : 0 Prove : If a line parallel to a side of a triangle intersects other sides in two distinct points, then the line divides those sides in proportion. Draw a triangle, right angled at such that, 3 cm and 4 cm. Now construct a triangle similar to, each of whose sides is 7 5 times the corresponding side of. cylinder of radius cm contains water upto depth of 0 cm. spherical iron ball is dropped into the cylinder and thus water level is raised by 6.75 cm. what is the radius of the ball? est Of Luck
04 00 Seat No. MT - MTHEMTIS (7) GEOMETRY - PRELIM II - (E) Time : Hours Prelim - II Model nswer Paper Max. Marks : 40.. ttempt NY FIVE of the following : Line D is a secant intersecting the circle at points and D and line is a tangent at D 6 4 D 36 4 D D 36 4 D D 9 units Radius (r) cm Length of arc (l) 3.4 cm rea of sector r l 3.4 3.4 cm The area of a sector is 3.4 cm. If the terminal arm of a directed angle lies along the co-ordinate axes, then it is called a quadrantal angle. (iv) Equation of the line is y 3x omparing the given equation with slope-intercept form y mx + c, c y intercept of the line is. (v) Radius of circle (r) 7 cm rea of the circle r
/ MT 7 7 7 54 cm The area of a circle is 54 cm. (vi) sec 3 ut, sec 30º sec sec 30º 3 [Given] 30º.. Solve NY FOUR of the following : D is a square. 6 cm To find : Side of a square D is a square[given] Let the sides of the square be x cm In, m 90º [ngle of a square] + [y Pythagoras theorem] 6 x + x 56 x x 56 x 56 x 6 [Taking square roots] The side of a square is 6 cm. x x 6 cm x D x In PQR, seg PM is the median [Given] PQ + PR PM + QM [y ppollonius theorem] 90 (9) + QM [Given] 90 (8) + QM 90 6 + QM P 90 6 QM 8 QM 9 8 QM Q R QM 64 M QM 8 units [Taking square roots]
3 / MT QM QR [ M is midpoint of side QR] 8 QR 8 QR QR 6 units In PT, m TP 60º [Given] m PT 90º [Radius is perpendicular to tangent] P m PT 30º [Remaining angle] PT is a 30º - 60º - 90º triangle y 30º - 60º - 90º triangle theorem T PT P [Side opposite to 30º] 6 P [Given] P 6 P cm (iv) L (Rough Figure) L M N N M mark for rough figure mark for drawing NM NLM mark for tangent at M.
4 / MT (v) L.H.S. sec + cosec sec, cos ec cos sin cos sin sin + cos cos. sin cos. sin [ sin + cos ] sec. cosec R.H.S. sec + cosec sec. cosec (vi) 60º sin ( ) sin sin ( 60) sin 60 sin ( 60) 3 sec ( ) sec sec ( 60) sec 60 sec ( 60) cos ( ) cos cos ( 60) cos 60 cos ( 60) tan ( ) tan tan ( 60) tan 60 tan ( 60) 3.3. Solve NY THREE of the following : D is a trapezium side side D [Given] O On transversal, D D [onverse of alternate angles test] O DO... [ - O - ] In O and OD, O DO [From ] O OD [Vertically opposite angles] O ~ OD [y test of similarity]
5 / MT O O O DO O O O DO [c.s.s.t.] [y lternendo] Let the radius of first circle be r and that of second circle be r ircles are touching externally r + r 4 cm r 4 r... ccording to given information (I ircle) + (II ircle) 30 cm r + r 30 (r + r ) 30 r + r 30 r + (4 r ) 30 [From ] r + 96 8r + r 30 r 8r + 96 30 0 r 8r + 66 0 (r 4r + 33) 0 r 4r + 33 0 r r 3r + 33 0 r (r ) 3 (r ) 0 (r 3) (r ) 0 r 3 0 or r 0 r 3 cm or r cm If r 3 If r then,r 4 3 then, r 4 r cm r 3 cm The radii of the circles is cm and 3 cm. (Rough Figure) R 6 cm 6.5 cm S 7 cm T
6 / MT R 6 cm 6.5 cm O S 7 cm T mark for rough figure mark for drawing DRST mark for drawing the angle bisectors mark for drawing the incircle (iv) Let (, 7) The y intercept of the line is 3 The line intersects the y-axis at point (0, 3) Let (0, 3) The line passes through point and The equation of the line y two point form x x y y x x y y x 0 y 7 7 3 x y 7 4 4 (x ) (y 7) 4x 8 y 4 4x y 8 + 4 0 4x y + 6 0 x y + 3 0 [Dividing throughout by ] The required equation of the line is x y + 3 0 (v) urved surface area of the frustum of a cone 80 cm Perimeters of circular bases are 8 cm and 6 cm r 8... r 6... dding and, we get
7 / MT r + r 8 + 6 (r + r ) 4 (r + r ) 4 (r + r )... urved surface area of the frustum of a cone (r + r ) l 80 (r + r ) l 80 l [From ] l 5 cm Slant height of the frustum of a cone is 5 cm..4. Solve NY TWO of the following : Let the centre of the inscribed circle be O Let P Q x... [The lengths of the two tangent P R y... segments to a circle drawn from R Q z... an external point are equal] a + b c + a + b c R + R + P + P (Q + Q) [ - R -, - P -, - Q - ] a + b c y + z + x + y (x + z) [From, and ] Q a + b c y + z + x + y x z a + b c y O P a + b c y a + b c P...(iv) [From ] In PRO R m OP m OR 90º [Radius is perpendicular to tangent] m PR 90º [Given] m POR 90º [Remaining angle] PRO is a rectangle [y definition] P OR...(v) [Opposite sides of a rectangle] a + b c OR [From (iv) and (v)] a + b c r Let, P (, ), Q (0, 3), R (, ), S (0, ) Slope of a line Slope of line PQ y x y x 3 0 ( ) 0 P (, ) S (0, ) Q (0, 3) R (, )
8 / MT Slope of line PQ Slope of line RS 0 Slope of line RS Slope of line PQ Slope of line RS line PQ line RS... Slope of line QR 3 0 Slope of line QR Slope of line PS 0 ( ) 0 Slope of line PS Slope of line QR Slope of line PS line QR line PS... In PQRS, side PQ side RS [From ] side QR side PS [From ] PQRS is a parallelogram [y definition] The points (, ), (0, 3), (, ) and (0, ) are the vertices of parallelogram. seg and D represents two poles. 8 m, D 7 m seg represent the length of the wire. m ED is a rectangle ( mark for figure) E D 7 m [Opposite sides of rectangle] E + E [ - E - ] 8 E + 7 8 m E 8 7 E 7 m E m D m
9 / MT In right angled E, E sin [y definition] sin sin... ut, sin 30º... sin sin 30º 30º The angle made by the wire with horizontal is 30º..5. Solve NY TWO of the following : Given : In, line DE side Line DE intersects sides and at points D and E respectively. To Prove : D D E E onstruction : Draw seg E and seg D. Proof : DE and DE have a common vertex E and their bases D and D lie on the same line. Their heights are equal D E ( mark for figure) ( DE) ( DE) D D... [Triangles having equal heights] DE and DE have a common vertex D and their bases E and E lie on the same line. Their heights are equal. ( DE) ( DE) E... [Triangles having equal heights] E line DE side [Given] DE and DE are between the same two parallel lines DE and. Their heights are equal. lso, they have same base DE.
0 / MT [ reas of two triangles having equal (DE) (DE)... basesand equal heights are equal ] ( DE) ( DE) ( DE) ( DE)...(iv) [From, and ] D D E E [From, and (iv)] P (Rough Figure) P 3 cm 3 cm 4 cm R 4 cm R 3 4 5 mark for mark for constructing 7 congruent parts mark for constructing 5 R 7 mark for constructing PR mark for required PR 6 7 Radius of the cylinder (r) cm spherical iron ball is dropped into the cylinder and the water level rises by 6.75 cm Volume of water displaced volume of the iron ball Height of the raised water level (h) 6.75 m Volume of water displaced r h
/ MT 6.75 cm 3 Volume of iron ball 6.75 cm 3 ut, Volume of iron ball 4 r 3 6.75 4 3 6.75 cm r3 6.75 3 0 cm r 4 r 3 3 6.75 3 r 3 3 3 3 4 6.75 r 3 3 3 3 7 r 3 3 3 3 3 3 3 r 3 3 r 9 3 [Taking cube roots] Radius of the iron ball is 9 cm.