BOARD ANSWER PAPER : MARCH 2014
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- Brice McLaughlin
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1 ORD NSWER ER : MRH 04 GEOMETRY. Solve any five sub-questions: i. R : K :...[Given] ( TR) R...[Ratio of the areas of two triangles having equal heights ( TK) K ( TR) ( TK) is equal to the ratio of their corresponding bases.] Geometry ii. If two circles touch internally, then the distance between their centres is difference of their radii. the distance between their centres 8 5 cm iii. θ 60...[Given] sin ( 60 ) sin 60...[ sin( θ) sin θ] sin( 60 ) iv. Let (, ) (x, y ), (4, 7) (x, y ) y y Slope of line x x v. Radius (r) 7 cm...[given] ircumference of a circle πr cm vi. The longest side is 0 cm. (0) 00.(i) Now, sum of the squares of the other two sides will be (6) + (8) (ii) (0) (6) + (8).[From (i) and (ii)] Yes, the given sides form a right angled triangle..[y converse of ythagoras theorem]. Solve any four sub-questions: i. In, ray D is the bisector of. D D.[y angle bisector property] 4.8 x x x x 6 cm
2 oard nswer aper: March 04 ii. QR 60...[Given] QM 90 RM 90 In QMR, QR + QM + QMR + RM 60...[Tangent is perpendicular to radius]...[ngle sum property of a quadrilateral] QMR QMR 60 QMR QMR 0 iii. Equation of the line in slope-intercept form is y mx + c Given equation of the line is x y 4 0 y 4 x y 4 x y x The equation of the line in slope-intercept form is y x. slope m and y-intercept c iv. sin θ...[given] We know that, sin θ + cos θ + cos θ 4 + cos θ cos θ 4 cos θ 4 cos θ...[taking square root on both sides] v. Slope of the line Q (m) and the line passes through point (5, ) (x, y ) Equation of line in slope point form is y y m(x x ) y ( ) (x 5) y + x 0 x y + 0 x y The equation of line Q is x y.
3 Geometry vi..5 cm For drawing the circle of radius.5 cm Extending the line passing through Drawing perpendicular at point []. Solve any three sub-questions: i. Given: In RST, line l side ST. Line l intersects side RS and side RT in points and Q respectively, such that R S and R Q T. R To prove : R S RQ QT onstruction: Draw seg SQ and seg T. Q l roof: S T In RQ and SQ, where R S, ( RQ) R... (i) [Ratio of the areas of two triangles having equal heights ( SQ) S is equal to the ratio of their corresponding bases.] In RQ and TQ, where R Q T, ( RQ) ( TQ) RQ QT... (ii) [Ratio of the areas of two triangles having equal heights is equal to the ratio of their corresponding bases.] ( SQ) ( TQ)...(iii) [reas of two triangles having equal bases and equal heights are equal.] ( RQ) ( SQ) ( RQ) ( TQ)... (iv) [From (i), (ii) and (iii)] R S RQ QT... [From (i), (ii) and (iv)]
4 oard nswer aper: March 04 ii. x x R y z y Q z Let R x...(i) Q y...(ii) Q R z...(iii) [The length of two tangent segments to a circle from an external point are equal.] erimeter of 40...[Given] ( + ) + (Q + Q) + (R + R) 40 x + y + y + z + z + x 40 x + y + z 40 x + y + z 0 x [ y + z] x [ 0] x 0 R 0 cm The length of the tangent segment from to the circle is 0 cm. iii. R.4 cm M 7. cm Q For drawing the circle of radius.4 cm lotting point Q such that Q 7. cm Drawing perpendicular bisector of seg Q [] Drawing circle with centre M Drawing tangent QR 4
5 Geometry iv. L.H.S. sec + cosec cos + sin sin + cos cos sin... [ sin + cos ] [] cos sin cos sin sec cosec R.H.S. sec + cosec sec cosec v.. Let (, 4) (x, y ), Origin O (0, 0) (x, y ) Equation of a line in two point form is x x x x y y y y x ( ) y x + y 4 4 4(x + ) (y 4) 4x + y + 4x + y + 0 4x + y 0 equation of the line is 4x + y 0.. Let (, 5) Equation of line parallel to X-axis and passing through (, 5) is y y-coordinate of i.e., y 5 equation of the line is y 5.. Equation of X-axis is y 0. Equation of Y-axis is x Solve any two sub-questions: i. Let represent the height of lighthouse. 90 m oint represent the position of the ship. Draw ray seg. is the angle of depression, m 60 [] lso,...[lternate angles] 60 5
6 oard nswer aper: March 04 In right angled, tan m the distance of the ship from the lighthouse is 5.9 m. [] ii. Given: D is a cyclic quadrilateral. O D To prove: D + D 80, + D 80 roof: m D m(arc D)...(i) [Inscribed angle theorem] m D m(arc D)...(ii) [Inscribed angle theorem] dding equations (i) and (ii), we get m D + m D m(arc D) + m(arc D) m D + m D [m(arc D) + m(arc D)] D + D 60...[Measure of a circle is 60 ] D + D 80...(iii) In D, D + D + + D 60.[Sum of measures of angles of a quadrilateral] D 60.[From (iii)] + D D 80 Hence, the opposite angles of a cyclic quadrilateral are supplementary. 6
7 Geometry iii. Diagonal of a cuboid l + b + h ut, length of the diagonal cm l + b + h cm...(i) l + b + h 8 cm...(ii) [Given] (l + b + h) (l + b + h) (l + b + h) (l + b + h) (l + b + h ) + (lb + bh + hl) (8) () + (lb + bh + hl)...[from (i) and (ii)] (lb + bh + hl) (lb + bh + hl) (lb + bh + hl) 960 ut, total surface area of a cuboid (lb + bh + hl) the total surface area of the cuboid 960 cm. 5. Solve any two sub-questions: i. b D c p a In, 90...[Given] seg D hypotenuse...[given] D D...(i) [y theorem on similarity in right angled triangle] []. D...[From (i)] D D...[ c.s.s.t.] b D a p c b a p c b cp ab...(ii). +...[y ythagoras theorem] c b + a c b a + a b a b a b...[dividing throughout by a b ] c + ( ab) a b...(iii) c + ( cp) a b...[from (ii) and (iii)] c + cp a b p a + b 7
8 oard nswer aper: March 04 ii. O 6 cm M For drawing of side 6 cm [] Drawing the perpendicular bisectors of side and side [] Drawing incircle Drawing circumcircle Radius of circumcircle (O).4 cm Radius of incircle (OM).7 cm Radius of circumcircle O.4 Radius of incircle OM.7 The ratio of radii of cicumcircle and incircle is :. [] iii. 8 Given: For the stair-step (cuboid): length (l) 50 cm, breadth (b) 5 cm height of st step (h ) cm height of nd step (h ) + 4 cm height of rd step (h ) cm For the brick (cuboid): length (l ).5 cm, breadth (b ) 6.5 cm, height (h ) 4 cm volumeof thestairsteps No of bricks used volumeof brick st nd rd volumeof step + volumeof step + volumeof step volume of brick l b h+ l b h + l b h l b h l b(h+ h + h ) l b h 50 5( ) bricks The number of bricks used are 88.
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
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