Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD

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Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions JEE MEDICL-UG ORDS KVPY NTSE OLYMPIDS SSC - ORD - 018 Date: 1.0.018 MTHEMTICS - PPER- - SOLUTIONS Q.

1 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions JEE MEDICL-UG ORDS KVPY NTSE OLYMPIDS SSC - ORD Date: MTHEMTICS - PPER- - SOLUTIONS Q.1 ttempt any FIVE of the following sub-questions : [5] (i) DEF MNK. If DE = 5 and MN = 6, then find the value of DEF DE MNK MN 5 6 DEF MNK (Ratio of areas of similar triangles is equal to squares of corresponding sides) Topic:Similarity_; Sub-topic:rea of similar triangle L-1 SSC oard Test_Mathematics. (ii) If two circles with radii 8 cm and cm respectively touch externally, then find the distance between their centres. C1C r1 r 8 11cm If two circles touches externally then distance between their centres is equal to sum of the radii. Topic:Circle_; Sub-topic:Properties of circle L-1 SSC oard Test_Mathematics (iii) Find the length of the altitude of an equilateral triangle with side 6 cm. In DC sin 60 D C C D 6 D cm D lternate Method : D 60 OR ltitude of equilateral triangle side = 6 cm Topic:Trigonometry_; Sub-topic:pplication of trigonometry L- SSC oard Test_Mathematics C 1 1

2 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions (iv) If 45, then find tan. Given 45 tan tan 45 1 Topic:Trigonometry_; Sub-topic:ngle L-1 SSC oard Test_Mathematics (v) Slope of a line is and y intercept is 4. Write the equation of a line. Given Slope of a line (m) = y intercept (c) = 4 Equation of line having slope (m) and y-intercept (c) is y mx c y x 4 Topic:Coordinate geometry_; Sub-topic:Equation of line(slope) L-1 SSC oard Test_Mathematics (vi) Using Euler s formula, find V, if E = 0, F = 1. Euler s formula F V E 1 V 0 V 1 0 Topic:Mensuration_; Sub-topic:Euler s formula L-1 SSC oard Test_Mathematics Q. ttempt any FOUR of the following subquestions : [8] (i) The ratio of the areas of two triangles with common base is 6 : 5. Height of the larger triangle of 9 cm, then find the corresponding height of the smaller triangle. Let the height of the larger triangle be h 1 and that of the smaller tringle be h, The ratio of the areas of two triangles with common base is equal to the ratio of their corresponding heights. Larger Smaller h h h...(substituting the given values) 6h h 6 h 7.5cm The corresponding height of the smaller traingles of 7.5 cm. Topic:Similarity_; Sub-topic:rea of similar triangle L-1 SSC oard Test_Mathematics

3 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions (ii) In the following figure, point is the centre of the circle. Line MN is tangent at point M. If N = 10 cm and MN = 5 cm, determine radius of the circle. M N Seg M is the radius and line MN is the tangent to the circle at the point M. MN 90...(Tangent is perpendicular to the radius) In right angled MN, by Pythagoras theorem, N M MN 10 M 5 M M 75 5 M 5 cm The radius of the circle is 5 cm. Topic:Circle_; Sub-topic:Tangent theorem L- SSC oard Test_Mathematics (iii) Draw C of measure 80 and bisect it. C O Topic:Construction_; Sub-topic:isector of angle L-1 SSC oard Test_Mathematics (iv) 5 If cos, where ' ' is an acute angle. Find the value of sin. 1 5 cos sin 1 cos sin 1 as is acute, therefore sin must be positive. 1 sin 1 Topic:Trigonometry_; Sub-topic:Identities L- SSC oard Test_Mathematics

4 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions (v) The volume of a cube is 51 cm. Find its side. Volume of cube = 51 cm x = 51 x = 8 cm Side is 8 cm. Topic:Mensuration_; Sub-topic:Volume L-1 SSC oard Test_Mathematics (vi) The radius and slant height of a cone are 5 cm and 0 cm respectively. Find its curved surface area..14 r = 5 cm, = 0 cm Curved surface area r cm Topic:Mensuration_; Sub-topic:Volume L-1 SSC oard Test_Mathematics Q. ttempt any THREE of the following subquestions : [9] (i) In the following figure, seg DH seg EF and seg GK seg EF. If DH = 18 cm, GK = 0 cm and D DEF 450cm, then find : (i) EF (ii) GFF (iii) DFGE E H K F (i) (ii) rea of a triangle 1 base height 1 EF EF DH EF...(Substituting the given values) 450 EF 18 EF 50 EF 50 cm DEF and GEF ahve the common bae EF. their areas are proportional to their corresponding heights. DEF DH GEF GK G GEF 0...(Substituting the given values) GEF 750 cm 18 GEF 750 cm 4 4

5 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions (iii) DFGE DEF GEF...(rea addition postulate) cm DFGE 100 cm Topic:Similarity_; Sub-topic:rea of similar triangle L- SSC oard Test_Mathematics (ii) In the following figure, ray P is a tangent to the circle at and PC is a secant. If P = 15, P = 10, then find C. C P In the given figure, P is a tangent segment and PC is the secant. P PC P 5 10 PC 15 PC.5 10 Now, P C PC...(P C) 10 C.5 C Therefore C = 1.5. Topic:Circle_; Sub-topic:Secant theorem L-1 SSC oard Test_Mathematics (iii) Draw the circle with centre C and radius.5 cm. Point is at a distance 7 cm from the centre. Draw tangents to the circle from the point. P C.5cm Q.5cm 5 5

6 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions (a) Take centre C (b) Draw circle of radius.5 cm (c) Extend C till C where C 7 (d) Take distance more than half in compass and mark two arcs from the ends to line C. (e) Now draw two more arc above and below the line keeping compass on these arcs. Topic:Construction_; Sub-topic:Tangent L-1 SSC oard Test_Mathematics (iv) Show that : cm 1 cos 1 cos cosec cot LHS 1 cos 1 cos 1 cos 1 cos 1 cos 1cos 1 cos 1 cos 1 cos 1 cos 1 cos cosec cot RHS sin sin sin sin Hence prove. Topic:Trigonometry_; Sub-topic:Indentities L- SSC oard Test_Mathematics (v) Write the equation of line passing through (, 4) and (4, 5) in the form of ax by c 0., 4 4,5 Let,4 x, y ; 4,5 x, y Now, x x1 y y1 x x y y 1 1 x y x 7 y 4 x 7 y x y x7 y 8 0 x 7 y 1 0 The equation of the line is x 7 y 1 0 Topic:Coordinate geometry_; Sub-topic:Equation of line L-1 SSC oard Test_Mathematics 6 6

7 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions Q.4 ttempt any TWO of the following subquestions : [8] (i) Prove that, the lengths of the two tangent segments to a circle drawn from an external point are equal. O P Given : O is the centre of the circle and P is a point in the exterior of the circle. and are the points of contact of the two tangents from P to the circle. To Prove : P = P. Construction : Draw seg O, seg O and seg OP. Proof : Line P radius O and line P radius O... (Tangent perpendicular to radius) PO PO 90 In right angled tringles, OP and OP, hypotenuse OP hypotenuse OP...(Common side) seg O seg O...(Radii of the same circle) OP OP...(Hypotenuse-side of theorem) seg P seg P...(c.s.c.t.) P P Topic:Circle_; Sub-topic:Theorem L- SSC oard Test_Mathematics (ii) tree is broken by the wind. The top of that tree struck the ground at an angle of 0 and at a distance of 0 m from the root. Find the height of the whole tree. 1.7 Let represents the unbroken part and C represent the broken part of the tree. The top of the tree (T) touches the ground at C. C 0 cm, C 0 Total height of the tree = + T = + C...(1) In right angles C, tan C tan0 C 1 0 C lso, cos C cos0 C 0 C C 60 C 0 m...() C 0 C C 60 T...() 7 7

8 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions Height of the tree = + T...[From (1)] [From () and ()] the height of the tree 0 m 01.7m m The height of the whole tree is m. Topic:Trigonometry_; Sub-topic:pp. of trigo L- SSC oard Test_Mathematics (iii) (5, 4), (, ) and C(1, 8) are the vertices of a triangle C. Find the equation of median D. Let 5,4 x, y ;, x, y and C 1, 8 x, y D(x, y) is the midpoint of C. 1 1 x x y y the coordinates of D, ,, 1, 5 Let D 1, 5 x, y 4 4 The equation of median D is x x1 y y1 x x y y x 5 y x 5 y 4 Multiplying both the sides by 6, x y x 5 y 4 x 5 y x 15 y 8 x y x y (, ) (5,4) D (x,y) C (1, 8) Topic:Coordinate geometry_; Sub-topic:Equation of line L- SSC oard Test_Mathematics Q.5 ttempt any TWO of the following subquestions : [10] (i) Prove that, in a right-angled triangle, the square of hypotenuse is equal to the sum of the square of remaining two sides. Draw perpendicular D from the vertex, to the side C. D C. In right angled C, seg. D hypotenuse C. by similarity in right angled triangles, C D DC

9 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions Now, C D C D...(c.s.s.t) C D...(1) lso, C DC. C DC C C...(c.s.s.t) C C DC...() From (1) and (), C C D C DC C D DC C C...( D C) C C. i.e., C C Topic:Similarity_; Sub-topic:Pythagorus theorem L- SSC oard Test_Mathematics (ii) SHR SVU, in SHR, SH = 4.5 cm, HR = 5. cm, SR = 5.8 cm and SH SV 5. Construct SVU. Topic:Construction_; Sub-topic:Construction of triangle L- SSC oard Test_Mathematics (iii) If V is the volume of a cuboid of dimensions a b c and S is its surface area, then prove that : V S a b c The volume (V) of a cuboid = abc (1) V abc The surface area (S) of a cuboid ab bc ca...() RHS S a b c ab bc ca S abc 9 9

10 Rao IIT cademy/ SSC - oard Exam 018 / Mathematics Code- / QP + Solutions ab bc ca S abc S S abc...[from ()] 1 1 abc V...[From (1)] V S a b c. Topic:Mensuration_; Sub-topic:Volume_L- SSC oard Test_Mathematics 10 10

BOARD ANSWER PAPER :OCTOBER 2014

BOARD ANSWER PAPER :OCTOBER 2014 BRD NSWER PPER :CTBER 04 GEETRY. Solve any five sub-questions: BE i. BE ( BD) D BE 6 ( BD) 9 ΔBE (ΔBD) ----[Ratio of areas of two triangles having equal base is equal to the ratio of their corresponding