XII HSC - BOARD

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS Date: 0.0.08 XII HSC - BOARD - 08 MATHEMATICS (40) - SOLUTIONS Q. (A) SECTION - I (i) If A 4, then adjoint of matri A is (A) 4 (A) (B) 4 (C) 4 (D) 4 A 4 adja 4 Topic: Matri; Sub-topic:Adjoint L- XII-HSC Board Test_Mathematics (ii) The principal solutions of sec are. (A), 6 (B) sec (B), 6 6 cos cos cos 6 6, 6 6 (C), 4 4 (D), 6 4 Topic: Trigo. function; Sub-topic:General solution_l- XII-HSC Board Test_Mathematics

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (iii) The measure of acute angle between the lines whose direction ratios are,, 6 and,, is. (A) cos 7 (D) (B) cos 5 8 (C) cos (D) cos 8 cos 6 6 6 8 8 49 9 7 cos 8 Topic: D Geometry; Sub-topic:Angle L- XII-HSC Board Test_Mathematics Q. (B) (i) Wirte the negations of the following statements : (a) (b) All students of this college live in the hostel. 6 is an even number or 6 is a perfect square. (a) p : All students of this college live in the hostel. Negation : p : Some students of this college do not live in the hostel. (b) p : 6 is an even number. q : 6 is a perfect square. Symbolic form : p q p q p q Negation : 6 is not an even number and 6 is not a perfect square. Topic: Logic; Sub-topic:Negation L- XII-HSC Board Test_Mathematics (ii) If a line makes angles,, with the co-ordinates aes, prove that cos cos cos 0. L.H.S : cos cos cos cos cos cos cos cos cos 0 = R.H.S Topic:D Geometry; Sub-topic: D L- XII-HSC Board Test_Mathematics

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (iii) Find the distance of the point (,, ) from the plane y 4z 0 0. Distance of the point D a by cz d a b c y z,, a, b, c 4 y z to plane a by cz d 0 is 4 0 7 7 D units 4 6 Topic:Plane; Sub-topic:Distance L- XII-HSC Board Test_Mathematics (iv) Find the vector equation of the lines which passes through the point with position vector 4iˆ ˆj kˆ and is in the direction of iˆ ˆj kˆ. Let a 4iˆ ˆj kˆ b ˆ i ˆj kˆ Equation of the line passing through point r a b 4ˆ ˆ ˆ ˆ ˆ ˆ Aa and having direction b is r i j k i j k Topic:Line; Sub-topic:Equation L- XII-HSC Board Test_Mathematics (v) If a iˆ ˆj 7 kˆ, b 5iˆ ˆj kˆ and c iˆ ˆj kˆ then find a b c. a b c a b c a b c a b c a b c 7 a b c 5 = ( + ) + ( 5 + ) +7(5 ) = 6 + 8 = 5 Topic: Vector; Sub-topic:Triple dot product L- XII-HSC Board Test_Mathematics

Q. (A) (i) Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions Using vector method prove that the medians of a triangle are concurrent. Let a,b,c,d,e the position vectors of the vertices A, B, C of ABC and d, e, f be the position vectors of the midpoints D, E, F of the sides BC, CA and AB respectively A F G E B D C Then by the midpoint formula, b c, c a, a d e f b d b c ; e c a ; f a b d a a b c e b a b c f c a b c d a e b f c a b c g lies on the three medians AD, BE and CF dividing each of them internally in the ratio :. Hence, the medians are concurrent at point G. let Topic: Vector; Sub-topic:Theorem L- XII-HSC Board Test_Mathematics (ii) Using the truth table, prove the following logical equivalence : p q p q p q Mark Mark 4 A 5 6 7 B 8 p q A B By column number and 8 p q p q p q Topic: Logic; Sub-topic:Truth table L- XII-HSC Board Test_Mathematics 4 4

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (iii) If the origin is the centroid of the triangle whose vertices are A(, p, ), B(q,, 5) and R( 5,, r), then find the values of p, q, r. Let a, b, c be the position vectors of ABC whose vertices are A(, p, ), B(q,, 5), C( 5,, r) a iˆ pj ˆ kˆ, b qiˆ ˆj 5 kˆ, c 5iˆ ˆj rkˆ Given that origin O is the centroid of ABC a b c O a b c O iˆ pj ˆ kˆ qj ˆ ˆj 5kˆ 5iˆ ˆj rkˆ O q 5i ˆ p ˆ j 5 r k ˆ 0i ˆ 0 ˆ j 0k ˆ by equality of vectors q 5 0 q p 0 p 5 r 0 r p, q and r Topic:Vector; Sub-topic:Section formula L- XII-HSC Board Test_Mathematics Q. (B) (i) Show that a homogeneous equation of degree two in and y, i.e., a hy by 0 represents a pair of lines passing through the origin is h ab 0. Consider a homogenous equation of degree two in and y a hy by 0... i In this equation at least one of the coefficients a,b or h is non zero. We consider two cases Case I : If b = 0, then the equation of lines = 0 and (a + hy) = 0 These lines passes through the origin. Case II : b 0, Multiplying both the sides of equation (i) by b, we get ab hby b y 0 b y hby ab To make L.H.S a complete square, we add h on both the sides. b y hby h ab h by h h ab by h h ab by h h ab 0 by h h ab by h h ab 0 5 5

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions It is the joint equation of two lines i.e. by h h ab 0 and by h h ab 0 h h ab by 0 and h h ab by 0 These lines passes through the origin. Topic: Pair of straight line; Sub-topic:Theorem L- XII-HSC Board Test_Mathematics C A c a B (ii) In ABC, prove that tan cot. c a In ABC, by sine Rule, a b c k sin A sin B sin C a k sin A, b k sin B, c k sin C Consider, c a k sin C k sin A c a k sin C k sin A sin C sin A sin C sin A C A C A cos sin C A C A sin cos cot C A tan C A tan B tan C A tan C A C a cot B C a Hence proved. Topic: Trigo. function; Sub-topic:Theorem L- XII-HSC Board Test_Mathematics 6 6

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (iii) Find the inverse of the matri, A 0 0 using elementary row transformations. A 0 0 4 0 A eists. We know, AA I 0 0 0 A 0 0 0 0 0 R R R 0 0 0 5 A 0 0 0 0 R R R 0 0 0 0 A 0 0 0 R R R and R R R 0 4 0 0 A 0 0 5 R R R 0 0 6 0 0 A 0 0 5 6 A 5 Topic: Matri; Sub-topic:Inverse L- XII-HSC Board Test_Mathematics 7 7

Q. (A) (i) Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions Find the joing equation of the pair of lines passing through the origin, which are perpendicular to the lines represented 5 y y 0. Given homogeneous equation is 5 y y 0 Which is factorisable 5 5y y y 0 5 y y y 0 y y 5 0 y 0 and 5 y 0 are the two lines represented by the given equation. Their slopes are and 5 Required two lines are respectively perpendicular to these lines. Slopes of required lines are and Their individual equations are y and y 5 and the lines pass through origin. 5 i.e., y 0 and 5y 0 Their joint equation is y y 5 0 y 5y 5y 0 y 5y 0 Topic: Pair of straight line_sub-topic:formulation of equation L- XII-HSC Board Test_Mathematics (ii) Find the angle between the lines y z and 4 8 Let a and b be the vectors in the direction of the lines respectively. a 4 i j 8k and b iˆ j k y z 4. y z and 4 8 y z 4 a b 4 8 8 8 8 and a 6 64 8 9 b 4 4 9 Let be the acute angle between the two given lines 8 8

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions a b 8 cos a b 9 cos Topic: Line; Sub-topic:Line L- XII-HSC Board Test_Mathematics (iii) Write converse, inverse and contrapositive of the following conditional statement : If an angle is a right angle then its measure is 90. Converse : If the measure of an angle is 90 then it is a right angle. Inverse : If an angle is not a right angle then its measure is not 90. Contra positive : If the measure of an angle is not 90 then it is not a right angle. Topic:Logic; Sub-topic:Condictional L- XII-HSC Board Test_Mathematics Q. (B) 56 (i) Prove that : sin cos sin 5 65 Let cos and let cos 5 sin sin y 5 sin y 5 4 cos y 5 using sin y sin cos y cos sin y 5 4 5 5 0 6 5 56 65 56 y sin 65 56 cos sin sin 5 65 Hence proved. Topic: Trigo. function; Sub-topic:ITF L- XII-HSC Board Test_Mathematics 9 9

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (ii) Find the vector equation of the plane passing through the points A(, 0, ), B(,, ) and C(4,, ). Let the p.v. of points A(, 0, ), B(,, ) and C(4,, ) be a i k, b i j k and c 4i j k b a j, c a i j k i j k 0 0 b a c a i k Equation of plane through A, B, C in vector form is r a b a c a 0 r a i j 0 r i j i k i j r i j Topic: Plane; Sub-topic:Equation of plane L- XII-HSC Board Test_Mathematics (iii) Minimize Z = 7 + y subject to 5 y 5, y, 0, y 0 Z 7 y Subject to 5 y 5 y 0, y 0 Line Inequation Points on Points on y Feasible region AB 5 y 5 A,0 B 0,5 Non - origin side CD y unit = cm both ais C,0 D0, Non - origin side 0 0

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions B(0, 5) 5 D(0, ) P, P A(, 0) AB C(, 0) CD common feasible region BPC Points Minimize z 7 y B 0,5 Z B 7 0 5 5 P 5, 5 5 Z P 7 6 C,0 Z C 7 0 Z is minimum at = 0, y = 5 and min (z) = 5 Topic: LPP; Sub-topic:Graphical solution L- XII-HSC Board Test_Mathematics SECTION - II Q.4 (A) Select and write the appropriate answer from the given alternatives in each of the following (i) sub-questions : [6] Let the p. m. f. of a random variable X be P( ) for,0,, 0 0 otherwise Then E( X ) is. (A) (B) (C) 0 (D) (C) 0 4 p 0 0 0 0 4. p 0 0 0 0. P 0 Topic: Probability distribution; Sub-topic:Epected value L- XII-HSC Board Test_Mathematics

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions k (ii) If, 8 6 then the value of k is. 0 (A) (B) (C) 4 (D) 5 (A) I k 0 6 tan k 0 6 tan k tan 0 k k 4 Topic:Definite integral; Sub-topic:Definite integral L- XII-HSC Board Test_Mathematics dy (iii) Integrating factor of linear differential equation y log is. (A) (D) (B) (C) (D) dy y log P I. F e e log Topic: Diferential equation; Sub-topic:LDE L- XII-HSC Board Test_Mathematics

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions Q.4 (B) Attempt any THREE of the following : [6] cos sin (i) Evaluate : e sin cos sin I e sin sin cot cosec cosec e f '( ) f ( ) e [ f ( ) f ( )] e f ( ) C I e cos ec C Topic: Integration; Sub-topic:Integration L- XII-HSC Board Test_Mathematics dy (ii) If y tan (log ), find. y [tan (log )] differentiate w.r.t. both side dy [tan (log )] sec (log ) dy 6 tan (log ) sec (log ) Topic: Differentiation; Sub-topic:Composite function L- XII-HSC Board Test_Mathematics (iii) Find the area of ellipse y. 4 Required area = 4 Area ( OAPB) 0 y y 4 y Required area 4 0 (0,) B O P (0,) A 8 sin 0

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions 8 0 sin () 0 8 sq.units Topic: Definite integral; Sub-topic:AOI L- XII-HSC Board Test_Mathematics (iv) Obtain the differential equation by eliminating the arbitrary constants from the following equation : y c e c e y c e c e differentiate w.r.t.. dy c e c e Again diff. w.r.t.. d y 4c e 4c e d y 4( c e c e ) 4 y 4y 0 Topic: Differential equation; Sub-topic:Formulation L- XII-HSC Board Test_Mathematics (v) Given X ~ B ( n, p) If n 0 and p 0.4, find E( X ) and Var. ( X ). n 0, p 0.4 q p 0.6 E( X ) np 00.4 4 V ( ) npq 40.6.4 Topic:Binomial distribution; Sub-topic:Mean L- XII-HSC Board Test_Mathematics 4 4

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions Q.5 (A) Attempt any TWO of the following: [6] (i) Evaluate: sin cos Let I sin cos Put tan t Then,, dt t t t sin and cos t t I dt /( t ) t t t t dt /( t ) ( ) 4 ( t t t ) dt t dt 4 4 ( ) t t t tan ( t ) c tan tan c Topic: Integration; Sub-topic:Method of substitution L- XII-HSC Board Test_Mathematics (ii) If a cos t, y asin t, show that We have, dy y dy dy / dt, / dt 0...() / dt Now, y a y asin t a (sin t) sin t dy d d a t a t t dt dt dt (sin ) (sin ) (sin ) a sin t cost...() Also, a acos t a(cos t) cost 5 5

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions d a t t dt dt cos (cos ) acos t ( sin t) a cos t sin t...() From (), () and (), dy a sin t cost sin t y a cos t sin t cost Topic:Differentiation; Sub-topic:Parametric function L- XII-HSC Board Test_Mathematics (iii) Eamine the continuity of the function: log00 log(0.0 ) f ( ) 00 for 0;at 0 f is continuous at 0 if lim f ( ) f (0). 0 00 R.H.S. f (0)...(given)......() log00 log (0.0 ) log( 00 ) L.H.S. lim f ( ) lim lim 0 0 0 [log( 00 )] 00 lim 00 0 00 From () and (), L.H.S. = R.H.S. lim f ( ) f (0). 0 f is continuous at 0. Topic: Continuity; Sub-topic:At a point L- XII-HSC Board Test_Mathematics Q.5 (B) Attempt any TWO of the following: [8] (i) Find the maimum and minimum value of the function: f ( ) 6 0 f ( ) 6 0 f ( ) ( ) ( ) 6() 0 6 4 6 6( 7 6) 6( )( 6) f has a maima/minima if f ( ) 0 i.e. if 6( )( 6) 0 i.e. if 0 or 6 0 6 6

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions i.e. if 0 or 6 Now, f ( ) 6( ) 4() 4 f () () 4 0 f () 0 Hence, f has a maimum at, by the second derivative test. Also, f (6) (6) 4 0 f (6) 0 Hence, f has a minimum at 6, by the second derivative test. Now, maimum value of f at, f () ( ) ( ) 6() 0 6 0 and minimum value of f at = 6, f (6) (6 ) (6 ) 6() 0 4 756 6 0 8 Topic: AOD; Sub-topic:Maima or Minima L- XII-HSC Board Test_Mathematics a (ii) Prove that: log a a a c a a a a a a a a a a a a a a log a log a c log a log a c a a a log a a c Topic: Integration; Sub-topic:Theorem L- XII-HSC Board Test_Mathematics 7 7

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (iii) Show that: a a f ( ) f ( ), if f ( ) is an even function a 0 0, if f ( ) is an odd function We shall use the following results : b a a b f f...() b a b a f f t dt...() If c is between a and b, then b c b f f f...() a a c Since 0 lies between -a and a, by (), we have, a 0 a f f f I I... Say a a 0 In I, put t. Then dt When a, t a t a When 0, t 0 t 0 0 0 0 f f t dt f t dt a a a a 0... f t dt By a 0... f By a a a a f f f 0 0 (i) If f is an even function, then f f in this case, a a a a a f f f f 0 0 0 (ii) If f is an odd function, then f f in this case, a a a a a f f f f f 0. a 0 0 0 0 Topic: Definite integral; Sub-topic:Theorem L- XII-HSC Board Test_Mathematics 8 8

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions Q.6 (A) Attempt any TWO of the following : [8] (i) If 9 f ( ), for 5, for, for is continuous at, find and. f is continuous at lim f ( ) lim f ()...() Now, 0 9 ( )( ) lim f ( ) lim lim ( ) 0 lim[( ) ] ( ) 6 lim ( ) lim () () 8 9 7 f and Also, f () 5...(given) From (), we get, 6 7 5 65 and 7 5 5 6 and 5 7 and Topic: Continuity; Sub-topic:At a point L- XII-HSC Board Test_Mathematics (ii) Find dy if 5 y tan 6 Let 5 y tan 6 5 6 tan 5 tan ( )( ) ( ) ( ) tan ( )( ) y Differentiate w.r.t. tan ( ) tan ( ) dy ( ) ( ) 9 9

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions 9 4 4 4 9 5 Topic:Differentiation; Sub-topic:Inverse function L- XII-HSC Board Test_Mathematics (iii) A fair coin is tossed 9 times. Find the probability that it shows head eactly 5 times. Let X = no. of heads shows. n 9 p q ( ) n n.( ) n P X C p q X 0,,..., n 5 4 9 P( X 5) C 5 9876 9 4 6 5 0.460 Topic:Binomial distribution; Sub-topic:Binomial distribution_l- XII-HSC Board Test_Mathematics Q.6 (B) Attempt any TWO of the following : [8] (i) Verify Rolle s theorem for the following function: f ( ) 4 0 on [0, 4] Since f ( ) is a polynomial, (i) It is continuous on [0, 4] (ii) It is differentiable on (0, 4) (iii) f (0) 0, f (4) 6 6 0 0 0 0 f (0) f (4) 0 Thus all the conditions on Rolle s theorem are satisfied. The derivative of f ( ) should vanish for at least one point c in (0, 4). To obtain the value of c, we proceed as follows f ( ) 4 0 f ( ) 4 ( ) f ( ) 0 ( ) 0 c in (0, 4) We know that (0, 4) Thus Rolle s theorem is verified. Topic:AOD; Sub-topic:Roll s theorem L- XII-HSC Board Test_Mathematics

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions (ii) Find the particular solution of the differential equation: y( log ) log 0 dy when y e and e. Given equation is y( log ) log 0 dy y( log ) log dy y( log ) log dy Separating the variables, log dy y log Integrating, we have, log dy y log log y log log log c log y log c log y c log is the general solution. Given : e, y e e c e c. e.log e c. e. e y e log Topic:Differential equation; Sub-topic:Variable separable method_l-_xii-hsc Board Test_Mathematics (iii) Find the variance and standard deviation of the random variable X whose probability distribution is given below : P( X ) 0 8 8 8 8

Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions p p p i i i i i i 0 8 0 0 8 8 8 6 8 8 8 9 8 8 8 Total 4 8 8 E( X ) pi i 8 Var( X ) p n i 9 4 i i 4 Var( X ) 4 Standard deivation of X 4 Topic: Probability distribution; Sub-topic:Epected value_l- XII-HSC Board Test_Mathematics