MATHEMATICS PAPER IA. Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS.

Size: px

Start display at page:

Download "MATHEMATICS PAPER IA. Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS."

Loreen James
5 years ago
Views:

1 MATHEMATICS PAPER IA TIME : hrs Mx. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0. If A = {,, 0,, } nd f : A B is surjection defined y f(x) = x + x +, then find B.. Find the domin of the function f (x) log(x 4x ).Find vector in the direction of vector i j tht hs mgnitude 7 units. 4.Let i 4 j 5k, i j k nd c j k, find the unit vector in the opposite direction of c. 5.Find the ngle etween the plnes r ( i j k) nd r (i 6 j k) prove tht cot cot cot cot cot Drw the grph of y tn x in, n 8. Prove tht hx sin hx h nx sin hnx 9.If I = 0 0 nd E = 0 0 0, show tht (I + E) = I + E. Solve the following system of homogeneous equtions. 0.. If A = 0, then find AAT. Do A nd A T commute with respect to multipliction of mtrices?

2 SECTION B SHORT ANSWER TYPE QUESTIONS ANSWER ANY FIVE OF THE FOLLOWING 5 X 4 = 0. If c d, c d nd,, c re non-coplnr vectors then show tht c d 0..If c 0,, 5 nd c = 7 then find the ngle etween nd Prove tht sin sin sin sin If, re solutions of the eqution sin c where,, c R nd if 0 nd sin sin then show tht c (iv)_ sin sin c Prove tht Tn Tn 6. If : : c = 7 : 8 : 9 then find A B C 7. If AB = I or BA = I, then prove tht A is invertile nd B = A. SECTION C LONG ANSWER TYPE QUESTIONS ANSWER ANY FIVE OF THE FOLLOWING 5 X 7 = 5 8.If f : A B, g : B C re two ijections then ( ) gof f og. 9. Prove tht upto n terms n n ( 6n )

3 0.If, nd c represent the vertices A, B nd C respectively of ABC, then prove tht ( ) ( c) (c ) is twice the re of ABC.. If A,B,C re ngles of tringle then prove tht A B C A B C sin sin sin c. Show tht cot A cot B c C R.. Prove tht det c c c c ( )( c)(c )( c c) 4. By Mtrix inversion method solve : x y + 8z = x + 4y + 5z = 8 5x y + 7z = 0 Solutions. If A = {,, 0,, } nd f : A B is surjection defined y f(x) = x + x +, then find B. Sol. Given tht f(x) = x + x + f( ) = ( ) + = 4 + = f( ) = ( ) + = + = f(0) = (0) 0 + = f() = + + = f() = + + = 7 Thus rnge of f, f(a) = {,, 7} Since f is onto, f(a) = B B = {,, 7}. Find the domin of the function f (x) log(x 4x ) x 4x 0 (x )(x ) 0 x or x

4 Since the coefficient of x is +ve Domin of f is R [, ].Find vector in the direction of vector i j tht hs mgnitude 7 units. Sol. The unit vector in the direction of the given vector is â ( i j) i j Therefore, the vector hving mgnitude equl to 7 nd in the direction of is i j i j Let i 4 j 5k, i j k nd c j k, find the unit vector in the opposite direction of c. Sol. c i 4 j 5k i j k j k i 6 j k Unit vector in the direction of c c c i 6 j k i 6 j k Find the ngle etween the plnes r ( i j k) nd r (i 6 j k) 4. Sol. Given r ( i j k) r (i 6 j k) 4 Given eqution r n p, r n q Let e the ngle etween the plnes. n n n n Then ( i j k) (i 6 j k) i j k i 6 j k

5 46 6. prove tht cot cot cot cot cot Sol. cot cot 9 0 tn 9 cot cot 7 0 tn 7 5 cot cot cot cot 6 cot(90 7 ) tn cot cot 8 cot(90 9 ) tn cot cot cot cot cot tn 7 tn 9 tn 9 tn 7 7. Drw the grph of y tn x in, 8. Prove tht hx sin hx n h nx sin hnx Solution: = hx sin hx e e e e x x x x x x x x n e e e e x nx e e nx nx nx nx e e e e RHS= hnx-sinhnx = = e e e e nx nx nx nx e nx n

6 9.If I = 0 0 nd E = 0 0 0, show tht (I + E) = I + E. Sol. I + E = (I + E) = (I + E) = I E Solve the following system of homogeneous equtions. 0.. If A = 0, then find AAT. Do A nd A T commute with respect to multipliction of mtrices? T 0 Sol. A AA T T A A Since AA T A T A, A nd A T do not commute with respect to multipliction of mtricesif c d, c d nd,, c re non-coplnr vectors then show tht c d 0. Sol. c d 0 () c d 0 () () ( ) c d 0 () () = ()

7 c d c d By sustituting = in () we get c d 0..If c 0,, 5 nd c = 7 then find the ngle etween nd. Sol. Given, 5, c 7 nd c 0 c Squring on oth sides c [ (, )] c 9 5 [.5(, )] 49 [5(, )] 49 4 (, ) (, ) 5 0 (, ) Angle etween nd is Prove tht sin sin sin sin Solution: sin sin sin sin sin sin sin sin sin sin sin sin sin sin 8 8

8 sin sin If, re solutions of the eqution sin c where,, c R nd if 0 nd sin sin then show tht Solution: c (iv)_ sin sin sin c sin c c c c c 0 c Since, re solutions, re roots of ove eqution c sum of roots Product of roots c Prove tht Tn Tn Solution: Let L.H.S tn tn 4 4 tn tn tn tn tn tn tn tn sec tn 6. If : : c = 7 : 8 : 9 then find A B C Soltion : -

9 A B c 64k 8k 49k 96k c 8k 9k 8k 9 k c 49k 8k 64k 66 c 7k 9k 6 c 49k 64k 8k k C 7k 8k 0k 8k 7 A B coc :: 6 7. If AB = I or BA = I, then prove tht A is invertile nd B = A. Sol. Given AB = I AB = I = A B = = A 0 A is non-singulr mtrix. nd BA = I BA = I B A = A 0 A is non-singulr mtrix. AB = I or BA = I, A is invertile. A exists. AB = I A AB = A I IB = A B = A B = A 8.If f : A B, g : B C re two ijections then ( ) gof f og. Proof: f : A B, g : B C re ijections gof : A C is ijection ijection. f : A B is ijection f : B A is ijection g : B C is ijection g : C B is ijection g : C B, g : B A re ijections f og : C A is ijection Let z C z C, g : B C is onto y B g( y) z g ( z) y y B, f : A B is onto x A f ( x) y f ( y) x ( gof )( x) g[ f ( x)] g( y) z ( gof ) ( z) x ( gof ) ( z) x f ( y) f [ g ( z)] ( f og )( z ) n n 9. Prove tht upto n terms Sol:,, 4 n terms tn ( n ) n ( 6n ) ( gof ) f og ( gof ) : C A is

10 , 4, 5.. n terms tn ( n ) n ( n )( n ) Let S n e the given sttement For n = L.H.S =. = 6 ( 6 ) R. H. S 6 L. H.S = R.H.S S() is true Assume S k is true n n ( 6n ) ( k )( k ) k( k 6k ) ( k )( k ) k( k 6k ) ( k 5k 6) k 9k 6k 8 ( k ) k 8k 8 k Sk is true Hence S ( n) is true for ll n N ( k ) ( k ) 6( k ) 0.If, nd c represent the vertices A, B nd C respectively of ABC, then prove tht ( ) ( c) (c ) is twice the re of ABC. Sol. A() B() Let O e the origin, OA,OB,OC c Are of ABC is C(c) (AB AC)

11 (OB OA) (OC OA) ( ) (c ) c c c c 0 c c c c. Hence proved.. If A,B,C re ngles of tringle then prove tht A B C A B C sin sin sin SOL. A+B+C = 80 0 A B C LHS sin sin sin A B C B C sin sin.sin A A B C sin sin 90.sin A A B C =-.sin A A B C sin A B C B C = 90 sin A B C B C sin sin A B C sin A B C sin RHS c. Show tht cot A cot B c C R. Sol. L.H.S. cot A cot B c C

12 A B C 4R sin A 4R sin B 4R sin C (y sine rule) sin A sin B sin C R (sin A A sin B B sin CC) R (sin A sin B sin C) R (4sin Asin Bsin C) (R sin A)(R sin B)(R sin C) R c R.H.S. R c. Prove tht det Sol. c c c ( )( c)(c )( c c) c c c c c c c c c c c c c c c c R R R = R R R ( c)( c) = R R R 0 c c 0 c c c c 0 c c c 0 c c c c c 0 c c c ( c)( c) 0 c c c c c 0 c c c ( c)( c)( ) 0 c c c

13 c c c ( c)( c)( ) c ( c)( c)( )( c c) 4. By Mtrix inversion method solve : x y + 8z = x + 4y + 5z = 8 5x y + 7z = 0 Sol. Mtrix inversion method : 8 x Let A = 4 5, X = y nd D = A B ( 5) C A ( 7 6) B (4 40) C ( 4 5) 5 8 A B (0 4) 4 5 C 8 4 A A A AdjA B B B C C C 6 Det A A B c C 8 ( )4 8( 6) z 0

14 8 9 7 AdjA A Det A X A D Solution is x =, y =, z =.

Board Answer Paper: October 2014

Board Answer Paper: October 2014 Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p