1. A 0,,,, : cos. g x x. 2. If f : R R, g : R R are defined by f x 3x 2 ( ( 3, 1 ( ( ) 1 1, ,

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1 If π π π π. A 0,,,, : cos 6 and f A B is a surjection defined by f then find B Given f ( ) cos, f (0) cos(0), f cos, f cos, f cos, f cos Rangeof B,,,,0 g. If f : R R, g : R R are defined by f, ( ) then findi)fog() ii)gof(a-) ( i) Given g( ) ( ii) Given f ( ) g() f (a ) (a ) g() f (a ) 6a 9 f ( g()) f () f (a ) 6a 0 f ( g()) ( ) [ f ( ) ] g( f (a )) g(6a 0) fog() g( f (a )) (6a 0) [ g( ) ] gof (a )) (6a 0a 0). If f ( ), g( ), h( ) for all R, then find ( fo( goh)){ } Given f ( ), g( ), h( ) ( fo( goh)){ } f { g[ h( )]} f { g[ ]} f { }. Find the domain of the real valued function f Given f ( )( )( ) 0,, Domain of ' f ' is R \,, ( )( ) ( )( )( ). If A -,-,0,, and f : A Bis a surjection defined by sol Given f f, then find B. ( ), f (0) 0 0, ( )( ) ) ) ) Rangeof B,,7 f f f f ( (, ( ( ), 7, 6. find thedomainof real valued function f ( ) 9 Given f ( ) 9 but f R ( ) ( )( ) 0 [,] Domain of f is[,]

2 y y y y y ( fog){ y} f { g[ y]} f y y y y y y y y y y 8. find the domain of real valued function f ( ) log e( ) sol Given f. ( ) log e( ) but f R ( ) 0 0 ( )( ) 0 (,) (, ) Domain of f is (,) (, ) 9. If f : Q Qis defined by f for all Qthen find f Given f let f ( ) y f ( y) () y f ( ) y y y () y y f ( y) f ( y) 0. find thedomainof real valued function f ( ) Given f ( ) but f ( ) R 0, 0, 0, 0 [, ] \{0} Domain of f is[,] \{0}. If f : R R is defined by f, thenthis functionis injuction or not? justify? Given f Suppose f ( ) f ( ) f ( ) f ( ) f is injective function. If the function f is defined by Then find the values of f, f.,,., f If, f ( )., f ( ) not define f () () 0. Find the domain of the real valued function f Given f 0 0 ( )( ) 0 (,) Domain of ' f ' is (,)

3 Lower triangular matrices : A square matrices A=[ a ij ] is said to be an lower triangular matrices if a ij =0 when i< j 0. Define skew symmetric matri, If A 0 is a skew symmetric matri, find? 0 skew symmetric matri : a matri' A' is said tobe skew symmetric matri if A A 0 0 Given A 0 T A T If Ais a skew symmetric matri, A A 0 0 ( ) If A 0 0 then finda? Given A 0 0 A 0 0 A I A ( I) A 8I A If A 0, B and X=A+B then find X? 0 Given A 0, B and X=A+B X= 0 X 0 7. Find the adjoint and inverseof the matri 6 8. If A and A 0, then k a b d b d b if A then A, adj A c d ad bc c a c a Given A and A 0 k AA 0 Given A A.6 ( ). A k k 0 0 Adjoint A 8 k 0 0 adj A 6 k k 0 0 A A A 8 k 0 k T find the valueof k?

4 T 0 Given A 0 A Trace : sum of theelements inthe principal diagonal of a square matri T 0 0 AA Given A, Tra( A) ( ) 0. Construct a X matri whose elements are defined by aij i j a a Given that X matri is A a a a a Now aij i j, i,,, j, a () a () 6 a () a () 6 a () 0 0 a () 6 therequired matriis A 0. Find therank of 0 0 Let A. If A. and det A, then find ' ' Given A A 0 [ rows areidentical] 6 det A = ( + )= 7 Take a min or, 0 [ rows areidentical] Rank of A y8. If, find, y, z and a z 6 a 8, y 8 y, z z, a 6 a 0.Define linear combination of vectors? Let a, a, a an are vectors and,, n are scalars. Then the vectors a a a nan is called a linear combination of vectors

5 where A, X y z A (6 ) ( )( ) ( ) 9 0 A 0 Rank of A the given system of equations havetrivial solution only y z 0 6. Define symmetric matri, If A 6 is a symmetric matri, find? 7 symmetric matri : a matri ' A' is said tobe symmetric matriif A T Given A 6 A T If Ais a symmetric matri, A A Find the area of parallelog ram whose diagonals arei j k and i j k Let a i j k, b i j k i j k ab i( 6) j( ) k( 9 ) i j 0k ab ( ) ( ) ( 0) 00 0 the area of parallelog ram whose diagonals are a, b ab.0 sq. units 8. Show that the points whose position vectors a b c, a b c, 7a c are collinear consider a b c 7 0 [ ( 0) ( ) (0 )] a b c (0) a b c 0 given points are collinear 9. Find the unit vector in the direction of i j k T A Given a i j k then a 9 a i j k unit vector inthe directionof a a

6 we have m 0, n m n m n. Find the vector equation of the line passing through the points a i j k and parallel to the vector b i j k The vector equation of the line passing through the points a and parallel to the vector b is r a tb given a i j k and b i j k Theline r i j k t( i j k). If the position vectors of the point s A, B and C are i j k, i j k and 6i j k respectively and AB AC, then find? Given OA i j k, OB i j k and OC 6i j k AB OB OA ( i j k) ( i j k) i j k AC OC OA (6i j k) ( i j k) 8i j k Given AB AC ( i j k) (8i j k) 8. Find the area of the parallelogram having i j and i k as adjacent sides Theareaof the parallelogram ab Given a i j and b i k i j k nowa b 0 i( 0) j( 0) k(0 9) i j 9k 0 ab sq. units. Find the area of the parallelogram having j k and i k as adjacent sides The area of the parallelogram ab Givena j k and b i k i j k nowa b 0 i( 0) j(0 ) k(0 ) i j k 0 ab 9 sq. units. If,, arethe angles madeby the vector i 6 j k with the positive direction of coordinate ais, then find cos,cos,cos? Sol Given a i j k a. 6, ( 6) then cos,cos,cos Find the angle between the vectors a i j k and b i j k We knowthat cos a. b a b giventhat a i j k a and b i j k b ( ) () ( ) () 7 cos

7 given a i j k, b j k and c i j Theline r ( t s)( i j k ) t ( j k ) s( i j ) 8. If a i j k and b i j k then find the projection of b on a? Given a i j k and b i j k b. a (i j k).( i j k) a ( ) ( ) a ( b. a) a the projection of b on a ( i j k ) a 9. Let a i j k and b i j. Find theunit vector inthe directionof a b? Given a i j k and b i j a b ( i j k) ( i j) i j k a b a b theunit vector inthe directionof a b ( i j k) a b 0. If the vectors a i λ j k and b i j k are perpendicular to each other, find λ If a b a. b 0 () ( ) ( ) 0. Find the vector equation of the line joining the points a i j k and b i j k The vector equation of the line joining the points a and b is r ( t) a tb given a i j k and b i j k Theline r ( t) i j k t( i j k). Find the angle between the planes r.(i j k) and r.(i 6 j k) Given planes are r.(i j k) () and r.(i 6 j k) () let be the angle between planes () and () we knowthat cos cos a a b b c c ( a b c )( a b c ). ( ).6. ( ( ) )( 6 ) 6 6 cos cos cos ( )(9 6 ) 6 6. If the vectors λ i j k and λi λ j k are perpendicular to each other, find Givena λ i j k and b λi λ j k a b a. b 0 ( ) ( )( ) ( ) 0 0 ( )( ) 0 or

8 . Find the period of the function f ( ) cos( ) 7 The period of the function cos a is a given f ( ) cos( ) 7 the period of f ( ) 6. If sin,sin and, are acute, then showthat? 0 Givensin sin 0 tan tan tan tan 6 tan( ) tan tan tan 6. 6 tan( ) tan 7. If sin and is not in the first quadrant, find the value of cos? 8. Pr ove that cos 8.cos 8 givensin 0, and is not in the first quadrant L. H. S cos 8.cos lies insec ond quadrant cos is negative [cos 8.cos] we knowthat cos sin [cos(8 ) cos(8 )] cos [cos60 cos6] cos, Hencecos R. H. S 8 cos9 sin 9 9. Pr ovethat cot 6 cos9 sin 9 sin 9 cos9 cos9 sin 9 cos9 tan tan 9 Given tan( 9) tan tan(90 6) cot(6) cos9 sin 9 sin 9 tan.tan 9 cos9 cos9 0. Find the period of the function f ( ) tan( 9 n ) The period of the function tan a is a given f n ( ) tan( 9 ) n( n )(n ) 6 6 the period of f ( ) nn ( )(n ) n( n )(n ) 6 f ( ) tan( n ) tan

9 we knowthat sin cos sin t sin t sin t sin t tan cos t. Find the ma imum and min imum value of cos sin Givencos sin Here a, b, c ma. value c a b ( ) min. value c a b ( ) Find the ma imum and min imum value of cos sin Givencos sin Here a, b, c 0 ma. value c a b min. value c a b E lim inate from a cos, y bsin Given a cos, y bsin y cos, sin a b y a b cos, sin y cos, sin a b y a b a y b cos sin 8 8. Find asin e function whose period as The period of sin ais a given a a a the sine function sin ( ). If A B then prove that ( tan A)( tan B) Given A B tan( A B) tan tan A tan B tan A.tan B tan A tan B tan A.tan B tan A tan B tan A.tan B Adding on both side, we have tan A tan B tan A.tan B tan A tan B( tan A) ( tan A)( tan B) 6. Pr ove that tan 70 tan 0 tan 0 tan60 tan0 7. If tan 0, then showthat We know tan60. tan0 tan(70 0) tan 0 Given tan 0 cot 0 tan 70 tan 0 tan 0 tan60 tan0 tan 70.tan 0 LHS tan60. tan0 tan 70 tan 0 tan 0 tan(60 0) tan 0 tan(90 0) cot 0 tan 70.cot 70 tan 70 tan 0 tan 0 cot 0 cot (0) cot 0 tan 70 tan 0 tan 0

10 Giventhat sin h 9. If cos sin cos, we knowthat cosh sinh 9 cosh cosh cosh( ) sinh sinh( ) sinh.cosh If cosh, then find the values of cosh( ),sinh( ) Givencosh we knowthat cosh sinh sinh cosh sinh sinh 6 nowcosh( ) cosh sinh and sinh( ) sinh. cosh.. 6. P. T cosh sinh cosh( n) sinh( n) n e e e e LHS cosh sinh n e e e e cosh sinh n e cosh sinh e RHS cosh( n) sinh( n) n n n e e e e e e e e e n e L. H. S R. H. S n n n n n n n n cosh sinh cosh( n) sinh( n) n n prove that cos sin sin Given cos sin cos sin cos cos sin cos multiplying with on both side sin cos sin sin ( ) cos sin cos sin 60. If sin, where, Evaluate cos givensin and lies insec ound quadrant we know cos sin 9 6 cos cos cos now cos cos cos cos 6 cos 6 00 cos 6. Showthat tan h loge Weknowthat tanh ( ) loge tanh log e loge 6. If sinh showthat log 0 sol We knowthat. sinh loge given that sinh sinh e e log log 0 e

CLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. Type of Questions Weightage of Number of Total Marks each question questions

CLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. Type of Questions Weightage of Number of Total Marks each question questions CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (Matrices and Determinants) (iii) Calculus 44 (iv) Vector and Three dimensional Geometry 7 (v) Linear Programming