Study Material Class XII - Mathematics

Transcription

1 Study Material Class XII - Mathematics & 2 MARKS QUESTIONS PREPARED BY KENDRIYA VIDYALAYA SANGATHAN TINSUKIA REGION

2 Study Material Class XII Mathematics & 2 MARKS QUESTIONS CHIEF PATRON SH. SANTOSH KUMAR MALL COMMISSIONER,KVS PATRON DR. D. MANJUNATH DEPUTY COMMISSIONER(I/C),KVS TINSUKIA REGION COORDINATOR SH. RAJIB DAS PRINCIPAL, KV DULIAJAN RESOURCE PERSONS PGT MATHS OF KVS TINSUKIA REGION 1 P a g e

3 TABLE OF CONTENTS TOPIC- RELATIONS AND FUNCTIONS... 3 TOPIC-: INVERSE TRIGONOMETRIC FUNCTIONS... 8 TOPIC-: MATRICES TOPIC-: DETERMINANTS TOPIC: CONTINUITY TOPIC- DIFFERENTIABILITY TOPIC-: APPLICATION OF DERIVATIVES TOPIC-: INDEFINITE AND DEFINITE TOPIC-: DIFFERENTIAL EQUATION TOPIC-: VECTOR ALGEBRA TOPIC-: 3 DIMENSIONAL GEOMETRY TOPIC-: LINEAR PROGRAMMING TOPIC-: PROBABILITY P a g e

4 TOPIC- RELATIONS AND FUNCTIONS Sample Questions on the Topic Relations and Functions (1 Mark Questions) 1 Find the number of One One functions from a finite set A to A. 2 Let A = {4, 5, 0}. Find the number of binary operations that can be defined on A. 3 Let f: R R be a function defined by f(x) = x 2 3x + 4, for all x R, find the value of f 1 (2) 4 Let be a binary operation on N defined by a b = a b, find the value of (3 2) 1. 5 Find the number of onto functions from the set {1, 2, 3, 4 n} to itself. 6 If f: R R is given by f(x) = (3 x 3 ) 1/3, then find f(f(x)). 7 Find fog(x), if f(x) = x and g(x) = 5x 2 8 If f(x) = x + 7 and g(x) = x 7, x R then find fog (7). 9 Let be the binary operation on N given by a b = LCM (a, b) for all a, b N Find The binary operation : RxR R is defined as a b = 2a + b. find (2 3) What is the range of the function f(x) = x 1 (x 1)? 12 Write fog if f: R R and g: R R are given by f(x) = 8x 3 and g(x) = x 1/3 13 If X and Y are two sets having 2 and 3 elements respectively then find the number of functions from X to Y. 14 Let be a binary operation on N given by a b = HCF of a, b where a, b N write the value of If f: R R defined by f(x) = 3x + 2 define f (f(x)). 16 Let A = {1,2,3}, B = {4,5,6,7} and let f = {(1,4), (2,5), (3,6)} be a function from whether f is one one or not. 17 If the binary operation on the set Z of integers is defined by a b = a + b 5, then write the identity element for the operation in Z. 18 If R = {(x, y): x + 2y = 8} is a relation on N write the range of R. 19 Find the gof if f: R R and g: R R are given by f(x) = cosx and g(x) = 3x If gof is onto then is it necessary that f and g both onto? OR what? 21 Give example of two functions f: N Z and g: Z Z such that gof is injective but g is not injective. 22 Give example of the functions f: N N and g: N N such that gof is onto but f is not onto. 3 P a g e

5 23 Let f be the exponential function and g be the logarithmic function. What is fog (1)? 24 Consider the binary operation on the set {1, 2, 3, 4, 5} defined by a b = min {a, b}. Write the operation table. 25 State that given function is surjective or not, f: N N, given by f(x) = x 2 Sample Questions on the Topic Relations and Function (2 Marks Questions) 1 State the reason why the relation R = {(a, b): a b 2 } on the set R of real no. is not reflexive. 2 If be a binary operation on the set R of real numbers. defined by a b = a + b 2, then find the identity element of the binary operation 3 Find fog when f: R R and g: R R are defined by f(x) = 2x + 1 3, g(x) = x If the binary operation is defined on Q as a b = 2a + b + ab, for all a, b Q find the value of 3 4 and Show that f: R R Defined by f(x) = sinx is neither one one nor onto. 6 If the binary operation defined on the R as a b = a + ab then find whether binary is commutative or not. 7 Check whether the solution R on R defined as R = (a, b): a b 3 ) is reflexive. 8 Let be a binary operation on the set of real numbers. If a b = a + b ab, 2 (3 x) = 7, find the value of x. 9 Show that the function f: R R given: f(x) = e x, is one one but not onto. 10 If f(x) = x 2 +3, g(x) = 3x + 2. Find gof. 11 Show that the logarithmic function f: R + R given by f(x) = logx, x > 0 is one one. 12 Determine whether the function defined by a b = a b is binary or not on N 13 Determine whether the relation R on Z = {x, y; x y is an integer} is reflexive 14 Show that the function f: Q Q defined by f(x) = 2x 6 is one one. 15 Let A = {1, 2,3} and defined R = {(a, b): a b = 12}, Show that R is an empty relation on set A. 16 Determine whether the function f(x) = x + x f R R is (a) One One (b) Onto 17 Check whether the relation R in the set {1,2,3} given by R = {(1,2), (2,1)} is transitive 18 If f(x) is an invertible function, then find the inverse of f(x) = 3x Show that f(x) = 2x is one one and ontofor x R 4 P a g e

6 20 If f(x) = log x 1, 1 < x < 1 then show that f( x) = f(x) x If f(x) = [x] and g(x) = x then evaluate fog ( 5 ) gof ( 5 ) Let f: f: R R be defined by f(x) = 3x + 4 find the inverse of f if exist. 23 Show that : R R R given by a b = a + 2b is not associative. 24 Show that subtraction and division are not binary on N. 25 Show that an onto function f: {1, 2, 3} {1,2,3}is always one one. 1 n!, where n(a)=n 5 P a g e RELATIONS AND FUNCTIONS (1 MARK QUESTIONS ANSWER) 2 The number of binary operations on the set consisting n elements is given by n n2, therefore required operations are or 2 4 (3 2) 1=9 5 n! 6 f(x) = (3 x 3 ) 1 3 then find f(f(x)) f [(3 x 3 ) 1 3] = [3 ((3 x 3 ) 1 3) 3 3 ] = (x 3 ) 1 3 = x 7 fog(x) = f(g(x)) = f( 5x 2 ) = 5x 2 8 fog(7) = f(g(7)) = f(7 7) = f(0) = = = LCM of 5 and 7 = As a b = 2a + b. find (2 3) 4=( ) 4 = 7 4 = = f(x) = x 1 (x 1) (x 1)if x 1 > 0 or x > 1 x 1 = f(x) = { (x 1)if x 1 < 0 or x < 1, For x > 1, f(x) = 1, For x < 1, f(x) = 1 Range of f(x) = { 1,1} 12 fog(x) = f(g(x)) = f (x 1 3) =8(x 1/ 3 ) 3 = 8x 13 Number of functions from X to Y is 3 2 = 9 14 As a b = HCF of a, b, then 22 4 = HCF of 22 and 4 is = 2 15 f(f(x) = f(3x + 2) = 3(3x + 2) + 2 = 9x f is one one because f(1) = 4, f(2) = 5, f(3) = 6, no two elements of A have same f image. 17 Let e Z be the required identity a e = a a + e 5 = a e = 5 18 {1,2,3} 19 gof(x) = 3cos 2 x

7 20 f is not necessarily be onto but g must be onto 21 f(x) = x, g(x) = x 22 x 1 if x > 1 f(x) = x + 1, g(x) = { 1 if x = not surjective * e=2 6 P a g e RELATIONS AND FUNCTIONS (2 MARKS QUESTIONS ANSWER) > 2 (1 2 )2 ( 1, 1 ) R, Hence R is not reflexive fog = 2x (i) 3 4 = = 22, (ii) 2 5 = = 19 5 Since f(0) = f(π) but 0 π Therefore f is not one one and Range of f = [ 1,1] R therefore f is not onto. 6 a b = a + ab and b a = b + ba, Therefore a b b a, so it is not commutative 7 Since a < a 3 is not true for a = 1, (a, a) R 2 8 x = 3 9 Let a, b R, f(a) = f(b), e a = e b a = b, so it is one one Range of f = R + codomain, So it is not onto 10 3x Show for one one, let a, b R + f(a) = f(b) log a = log b a = b 12 a b = a b, 5,2 N, but 2 5 = 3 N so is not binary. 13 For (a, a) Z, a a = 0 is an integer. So it is reflexive. 14 Let x 1, x 2 Q, f(x 1 ) = f( x 2 ) 2x 1 6 = 2 x 2 6 x 1 = x 2, so it is one one. 15 There does not exist any elements in A for which a b = 12 so it is an empty relation.

8 ie. 1 1 = 0 12 (1, 1) R, 1 2 = 1 12 (1, 2), (2, 1) R, 1 3 = 2 12 (1, 3), (3, 1) R, 2 3 = 1 12 (3, 2), (2, 3) R, 16 Neither One One nor Onto 17 R is not transitive as (1,2) R, (2,1) R but (1,1) R So it is not transitive 18 Let y = f(x) = 3x 2 5., then D f = R, R f = R 5y=3x-2, x = 5y+2, x, y R f 1 (y) = 5y Let a, b R, f(a) = f(b) 2a = 2b a = b, So it is one one Let y R, y = 2x x = y R (domain), So f is onto f( x) = log 1 + x log 1 x = log 1 x log 1 + x = f(x) 21. fog( 5 ) = 2, gof( 5) = 3, Ans is Show one one and onto then f -1 (x) = x 4 23 Take 3, 5, 8 as real no. then(3 5) 8 = 29 and 3 (5 9) = 45, hence is not associative 24 : N N N is given by x y = x y which is not a binary operation if x < y (for eg. x = 4, y = 6) 3 Similarly for division (x, y) = x y may not be a natural no. (for eg. x = 4, y = 6) 25 Suppose f is not one one. Then there exists two elements say 1 and 2. Also the image of 3 under f can be only one element. Therefore the range set can have at most two elements of the codomain {1, 2, 3}. showing that f is not onto a, contradiction. Hence f must be one one. 7 P a g e

9 Q.NO TOPIC-: INVERSE TRIGONOMETRIC FUNCTIONS 1 MARK PROBLEMS 1. If y = sin 1 x then write the range of y 2. Find the principal value of sin 1 ( 1 2 ) + sin 1 ( 1 2 ) 3. Find the principal value of cos 1 ( 3 2 ) + cos 1 ( 1 2 ) 4. Find the principal value of tan 1 ( 3) cot 1 ( 3) 5. Find the principal value of tan 1 ( 1) + sec 1 ( 2) 6. Find the principal value of cosec 1 (2) cosec 1 ( 2) 7. Find the principal value of sec 1 ( 2 3 ) 8. Find the principal value of cot 1 ( 1 3 ) 9. Find the value of tan 1 (1) + cos 1 ( 1 2 ) + sin 1 ( 1 2 ) 10. Find the value of cos ( π 2 sin ) 11. Find the value of sin 1 (sin 3π 5 ) 12. Find the value of cos 1 (cos 13π 6 ) 13. Find the value of cos 1 (cos 7π 6 ) 14. Find the value of tan 1 (tan 7π 6 ) 15. Find the principal value of cos 1 (cos( 680 )) 16. Find the value of tan 1 (tan 3π 4 ) 17. Find the value of sin 1 (cos ( 43π 5 )) 18. If sec 1 (2) + cosec 1 y = π If tan cot 1 x = π 2 then find the value of y. then find the value of x 20. Find the value of sin ( π 3 sin 1 ( 1 2 )) 21. If sin (sin cos 1 x) = 1, then find the value of x 22. If tan 1 x + tan 1 y = π 4 ; where xy <1,find the value of x + y + xy 8 P a g e

10 23. Find the value of sin 1 ( 3 2 ) + tan 1 ( 3) 24. Find the value of tan 1 [2 cos (2 sin )] 25. Find the value of tan (2 tan 1 ( 1 5 )) INVERSE TRIGONOMETRIC FUNCTIONS Q.NO 2 MARK PROBLEMS 1. Show that tan 1 ( 1 2 ) + tan 1 ( 2 11 ) = tan 1 ( 3 4 ) 2. Show that tan 1 ( 2 11 ) tan 1 ( 7 24 ) = tan 1 ( ) 3. Show that 2 tan 1 ( 1 2 ) + tan 1 ( 1 7 ) =tan 1 ( ) 4. Show that 2 sin 1 ( 3 5 ) = tan 1 ( 24 7 ) 5. Show that sin 1 (2x 1 x 2 ) = 2 sin 1 x = 2 cos 1 x 6. Prove that 3 sin 1 x = sin 1 (3x 4x 3 ), x [ 1 2, 1 2 ] 7. Prove that 3 cos 1 x = cos 1 (4x 3 3x), x [ 1 2, 1] 8. Prove that tan 1 (x) + tan 1 ( 2x 1 x 2) =tan 1 ( 3x x3 ), 1 3x 2 x < Prove that tan 1 ( x y ) tan 1 ( x y x+y ) = π Prove that tan 1 x = 1 2 cos 1 ( 1 x ), x [0,1] 1+x 11. Find the value of tan (sin cot ) Find the value of tan 1 2 [sin 1 2x 1+ x Simplify tan 1 ( 1+ x2 1 ), x 0 x 1 y2 2 + cos 1 2], x < 1, y > 0 and xy < 1 1+y 14. Simplify tan 1 ( x a 2 x 2) 15. Simplify tan 1 ( 3a2 x x 3 a 3 3ax 2), x < a a, a > 0; < x < a Simplify cot 1 ( 1 2 1), x > 1 x 9 P a g e

11 17. Simplify tan 1 1 cos x ( ) 1+cos x, 0 < x < π 18. Simplify tan 1 cos x ( ), 3π < x < π 1 sin x Simplify tan 1 ( cos x sin x cos x+sin x ), 0 < x < π 20. Simplify tan 1 a cos x b sin x ( ),if a tan x > 1 b cos x +a sin x b 21. Solve 2 tan 1 (cos x) = tan 1 (2 cosec x) 22. Solve 2tan 1 (sin x) = tan 1 (2 sec x) 23. Solve tan 1 (2x) + tan 1 (3x) = π Solve cos(tan 1 x) = sin (cot 1 ( 3 4 )) 25. If tan 1 ( x 1 ) + x 2 tan 1 x +1 ( ) = π x + 2 4,then find the value of x INVERSE TRIGONOMETRIC FUNCTIONS Q.NO SOLUTIONS FOR 1 MARK QUESTIONS 1. π y π OR y [ π, π ] sin 1 ( 1 2 ) + sin 1 ( 1 2 ) = sin 1 ( sin π 6 ) + sin 1 (sin π 4 ) = π 6 + π 4 = π cos 1 ( 3 2 ) + cos 1 ( 1 2 ) = cos 1 (cos π 6 ) + cos 1 ( cos π 4 ) = π 6 + 3π 4 = 11π 12 tan 1 ( 3) cot 1 ( 3) = tan 1 (tan π 3 ) cot 1 ( cot ( π 6 )) = π 3 5π 6 = π 2 tan 1 ( 1) + sec 1 ( 2) = π 4 + 2π 3 = 5π 12 cosec 1 (2) cosec 1 ( 2) = π 6 + π 4 = 5π 12 sec 1 ( 2 3 ) = π 6 cot 1 ( 1 3 ) = cot 1 ( cot π 3 ) = cot 1 (cot (π π 3 )) = 2π 3 tan 1 (1) + cos 1 ( 1 2 ) + sin 1 ( 1 2 ) = π 4 + 2π 3 π 6 = 3π 4 10 P a g e

12 cos ( π 2 sin ) = cos (cos ) = 3 7 sin 1 (sin 3π 5 ) = sin 1 (sin (π 3π 5 )) = 2π 5 cos 1 (cos 13π 6 ) = cos 1 (cos (2π + π 6 )) = π 6 cos 1 (cos 7π 6 ) = cos 1 (cos (2π 5π 6 )) = 5π 6 tan 1 (tan 7π 6 ) = tan 1 (tan (π + π 6 )) = π cos 1 (cos( 680 )) = cos 1 (cos(680 )) = cos 1 (cos(4π 40 ))= tan 1 (tan 3π 4 ) = tan 1 (tan (π π 4 )) = tan 1 (tan ( π π )) = 4 4 sin 1 (cos ( 43π 5 )) = sin 1 (cos ((8π + 3π 5 ))) = sin 1 (cos ( 3π 5 )) = sin 1 (sin ( π 2 3π 5 )) = π If sec 1 (2) + cosec 1 y = π If tan cot 1 x = π 2 then y = 2 then x = sin ( π 3 sin 1 ( 1 2 )) = sin (π 3 + π 6 ) = sin π 2 = If sin (sin cos 1 x) = 1,then x = tan 1 x + tan 1 y = π 4 tan 1 ( x+y 1 xy ) = π 4 x+y = 1 x + xy + y = 1 1 xy sin 1 ( 3 2 ) + tan 1 ( 3) = π 3 π 3 = 2π 3 tan 1 [2 cos (2 sin )] = π 4 tan (2 tan 1 ( 1 5 )) = tan (2 tan 1 ( 1 5 )) = tan θ where θ 2 = tan Now tan θ = 2 tanθ 2 1 tan 2θ 2 = P a g e

13 Q.NO SOLUTIONS FOR 2 MARKS QUESTIONS L. H. S = tan 1 ( 1 2 ) + tan 1 ( ) = tan 1 ( ) = tan 1 ( 3 ) = R. H. S 4 11 (formula: tan 1 x + tan 1 y = tan 1 x + y ( 1 xy ) ) tan 1 ( 2 11 ) tan 1 ( ) = tan 1 ( ) = tan 1 ( 29 ) = R. H. S (formula: tan 1 x tan 1 y = tan 1 x y ( 1 + xy ) 3. L.H.S = 2 tan 1 ( 1 ) + 2 tan 1 ( 1 ) 7 = tan 1 ( ( 1 2 )2) + tan 1 ( 1 7 ) apply formula: 2 tan 1 (x) = tan 1 ( 2x 1 x 2) tan 1 ( 4 3 ) + tan 1 ( 1 7 ) =tan 1 ( ) apply formula tan 1 x + tan 1 y = tan 1 ( x+y 1 xy ) =R.H.S 4. 2 sin 1 ( 3 5 ) = sin 1 ( ( 3 5 ) 2 ) = sin 1 ( ) = tan 1 ( 24 7 ) ( formula 2 sin 1 (x) = sin 1 (2x 1 (x) 2 ) ) 5. Let x = sin θ then θ = sin 1 x now sin 1 (2x 1 x 2 ) = sin 1 (2 sin θ 1 sin 2 θ) = sin 1 (2 sin θ cos θ) = sin 1 (sin 2θ) = 2θ = 2 sin 1 x Similarly By taking x = cos θ, we get sin 1 (2x 1 x 2 ) = 2 cos 1 x 6. As above problem by taking x = sin θ we will get 3 sin 1 x = sin 1 (3x 4x 3 ) Use formula sin 3θ = 3 sin θ 4 sin 3 θ 7. As above problem by taking x = cos θ we will get 3 cos 1 x = cos 1 (4x 3 3x) Use formula cos 3θ = 4 cos 3 θ 3 cos θ 8. let x = tan θ then θ = tan 1 x 12 P a g e

14 tan 1 3x x3 ( 1 3x 2) = tan 1 ( 3 tan θ tan3 θ 1 3 tan 2 θ ) = tan 1 (tan 3θ) = 3θ = 3 tan 1 x 9. Apply the formula tan 1 x tan 1 y = tan 1 ( x y ) and then simplify 1+xy R. H. S = 1 2 cos 1 ( 1 x 1 + x ) = 1 2 cos 1 ( 1 ( x)2 1 + ( x) 2) = tan 1 x = tan 1 x = LHS tan (sin cot 1 2 ) = tan 3 (tan tan 1 ) = tan (tan ) = 17 6 tan 1 2x 1 y2 2 [sin 1 + cos x2 1 + y 2] =tan 1 2 [2 tan 1 x + 2 tan 1 y] = tan[tan 1 x + tan 1 y] = tan (tan 1 ( x+y 1 xy )) = x+y 1 xy 13. let x = tan θ then θ = tan 1 x tan 1 ( 1 + x2 1 x ) = tan 1 ( 1 + tan2 θ 1 1 cos θ 1 ) = tan = tan 1 (tan θ tan θ sin θ 2 ) = θ 2 = 1 2 tan 1 x 14. Let x = a sin θ then θ = sin 1 x a tan 1 x ( a 2 x 2) = a sin θ tan 1 ( a 2 a 2 sin 2 θ ) = tan 1 sin θ ( cos θ ) = tan 1 (tan θ) = θ = sin 1 x a 15. ANS: 3 tan 1 x a 13 P a g e Hint:By taking x = a tan θ then θ = tan 1 ( x ) and then substitute and simplify as a above problem 16. Ans: sec 1 x By taking x = sec θ then θ = sec 1 x and then substitute and simplify

15 tan 1 1 cos x ( 1 + cos x ) = tan 1 ( 2 x sin2 2 2 cos 2 x = tan 2) 1 (tan x 2 ) = x 2 tan 1 cos x ( 1 sin x ) = tan 1 ( sin ( π 2 x) 1 cos ( π 2 x)) = tan 1 (tan ( π 4 + x 2 )) = π 4 + x 2 tan 1 cos x sin x ( cos x + sin x ) by dividing numerator denominator with cos x tan 1 1 tan x ( ) 1+tan x = tan 1 (tan ( π 4 x)) = π 4 x 20. tan 1 a cos x b sin x ( b cos x + a sin x ) Divide with b cos x both numerator and denominator we get a tan x b tan 1 ( 1+ a b tan x) formula: tan 1 x + tan 1 y = tan 1 ( x+y ) ) 1 xy =tan 1 a b tan 1 (tan x) =tan 1 a b x tan 1 (cos x) = tan 1 (2 cosec x) tan 1 2 cos x ( 1 cos 2 x ) = tan 1 (2 csc x) tan 1 2 cos x ( sin 2 x ) = tan 1 ( 2 sin x ) cot x = 1 x = π tan 1 (sin x) = tan 1 (2 sec x) tan 1 2 sin x ( 1 sin 2 x ) = tan 1 ( 2 cos x ) 14 P a g e

16 tan x = 1 implies x = π tan 1 (2x) + tan 1 (3x) = π 4 5x tan 1 ( 1 6x 2) = π 4 By simplifying x = 1 6 or x = 1 formula: tan 1 x + tan 1 y = tan 1 ( x+y 1 xy ) ) x = 1 does not satisfying the given equation Therefore x = cos(tan 1 x) = sin (cot 1 ( 3 4 )) tan 1 x = π 2 cot 1 ( 3 4 ) π 2 cot 1 x = π 2 cot 1 ( 3 4 ) x = tan 1 ( x 1 x 2 ) + tan 1 ( x + 1 x + 2 ) = π 4 formula: tan 1 x + tan 1 y = tan 1 x + y ( 1 xy ) After applying formula and simplify then we get 2x 2 4 = 3 2x 2 = 1 x = ± P a g e

17 TOPIC-: MATRICES 1 mark questions: 1. If A is the square matrix of order 3 and 2A = k A. Find the value of K? 2. Write the element a 12 of the matrix A = [a ij ] 2x2 Whose elements a ij are given 3. by a ij = e 2ix. sinjx If A AdjA = [ 0 7 0] then find AdjA If A is an invertible matrix of order 3x3 and ӀAӀ =9. Then find Adj AdjA 5. If a matrix has 18 elements, what are the possible orders it can have? 6. If [ x ] [x 1 ] = [19 ], find the value of x Find the value of x and y that makes the following pair of matrices equal. x + 3y y 1 [ ] = [4 7 x ] 8. If 2 [ 3 4 y ] + [1 5 x 0 1 ] = [ 7 0 ], then find x y If A is a 3x3 matrix, whose elements are given by a ij = 1 I-3i + ji, then write the value of 3 a If matrix A = [1 2 3] write AA. Where A is the transpose of matrix A If A= [ ] and B= [ ] then find a 22 +b If A =[aij] is a 2x2 matrix such that aij = i 2 j Let A =[ ] and C = [ 2 ] find 3A C x 1 If [ 0 1 0] [ y] = [ 0], find x,y,z z If A T = [ ] and B = [ 1 ], find ( A + 2B )T If A = [ 2 3 ], find A + AT If A = [ a c b d ],Verify that (A - AT )is a skew symmetric matric. 18. If A = [ ], find k such that A2 = 8A + KI 19. If A = [ ], find f(a), where f(x) = x2-5x P a g e

18 If A = [ 2 1 3] then find the value of A 2 3A + 2I Answers of 1 mark questions 1. k = a 12 = e 2x sin2x 3. A AdjA = 7I 4. AdjA = IAI n-1 = = x1, 9x2, 6x3, 3x6, 2x9, 1x x=5 7. x= 7, y = a 23 = AA = [1 2 3] [ 2] = [1 4 9] 3 11 a 22 +b 21 = 4-3=1. 12 [ ] [ 3 2 ] - [ ] = [ ] 14 x 1 [ y] = [ 0] Implies x=1, y =0, z= 1. z 1 15 A + 2B = [ ] + 2[ ] = [ ] ( A + 2B ) T =[ ]. 16 A + A T = [ ] + [ ] = [ ] 17 Let (A - A T ) = B, show that B T = -B. 18 K = P a g e

19 19 [ ] [ 2 1 3] MATRICES 2 mark questions: 1. Shows that skew symmetric matrix of odd order is always singular. 2. cosx sinx If A = [ ] then for what value of x, A is an identity matrix. sinx cosx 3. secx tanx secx Simplify tanx [ ] +secx [ tanx tanx secx secx tanx ] For what value of x, is the matrix A= [ ] a skew symmetric matrix? x Construct a 2 2 matrix where, a ij = I -2i +3j I. 6. x 3x y If [ 2x + z 3y w ] = [3 2 ], find x,y,z,w If A and B are symmetric matrices, show that AB is symmetric, if AB = BA. 8. Using elementary transformations, find the inverse of the matrix A= [ ] 9. Construct a 2x2 matrix, A = [aij] Whose elements are given by aij = 10. If A = [ 3 0 ], find A4 0 3 ( i + j)2 11. If A is a square matrix such that A 2 = A then write the value of (I + A ) 3 7A. 12. If A = [ ] and B = [ 3 ], find the matrix C such that A + B + C is a null matrix Express the matrix A = [ ] as the sum of a symmetric and a skew symmetric matrix. 14. If A =[ ], find k, so that KA2 5A 6I2= Given matrix A = [ 4 ] and B = [1 3 6] verify that (AB) T = B T A T 5 2a ij 18 P a g e

20 16. Given matrix A =[ ], find f (A), if f(x) = 2x 2-3x Find the matrix X such that [ 0 1 ] X = [ ] cos Ѳ i sinѳ If A = [ ] where i = 1, By the principle of Mathematical induction i sinѳ cos Ѳ prove that A n =[ cosn Ѳ i sinnѳ i sinnѳ cosn Ѳ ]. 19. Construct a matrix A = [aij] 2x2 whose elements aij are given by aij are given by aij e 2ix sinjx. 20. If A = [ 2 k k ] is a singular matrix, then find the value of 5k k2? 1. Let A be a skew-symmetric matrix of order n Then, A = -A IA I =I AI IAI = (-1) n IAI IAI = -IAI => 2IAI = 0 IAI=0, Hence A is a singular matrix. 2. cosx sinx If A = [ sinx cosx ] = I Answers of 2 mark questions If A = [ cosx sinx sinx cosx ] = = [ ] Cosx = 1, sinx = 0 X= 0 o. 3. [ ] 4. For a skew symmetric matrix, aij = aji, x = ( 2) = Required matrix = [ ] 6. x=3, y=7, z=-2, w= A =A, B =B and if AB is symmetric, then (AB) = AB (i) Also, (AB) = B A = BA (ii) From (i) and (ii), AB= BA 8. Using A = IA 19 P a g e

21 A -1 = [ ] 9 a11 = (1+1) 2 /2 = 2, a12 = (1+2) 2 /2 = 9/2, a21= (2+1) 2 /2 =9/2, a22 = (2+2) 2 /2 = 8 [ 2 9/2 9/2 8 ] 10. A 2 = [ ] [ ] = [ ] A 4 = [ ] [ ] = [ ] 11. ( I + A ) 3 7A = I 3 + 3I 2 A + 3IA 2 + A 3 7A = I +3A + 3A + A 7A = I 12. C = O ( A + B ) = [ ] - [ ] = [ ] / /2 1 Symmetric part = [ 5/2 5 5/2], Skew-symmetric part = [ 1/2 0 9/2] 0 5/ / K = Show that LHS = RHS [ ] X = [ ] 18. By using principle of mathematical induction get the result of A n. 19. A= [ e2x sinx e 2x sin2x e 4x sinx e 4x sin2x ] 20 5k- k 2 =4. 20 P a g e

22 TOPIC-: DETERMINANTS ONE MARK QUESTIONS: 1. Evaluate the following determinants: (one mark each) (i) sinθ cosθ (ii) cosθ sinθ cos15 sin15 (iii) sin75 cos75 cos80 cos10 (iv) sin80 sin10 x x + 1 (v) x 1 x a + ib c + id (vi) c + id a ib (vii) (viii) (ix) sinα cosα (x) sinα 0 sinβ cosα sinβ 0 2. If = 2x 4, Then find the value(s) of x. 6 x x If = 3, Then find the integral value(s) of x. 3x 2x 4. If 2x = 0, find x. 5x P a g e

23 5. Find the value of x for which 3 x x 1 = x x 6. If 1 x = 3 4, write the positive value of x Write the minor and cofactor of the element a21 of [ ] Find the value of a11a11 + a12a12 of [ ] Check whether the following matrix is singular [ 0 2 1] x Find the value(s) of x for which the matrix A = [ 1 0 3] is singular For what value of x, the matrix A = [ ] is singular? x Using determinants, find the area of the triangle whose vertices are (0, 0), (4, 3) and (8, 0). 13. Show that the points (1, 0), (6, 0) and (0, 0) are collinear. 14. Let A be square matrix of order 3 X 3. Write the value of 2A, where A = Let A is square matrix of order 3, A 0 and 3A = k A, then write the value of k. 16. If A and B are square matrices of order 3 such that A = 1 and B = 3, then find the value of 7AB. 17. If a matrix A of order 3 X 3 has determinant 2, then find the value of A (8I). 18. If A is a skew-symmetric matrix of order 3, write the value of det(a). 19. If A is a square matrix and A = 2, then write the value of AA, where A is the transpose of matrix A. 20. If A is a non-singular matrix such that A = 5, then find A If A is an invertible matrix of order 3 X 3 and A = 5, then find adj A. 22. If A is an invertible matrix of order 3 and adj A = 64, then find A. 22 P a g e

24 23. If A is a square matrix such that A(adj A) = 8I, where I denotes the identity matrix of the same order, then find the value of A. 24. If A is a square matrix of order 3 such that A = 3, then find A(adjA). 25. If A is an invertible matrix of order 3 X 3 and A = 7, then find adj (adja). 26. If A is an invertible matrix of order 3 X 3 and A = 4, then find adj(adja) 27. Use matrix method to examine the given system of equations for consistency or inconsistency. 5x + 2y = 2, 3x + 2y = Using properties of determinants, prove that = Using properties of determinants, prove that = Using properties of determinants, prove that = ANSWERS: (1 MARK QUESTIONS) 1. (i) 2 4 = (2)(2) ( 1)(4) = sinθ cosθ (ii) cosθ sinθ = (sinθ)(sinθ) (cosθ)( cosθ) = sin2 θ + cos 2 θ = 1. cos15 sin15 (iii) = (cos15 )(cos75 ) (sin15 )(sin75 ) = cos( ) = sin75 cos75 cos90 = 0. cos80 cos10 (iv) = (sin10 )(cos80 ) + (sin80 )(cos10 ) = sin( ) = sin90 = 1. sin80 sin10 x x + 1 (v) x 1 x = (x)(x) (x 1)(x 1) = x2 (x 2 1) = 1 a + ib c + id (vi) c + id a ib = (a + ib)(a ib) (c + id)( c + id) = (a2 i 2 b 2 ) ( c 2 + i 2 d 2 ) = (a 2 + b 2 ) ( c 2 d 2 ) = a 2 + b 2 + c 2 + d 2. (vii) = [(2)( 5) (4)(3)] 2 = ( 22) 2 = P a g e

25 2 1 2 (viii) = 2(0 5) + 1 (0 + 3) 2 (0 6) = (ix) = [2 (0 2) + 1 ( 1 4) + 3 ( 2 0)] 2 = ( 3) 2 = sinα cosα (x) sinα 0 sinβ = 0 sinα (0 cosα sinβ) cosα (sinα sinβ 0) cosα sinβ 0 = sinα cosα sinβ cosα sinα sinβ = Given, = 2x 4 6 x 18 = 2x x 2 = 6 x = ± 3 x Given, 3x 2x = 3 2x 2 4x + 9x = 3 2x 2 + 5x 3 = 0 (2x 1) ( x + 3) = 0 24 P a g e x = 1 2, 3 4. Given, 2x x = 0 (9)(2x + 5) 3(5x + 2) = 0 18x x 6 = 0 3x + 39 = 0 x = As 3 x x 1 = x 2 = 3 8 x 2 = 8 x = ±2 2 x x 6. As 1 x = x 2 x = 6 4 x 2 x 2 = 0 (x 2)(x + 1) = 0 x = 2, - 1 So, x = 2 (as we are asked only the positive value) 7. Minor of a21 = M21 = 2 Cofactor of a21 = A21 = ( 1) 2+1 M21 = 2.

26 8. a11a11 + a12a12 = (2)( 20) + ( 3)(46) = = let A = [ 0 2 1] P a g e then A = = 2(0 5) + 1(0 + 3) 2( 0 6) = 5 0. Therefore given matrix is nonsingular x Given that A = [ 1 0 3] is singular. A = x = x(0 + 3) 1(4 15) + 2( 1 0) = 0 3x + 9 = 0 x = Given that A = [ ] singular A = 0 x = 0 x 2 3 1( 6 2) + 2( 3 x) + 3(2 2x) = x + 6 6x = 0 8x = 8 x = Given vertices of triangle are (0,0), (4, 3) and (8, 0) Then, = = 1 [ (0 24)] = Hence, area of triangle = = 12 sq units 13. Given points are (1, 0), (6, 0) and (0, 0) Then, = = 1 [1( 0 0) (0 0)] = Hence, given points are collinear. 14. Given that A is a matrix of order 3 X 3 and A = 4. Then, 2A = 2 3 A [ ka = k n A, where A is n X n matrix] = (8)(4) = 32.

27 15. Given that A is a matrix of order 3 X 3 and 3A = k A We have, 3A = k A 3 3 A = k A [ ka = k n A, where A is n X n matrix] 27 A = k A k = Given that A and B are square matrices of order 3 such that A = 1 and B = 3 Then, 7AB = 7 3 AB [ ka = k n A, where A is n X n matrix] = 7 3 A B = (343) ( 1) (3) = Given that A is a square matrix of 3 X 3 and A = 2 Then, A(8I) = 8AI = 8A = 8 3 A = (512) (2) = Given that A is a skew symmetric matrix of order 3. A T = A A T = A A = ( 1) 3 A [ A T = A ] A = A 2 A = 0 A = Given that A is a square matrix such that A = 2 Then, A A = A A = A A = (2) (2) = A 1 = 1 A = Given, A is an invertible matrix of order 3 X 3 and A = 5 Now, adja = A n 1, where n = 3 = 5 2 = Given, A is an invertible matrix of order 3 X 3 and adja = 64 Now, adja = A n 1, where n = 3 64 = A 2 A = ± Given A is a square matrix of order 3 X 3 such that A (adja) = 8I Then, A I = 8I [ A(adjA) = A I] A = Given A is a square matrix of order 3 X 3 such that A = 3 26 P a g e

28 Now, A (adj A) = A n, where n = 3. = (3) 3 = Given, A is an invertible matrix of order 3 X 3 and A = 7 Now, adj (adj A) = A n 2 A, where n = 3 = (7) (3 2) A = 7A. 26. Given, A is an invertible matrix of order 3 X 3 and A = 4 Now, adj (adja) = A (n 1)2, where n = 3 = (4) (3 1)2 = 4 4 = Given system of equation is 5x + 2y = 2 3x + 2y = 5 Which can be written in the matrix form as AX = B where A =[ ], X = [x y ] and B = [2 5 ] Now, A = 5 2 = 4 0. So, the given system of equations is consistent L.H.S. = = (On taking 2 common from R2) =2(0) [as R1 and R2 are identical] = 0 = R.H.S L.H.S. = = (On taking 4 common from C1) = 4 (0) [as C1 and C2 are identical] = 0 = R.H.S L.H.S. = = [On applying R1 R1 + R2] = 0 [as R1 and R3 are identical]. TWO MARKS QUESTIONS: a b c 1. Without expanding, show that a + 2x b + 2y c + 2z = 0. x y z 27 P a g e

29 x sinθ cosθ 2. Prove that the determinant sinθ x 1 is independent ofθ. cosθ 1 x 3 y 3. If x 1 = 1 4, then find the possible values of x and y, where x, y N Using determinants, show that the points (a, b + c), (b, c + a) and (c, a + b) are collinear. 5. Using determinants, find a so that points (a, 2), (1, 5) and (2, 4) are collinear. 6. Using determinants, prove that α + β = αβ so that points(α, 0), (0, β) and (1, 1) are collinear. 7. Using determinants, find the equation of the line joining the points (1, 2) and (3, 6). 8. If A and B are square matrices of same order such that A = 6 and AB = I, then find the value of B 9. If A is a square matrix such that A T A = I, write the value of A. 10. IfA = [ x 2 2 x ] and A4 = 625, find the value(s) of x. 1 sinθ If = sinθ 1 sinθ, then prove that2 4, for all θ. 1 sinθ Find the adjoint of the matrix [ ]. 13. Write A 1 for A = [ ] If A is a matrix of order 3 X 3 and A(adjA) = [ 0 5 0], then find adja If A is a square matrix of order 3 X 3 such that adj(4a) = k adja, then find k. 16. If A is a non-singular matrix such that A 1 = [ ], then find (AT ) Use matrix method to examine the given system of equations for consistency or inconsistency: x+ 3y = 5 ; 2x + 6y = 8 28 P a g e

30 Without expanding prove that = x + y y + z z + x 19. Using properties of determinants, prove that z x y = Find the area of the triangle formed by (2, - 6), (5, 4) and (12, 4). ANSWERS (TWO MARKS QUESTIONS): a b c 1. L.H.S. = a + 2x b + 2y c + 2z x y z On applying R2 R2 R1 a b c = 2x 2y 2z x y z Taking 2 common from R2. a b c = 2 x y z x y z = 2(0) [as R2 and R3 are identical] x sinθ cosθ 2. We have, sinθ x 1 = x (x 2 1) sinθ ( x sinθ cosθ) + cosθ ( sinθ + xcosθ) cosθ 1 x = x 3 x + xsin 2 θ + sinθ cosθ sinθ cosθ + x cos 2 θ = x 3 x + x (sin 2 θ + cos 2 θ) = x 3 x + x = x 3, which is independent of θ. 3 y 3. Given, x 1 = xy = 3 8 xy = 8 Also, given that x and y are natural numbers. Possible values of x and y are x = 1, y = 8 or x = 2, y = 4 or x = 4, y = 2 or x = 8, y = Given points are (a, b + c), (b, c + a) and (c, a + b) a b + c 1 Then, = 1 b c + a 1 2 c a + b 1 [a(c + a a b) (b + c)(b c) + 1(ab + b2 c2 ac)] 29 P a g e = 1 2 = 1 2 [ac ab b2 + c 2 + ab + b 2 c 2 ac] = 0 Hence, given points are collinear.

31 5. Given points (a, 2), (1, 5) and (2, 4) are collinear a 2 1 Then, = = [a( 5 4) 2( 1 2) + 1( 4 10)] = 0 30 P a g e a = 0 a = 4 6. Given points(α, 0), (0, β) and (1, 1) are collinear. α 0 1 Then, = 1 0 β 1 = [α(β 1) 0 + 1(0 β)] = 0 [αβ α β]= 0 α + β = αβ 7. Let (x, y) be any point on the line joining (1, 2) and (3, 6) So the points (x, y), (1, 2) and (3, 6) are collinear. x y 1 Hence = = [x (2 6) y (1 3) (6 6)] = 0 [ 4x + 2y] = 0 2x y = 0 is the equation of the line joining (1, 2) and (3, 6). 8. Given A and B are square matrices of same order such that A = 6 and AB = I. Now, AB = I AB = I A B = 1 [ I = 1] 6 B = 1 B = Given that A is a square matrix such that A T A = I Now, A T A = I A T A = I A T A = 1 [As I = 1] A A = 1 [As A T = A ] A 2 = 1 A = ±1 10. Given that, A = [ x 2 2 x ] and A4 = 625 A 4 = 625 A 4 = 625 A = ±5

32 [ x 2 2 x ] = ±5 x 2 4 = ±5 x 2 4 = 5 or x 2 4 = 5 x 2 = 9 or x 2 = 1 x = ±3 1 sinθ Given, = sinθ 1 sinθ = 1(1 + sin 2 θ) sinθ ( sinθ + sinθ) + 1(sin 2 θ + 1) 1 sinθ 1 = 1 + sin 2 θ 0 + sin 2 θ + 1 = sin 2 θ We know that 1 sinθ 1 0 sin 2 θ 1 θ sin 2 θ 2 θ sin 2 θ 4 θ 2 4 θ θ 12. Let A = [ ] A11 = 3, A12 = 4, A21 = 1, A22 = 2 Hence adj A = [ ] 13. As A = [ ] Then, A = = 6 5 = 1 Also, A11 = 3, A12 = 1, A21 = 5, A22 = 2 So, adj A = [ ] Hence A 1 = 1 (adj 3 A) = [ 5 A 1 2 ] Given, A(adjA) = [ 0 5 0] 0 Then, A(adjA) = 5I 0 5 A I = 5I [ A(adjA) = A I] A = 5 now, adja = A n 1 adj A = 5 2 = Given that A is a square matrix of order 3 X 3 such that adj(4a) = k adja adj(4a) = k adj A 4 n 1 adja = k adja adj(λa) = λ n 1 adja 4 n 1 = k = k k = P a g e

33 16. Given, A 1 = [ ] Then, (A T ) 1 = a 1T = [ 15 3 T 2 51 ] = [ ] 17. Given system of equation is x + 3y = 5 2x + 6y = 8 Which can be written in the matrix form as AX = B where A =[ ], X = [x y ] and B = [5 8 ] Now, A = = 0. So, we will calculate (adja)b. A11 = 6, A12 = 2, A21 = 3, A22 = 1 So, adja = [ ] Now, (adja)b = [ ] [5 8 ] = [ 6 2 ] O Therefore, given system of equations is inconsistent and has no solution L.H.S. = C1 C1 C = Taking 8 common from C = = 8(0) [as C1 and C3 are identical] = 0 = R.H.S. x + y y + z z + x 19. L.H.S. = z x y On applying R1 R1 + R2 x + y + z x + y + z x + y + z = z x y Taking (x + y + z) common from R = (x + y + z) z x y = (x + y + z) (0) [as R1 and R3 are identical] = Let the given points are A (2, - 6), B (5, 4) and C (12, 4). 32 P a g e

34 2 6 1 So, ar( ABC) = [2(4 4) + 6(5 12) + 1(20 48)] = 1 2 = 1 2 [ ] = 1 2 [ 70] = [ 35] Since area of a triangle can t be negative So ar( ABC) = 35 sq. units 33 P a g e

35 TOPIC: CONTINUITY SAMPLE QUESTION OF 1 MARK 1 Prove that the functionf(x) = 5x 3 is continuous at x = Find the point of discontinuity of the function defined by f(x) = x Check the continuity of the function f(x) = x n is at x = n where n is a positive integer. tan2x if x 0 4. For what value of k is the function f(x) = { x k ifx = cosx, when x 0 A function f is defined by : f(x) = { x 2 A, when x = 0 Find the value of A. sin 1 x A function f is defined by f(x) = { + e x when x 0 x 2 when x = 0 Show that f is continuous at x = 0 x+5 continuous at x = 0. if x < 0 7. Find all points of discontinuity of f, where f is defined by:f(x) = { x 1 if x 0 8. Find all points of discontinuity of f, where f is defined by: f(x) = { x3 3 if x 2 x if x > 2 9. x Find all points of discontinuity of f, where f is defined by:f(x) = { x if x 0 0 if x = Find all points of discontinuity of f, where f is defined by:f(x) = { x10 1 if x 1 x 2 if x 1 2x if x < Is the defined by: f(x) = { 2 if x = 2 x 2 ifx > 2 discontinuous at x = Is the function continuous at x = 1defined by f(x) = 2x Is the function continuous at x = 0 defined by f(x) = x Is the function continuous at x = 0 defined by f(x) = x 15. Is the function defined by f(x)=x 3 + x 2 1 for all R. 16. What is the interval over which f(x) = sin x is continuous. 17. At what point function is discontinuous;f(x) = { 3x 2 x 0 x + 1 x > 0 x 34 P a g e

36 5x 4 if x Is the function continuous at x = 1: f(x) = { 4x 2 3x if x > Is the function continuous at x = 3: f(x) = { 2x2 5x 3 x 3 0 x = Is the function continuous at x = 0: f(x) = { x x 0 x 2 x 0 ANSWERS OF 1 MARK QUESTIONS 1 LHL=RHL= f(x)=22 2 Function is continuous at each real number other than( 5) 3 Continuous at x = n 4 K=2 5 A=1/2 6 LHL=RHL= f(0)=2 7 No point of discontinuity 8 No point of discontinuity 9 Discontinuous at x = 0 10 Discontinuous at x = 1 11 Discontinuous at x = 2 12 Continuous at x = 1 13 Continuous at x = 0 14 Continuous at x = 0 15 Function is continuous 16. (, ) 17. At x = Yes, continuous at x = Yes, continuous at x = Yes, continuous at x = 0 SAMPLE QUESTION OF 2 MARKS: 1 Is the function defined by f(x) = x 2 sinx + 5 continuous at x = π? 2 Discuss the continuity of the function f(x) = sin x + cos x 35 P a g e

37 3 Discuss the continuity of the function f(x) = sin x cos x 4 Discuss the continuity of the function f(x) = sin x. cos x x + k x < 3 5 If f(x) = { 4 x = 3 is continuous at x = 3, then find the value of k. 3x 5 x > 3 6 If f(x) = { x 3 2x + k 36 P a g e x 2 9 if x 3 otherwise is continuous at x=3, then find the value of k. sin x 7 Examine the continuity of the function: f(x) = { x 0 x 1 x = 0 8 Check the continuity of the functionf(x) = x 5 at x = 5 ax + 5 ifx 2 9 Find the value of a, so that f(x) = { x 1 if x > 2 at x = 0 is continuous at x = 2. x + 2 if x 1 10 Find the point of discontinuity of the function f defined by f(x)= { x 2 if x > x +x 2 For f(x) = { x x 0 0 x = 0 12 examine the continuity of f(x) at x=0. Discuss the continuity of the function at the point a defined by 1 cos x x 0 13 If f(x) = { x 2 k x = 0 14 f(x) = x 3 a 2 a { 2a a3 x 2 0 < x < a x = a x > a is continuous at x = 0, then find the value of k x + 5 if x 1 Is the function defined by f(x) = { a continuous function? If not, x 5 ifx > 1 find the point(s) of discontinuity. 15 Examine the continuity of the function f(x) = x at x = 1 16 Examine the continuity of the function f(x) = 1 17 x+3, x R. Give an example of the function which is continuous at x = 1, but not differentiable at x = 1 18 State the points of discontinuity for the function f(x) = [x], in 3 < x < Is the function f(x) = 2x x is continuous at x = For what value of k is the function f(x) = { x2 3 if x 3 k if x = 3 continuous at x = 3

38 ANSWERS OF 2 MARKS QUESTIONS: 1 Yes, function is continuous at x = π 2 Function is continuous for all R. 3 Function is continuous for all R. 4 Function is continuous for all R. 5 K=1 6 K=0 7 Function is discontinuous at x = 0 8 Function is continuous at x = 5. LHL=RHL=f(5) = 0, 9 LHL=RHL= f(2), 2a + 5 = 1 = 2a + 5 therefore a = ( 2) 10 x = 1 is the only point of discontinuity of f. 11 Function is discontinuous at x = LHL=RHL= f(a)=a Hence f(x)is continuous at x = a. 13 k = Function is not continuous at x=1. 15 Function is continuous. 16 For x = 3 function is not defined. Hence, not continuous for x R. 17 Absolute value function f(x) = x 1 is continuous at x = 1 but not differentiable at x = f(x) = [x] is not continuous for integers. Hence, not continuous at x = ±2, ±1,0 19 LHL=RHL= f(0), function is continuous at x = k = 6 ********************************************************************************** 37 P a g e

39 S.No. TOPIC- DIFFERENTIABILITY 1 Marks Questions Question 1. Differentiate: e logx2 with respect to x. 2. Differentiate: 5 sin x w.r.to x 3. For what value of x, f(x) = 2x 7 is not derivable? 4. Write the derivative of log x 5, x > Write the derivative of Sin (x 7 +1) w.r. to x. 6. Differentiate Sin (log x), w. r. to x. 7. Differentiate cos 1 x, w.r. to x. 8. Differentiate e m tan 1 x, w.r. to x. 9. Differentiate e x+3, w.r. to x. 10. Differentiate cos x, w. r. to e x 11. Differentiate log 7 (logx), w.r. to x. 12. Write the Second derivative of log x, with respect of x. 13. For what value of x, f (x) = x is not differentiable? 14. For what value of x in (0, 2), f (x) = [x] is not differentiable? 15. Write an example of a function which is continuous at any point but not differentiable at same point. 16. Write the statement of Rolle s Theorem. 17. Write the statement of Mean Value Theorem. 18. Differentiate cos 1 (sin x), w.r. to x. 19. Find dy dx if x=t2, and y= t What is the derivative of sin x w. r. t. Cos x? 21. If y = cos(logx + e x ) find dy dx 22. If f(x) = cos x, find f ( 3π 4 ) 23. If f(x) = cos x sinx, find f ( π 6 ) 38 P a g e

40 24. If x 2 + y 2 = c 2 then write the value of dy 25. If y = tan x then find its derivative. dx ANSWERS FOR 1 MARK SAMPLE PROBLEMS 1. 2x 2. 5 sin x log e 5 cosx 3. X = log5 x(logx) x 2 cos(x 7 + 1) 6. cos(logx) x 10 e x Sinx x log7logx 1 2 x 1 x x 2 8. me m tan 1 x 1+x 2 9. e x+3 2 x X=0 14. X=1 15. F(x) = x at x= dy t 20. cot x 21. = 2 dx (1 + x ex ) sin(logx + e x ) dy (1 + 3) 24. = x 2 dx y 25. dy = sec2 x dx 2 x 2 Marks Questions 1 If y= sin 1 [x 1 x x 1 x 2 ] and 0<x<1, then find dy dx. 2 Differentiate, 2 cos2 x w.r.t. x 3 Differentiate, sin 1 ( )w.r.t. x x Find dy dx when x and y are connected by the relation given as tan 1 (x 2 + y 2 ) = a. 5 Find dy dx 39 P a g e Sin x= of a function expressed in parametric form 2t, tan y= 2t 1+ t 2 1 t 2 6 If x= e x y, Prove that dy = x y dx x log x 7 If y =tan 1 x, find d2 y in terms of y alone. 2 8 If x= 4t and y = 4 t dx then find dy dx. 9 Differentiate 9 9x w.r. t. x

41 10 Differentiate cos 1 1 cos x w.r.t. x 2 11 Differentiate cot 1 ( a+x a x ) w.r.t. x Differentiate tan 1 ( 1+x2 1 ) w.r.t. x x Differentiate tan 1 x + a ( ) w.r.t. x 1 ax 14 Differentiate tan 1 ( 4 x 1 4x ) w.r.t. x 15 If y= tan 1 5x, 1 <x< 1 dy then prove that = x dx 1+4x 2 1+9x 2 16 Find second order derivative of e x sin 5x 17 If y= A sin x + B Cos x, then find the value of d2 y dx 2 + y 18 Find dy for the function 2x + 3y = sin y dx 19 If y = tan x + tan x + tan x + then find dy dx 20 Examine the applicability of Rolle s Theorem for the function f(x) = x 2 4 on [ 2,2] 21 If x = a cos θ and y = b sin θ find dy dx at θ = π 3 22 If e x + e y = e x+y Prove that dy dx = ey x 23 Find the value of C in the Rolle s Theorem for the function f(x) = x 3 3x in x [0, 3] 24 For the function f(x) = x + 1 x Theorem. in x [1,3].Find the value of C for Mean Value 25 For the function f(x) = x 3 5x 2 3x in x [1,3].Find the value of C for Mean Value Theorem. 1 Y= sin 1 x- sin 1 x 40 P a g e dy = 1-1 dx 1 x 2 2 x 1 x 2 dy dx = -2cos2x sin2x log2 Answers for 2 Marks Questions

42 3 dy dx = 1 2 x(1 + x) 4 dy dx = x y 5 Hint; Put t= tan θ then find dx dt and dy dt. dy dx = 1 6 Take log on both sides then it will be y = x 7 d 2 y dx 2 = 2 sin ycos3 y 8 dy = 4, dx dy =4 then dt t 2 dt dx = 1 t 2 logx then diff w. r. t. x 9 dy dx = 99x 9 x (log 9) 2 10 dy = 1 dx 2 11 dy dx = a a 2 +x 2 Hint; cot 1 ( a+x a x ) = tan 1 ( a x a+x ) = tan 1 1 tan 1 x a 12 dy dx = 1 2(1+x 2 ) Hint; Put x= tan θ then simplify 13 dy dx = 1 2 x(1+x) Hint; tan 1 x + a ( ) = 1 ax tan 1 x + tan 1 a 14 dy = 2 Hint; tan 1 ( 4 x ) = dx x(1+4x) 1 4x 2 tan 1 2 x then differentiate 15 Hint y= tan 1 3x+tan 1 2x then prove it 16 d 2 y dx 2=2ex (5cos 5x- 12 sin 5x) 41 P a g e

43 dy dx = 2 cos y 3 19 dy = sec2 y dx 2y 1 20 Show all the three conditions 21 dy dx = b 3a 23 C = 1 24 C = 3 25 C = P a g e

44 TOPIC-: APPLICATION OF DERIVATIVES 1 MARK QUESTION: 1 The side of a square is increasing at a rate of 0.2 cm/sec. Find the rate of increase of perimeter of the square. 2 The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference? 3 If the radius of a soap bubble is increasing at the rate of 1 cm / sec. At what rate 2 its volume increasing when the radius is 1 cm. 4 A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing? 5 The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x x Find the marginal revenue when x = 7. 6 If a manufacturer s total cost function is C(x) = x + x 2, where x is the output, find the marginal cost for producing 20 units. 7 If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere. 8 Find the rate of change of the area of a circle with respect to its radius r when r = 4cm. 9 Find the value of k for which the function f(x) = x 2 kx + 6, x > 0 is strictly increasing. 10 Write the interval for which the function f(x) = cos x, 0 x 2π is decreasing. 11 What is the interval on which the function f(x) = log x is increasing? 12 For which values of x, the functions y = x x3 is increasing? 13 Write the interval for which the function f(x) = 1 is strictly decreasing. x 14 Find the sub-interval of the interval (0, π ) in which the function f(x) = sin 3x is 2 increasing. 15 It is given that at x = 1, the function f(x) = x 4 62 x 2 + kx + 9 attains its maximum value is the interval [0, 2]. Find the value of k. 16 What is the slope of the tangent to the curve f = x 3 5x + 3 at the point whose x co-ordinate is 2? 17 At what point on the curve y = x 2 does the tangent make an angle of 45 with the x-axis? x 43 P a g e

45 18 Find the equation of the normal to curve x 2 = 4y which passes through the point (1, 2). 19 What is the slope of the normal to the curve y = 5x 2 4 sin x at x = If the curves y = 2e x and y = ae x intersect orthogonally (cut at right angles). What is the value of a? 21 Find the slope of the normal to the curve y = 8x 2 3 at x = For the curve y = (2x + 1) 3 find the rate of change of slope as x = Find the slope of the tangent to the curve y = 3x 4 4x at x=4. 24 If y = log e x, then find y when x = 3and x = Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%. 1 MARK QUESTION ANSWER: cm/sec cm/sec. 3 2 cm 3 /sec cm 2 /sec. 5 Rs Rs π 8 8πcm 2 /cm 9 k 0 10 (0, π] 11 (0, e] 12 x 1 13 R 14 (0, π 6 ) 15 k= P a g e

46 17 ( 1 2, 1 4 ) 18 x + y -3 = x 3 m 3 2 MARKS QUESTION: 1 Find the rate of change of the total surface area of a cylinder of radius r and height h with respect to radius when height is equal to the radius of the base of cylinder. 2 Find the rate of change of the area of a circle with respect to its radius. How fast is the area changing w.r.t. its radius when its radius is 3 cm? 3 An edge of a variable cube is increasing at the rate of 5 cm per second. How fast is the volume increasing when the side is 15cm? 4 Find the least value of μ such that the function x 2 + μx + 1 is increasing on [1, 2]. 5 Find the maximum and minimum values of function f(x) = sin 2x Find the maximum and minimum values if any of the function f(x) = x 1 + 7, x R. 7 Without using derivatives, find the maximum and minimum value of y = 3 sin x Find a for which f(x) = (x + sin x) + a is increasing. 9 If y = a log x + bx 2 + x has its extreme values at x = -1 and x = 2, then find a and b. 10 Find the maximum and minimum values of f, if any, of the function given by f(x) = x, x R 11 Find the local minimum value of the function f given by f(x) = 3 + x, x R 12 What is the maximum value of the function sin x + cos x? 13 Show that the function g(x) = log x do not have maximum or minima. 45 P a g e

47 14 Find the absolute maximum and the absolute minimum value of f(x) = 4x x2, x [ 2, 9 2 ] 15 Find the point on the curve y = 3x 2 12x 9 at which the tangent is parallel to x-axis. 16 Find the point on the curve y = 3x at which the tangent is perpendicular to the line with slope Find the point on the curve y = x 2 where the slope of the tangent is equal to the y coordinate. 18 Find the slope of the normal to the curve x = 1 a sinθ; y = bcos 2 θ at x = π At what points on the curve x 2 + y 2 2x 4y + 1 = 0, the tangents are parallel to y- axis? 20 Find a point on the curvey = x 3 3x, where tangents is parallel to the chord joining (1, -2) and (2, 7). 21 Find the value of m for which the line y = mx + 1 is a tangent to the curve y 2 = 4x. 22 Find the points on the curve 9y 2 = x 3, where the normal to the curve makes equal intercepts with the axes. 23 Using differentiations, find the approximate value of Using differentiations, find the approximate value of Use differential to approximate(25) Find the approximate value of f (3.02), where f(x) = 3 x 2 + 5x If the radius of a sphere is measured as 9cm with an error of 0.03 cm, then find the approximate error in calculating its volume. 2 MARKS QUESTION ANSWER: 1 8πR 2 6π cm 2 /cm cm 3 /sec Minimum value = 4, maximum value = 6. 6 Maximum value = 7, minimum value does not exist. 7 Maximum value = 4, minimum valve = 0. 8 a > 0 46 P a g e

48 9 a = 2, b = Minimum value 0, no maximum value in R Absolute minimum value = -10, absolute maximum value = 8 15 (2, 3) 16 ( 1, 7) 17 ( 0, 0), (2, 4) 18 a 2b 19 (-1, 2) and (3, 2) 20 (2, 2) and (-2, -2) (4, ± 8 3 ) π cm 3 gsgsgsgsggsgsggsgsgsgsgsggsgsgsgsgsgsggsgsgsgsgsggsgsgsgsgsgsgsggsgsgsgsgsggsgsggsgsgsgsgsgsgsggsg 47 P a g e

49 TOPIC-: INDEFINITE AND DEFINITE SAMPLE QUESTION OF 1 MARK 1. Evaluate : (sin 1 x +cos 1 x) dx 2. Evaluate : cosecx(cosecx + cotx)dx 3. Evaluate : dx 1 sin 2 x 4. 1 sinx Evaluate : dx cos 2 x 5. Evaluate : (8 x + x 8 )dx 6. Find the anti derivatives of ( x+ 1 x ) 7. Find the Integration : 1 1 dx 1+4x 2 8. Evaluate : e alog x + e xlog a dx 9. Find the Integration : etan 1 x dx 1+x Evaluate : cos2x+2sin2 x cos 2 x 11. Evaluate : dx sin 2 xcos 2 x 12. Evaluate e log2x dx 13. (1+log x)2 Evaluate dx x 14. Find logx dx 1 dx Evaluate e x (tan 1 x x2) dx 4 Evaluate x(x 1) dx Evaluate 4 + 9x 2 dx 0 48 P a g e

50 18. π 4 Evaluate sinx dx π If (3 x 2 + 2x + k)dx = 0, then find the vlaue of K. 20. Evaluate: 21. e 2 e Evaluate 1 xlogx dx π 4 π 4 sin 3 x dx 22. Write the value of x+cos6x 3x 2 +sin6x dx 23. Evaluate sec2 x dx x 24. Evaluate e2x e 2x e 2x +e 2xdx 25. Evaluate x sin 1 x dx 26. Evaluate: x3 sin (tan 1 x 4 ) 1+ x 8 dx 27. Evaluate: 28. Find 1 dx 2x x 2 1 dx (x+1)(x+2) 29. Find e x sinx dx 30. Evaluate : sec x dx 31. Evaluate : 1 x x dx 32. Evaluate : 1 e x 1 dx 33. Evaluate : sin 1 ( 2x 2) dx 34. Evaluate : π 2 π 2 SAMPLE QUESTION OF 2 MARKS 1+x sin 7 x dx 49 P a g e

51 35. Evaluate : 1 cosx 1+cosx dx 36. Evaluate : 1 dx x 2 +5x If K x 2 dx = π 8 then find the value of K. 38. Evaluate : cos 1 (sinx)dx 39. Evaluate : (1 x) x dx 40. Evaluate : x3 x 2 +x 1 dx 41. Evaluate : 0 x 1 1 a 2 +(bx) 2 dx π 2 Evaluate: 1 + sin x [x 2 ]dx dx Where [ ] is greatest integer function. 44. Find e [log(x+1) log x] dx 45. Evaluate : dx x(1+x 7 ) 46. Evaluate : 47. a a 1 x a 2 x 2 dx π Evaluate : 2 x 2 sinx Evaluate : 1 4 x 2 dx 49. log (sinx) Evaluate: tanx 50. Evaluate : ex dx x a dx dx π x 2 ANSWER OF 1 MARK + c 2. cotx cosecx + c 3. tanx + c 4. tanx secx+ C 8 x log8 + x9 9 + C x x C log 2x x2 + C 50 P a g e