Functions JEE(main) exam

Question 1 :

Let f : R → R be a continuous function such that ƒ(3x) − f(x) = x. If ƒ(8) = 7, then f(14) is equal to :

1 ) 4

2 ) 10

3 ) 11

4 ) 16

Question 2 :

The number of bijective functions f: {1,3,5,7,..., 99}{2, 4, 6, 8,....100}, such that ƒ(3) ≥ ƒ(9) ≥ ƒ(15) ≥ ƒ(21) ≥ ..... f(99), is

1 ) ⁵⁰P₁₇

2 ) ⁵⁰P₂₂

3 ) 33! × 17!

4 ) 25 × 49!

Question 3 :

Let f,g: N - {1} : → N be functions defined by f(a) = x, where x is the maximum of the powers of those primes p such that p^x divides a, and g(a) = a + 1, for all a ∈ N – {1}. Then, the function ƒ + g is

1 ) one-one but not onto

2 ) onto but not one-one

3 ) both one-one and onto

4 ) neither one-one nor onto

Question 4 :

The total number of functions, f: {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that ƒ(1) + ƒ(2) = ƒ(3), is equal to :

1 ) 60

2 ) 90

3 ) 108

4 ) 126

Question 5 :

Let f(x) = ax² + bx + c be such that f(1) = 3, f(-2) = y and f(3) = 4. If f(0) + f(1) + f(-2) + f(3) = 14, then y is equal to:

1 ) -4

2 ) 13/2

3 ) 23/2

4 ) 4

Question 6 :

The function f(x) = xe^x(1-x), x ∈

1 ) (-1/2,1)

2 ) (1/2,2)

3 ) (-1,-1/2)

4 ) (-1/2,1/2)

Question 7 :

Let f : R → R be a continuous function such that ƒ(3x) − f(x) = x. If ƒ(8) = 7, then f(14) is equal to :

1 ) 4

2 ) 10

3 ) 11

4 ) 16

Question 8 :

The number of bijective functions f: {1,3,5,7,..., 99}{2, 4, 6, 8,....100}, such that ƒ(3) ≥ ƒ(9) ≥ ƒ(15) ≥ ƒ(21) ≥ ..... f(99), is

1 ) ⁵⁰P₁₇

2 ) ⁵⁰P₂₂

3 ) 33! × 17!

4 ) 25 × 49!

Question 9 :

Let f(x) =

\frac{x - 1}{x+1}

∈ R - {0,-1,1}. If f ⁿ⁺¹(x) = f(f ⁿ(x)) for all n ∈ N, then f ⁶(6) + f ⁷(7) is equal to:

1 ) 7/6

2 ) -3/2

3 ) 7/12

4 ) -11/12

Question 10 :

Let f(x) = 2cos^-1 x + 4cot^-1 x -3x² -2x +10, x ∈ [-1,1]. If [a, b] is the range of the function f, then 4a - b is equal to :

1 ) 15 - π

2 ) 11

3 ) 11- π

4 ) 11 + π

Question 11 :

Let f : R → R be defined as f(x) = x³ + x - 5. If g(x) is a function such that, f(g(x)) = x, ∀ x ∈ R, then g'(63) is equal to :

1 ) 1/49

2 ) 3/49

3 ) 43/49

4 ) 91/49

Question 12 :

For the function f(x) = 4log_e( x - 1 ) - 2x ² + 4x +5, x > 1, which one of the following is not correct?

1 ) f is increasing in (1,2) and decreasing in (2, ∞)

2 ) f(x) = -1 has exactly two solutions

3 ) f ^' (e) - f ^'' (2) < 0

4 ) f(x) = 0 has a root in the interval (e, e+1)

Question 13 :

The sum of absolute maximum and absolute minimumn values of the function f(x) = |2x ² + 3x - 2| +sin x cos x in the interval [0,1] is:

1 ) 3+1/2 sin(1) cos ² (1/2)

2 ) 3 + 1/2 sin (1) +sin (1) cos (1)

3 ) 5 + 1/2[sin 1 + sin 2]

4 ) 2 + sin (1/2) cos (1/2)

Question 14 :

Let f : N → N be a function such that f(m + n) = f(m) + f(n) for every m, n ∈ N. If f(6) = 18, then f(2) . f(3) is equal to :

1 ) 6

2 ) 54

3 ) 18

4 ) 36

Question 15 :

Consider function f : A → B and g : B → C (A, B, C ⊆ R) such that (gof)^-1 exists, then :

1 ) f and g both aare one -one

2 ) f and g both are onto

3 ) f is one-one and g is onto

4 ) f is onto and g is one-one

Question 16 :

Let [x] denote the greatest integer less than or equal to x. Then, the values of x ∈ R satisfying the equation [e ^x ] ² + [e ^x + 1] lie in the interval :

1 ) [0, 1/e)

2 ) [ln 2, ln 3)

3 ) [1,e)

4 ) [0, ln 2)

Question 17 :

1 ) (0,2)

2 ) (-4,0)

3 ) (-4,4)

4 ) (0,4)

Question 18 :

1 ) (- ∞ , 1/4]

2 ) [-1/4, ∞ ]

3 ) [-1/3, ∞ ]

4 ) (- ∞ , 1/3]

Question 19 :

1 ) [1, ∞ )

2 ) [-1,2]

3 ) [-1, ∞ )

4 ) (- ∞ , 2]

Question 20 :

1 ) f(50) = 1

2 ) f(-50) = -1

3 ) f(50) = -501

4 ) f(-50) = 501

Question 21 :

Let f(x) = x ² , x ∈ R. For any A ⊆ R, define g (A) = { x ∈ R : f(x) ∈ A}. If S = [0,4], then which one of the following statements is not true ?

1 ) g(f(S)) ≠ S

2 ) f(g(S)) = S

3 ) f(g(S)) ≠ f(S)

4 ) g(f(S))= g(S)

Question 22 :

Let ƒ(x) = a ^x (a > 0) be written as ƒ(x) = ƒ₁ (x) + ƒ₂ (x), where ƒ₁ (x) is an even function of ƒ₂ (x) is an odd function. Then ƒ₁ (x + y) + ƒ₁ (x – y) equals

1 ) 2ƒ₁ (x)ƒ₁ (y)

2 ) 2ƒ₁ (x + y)ƒ₁ (x - y)

3 ) 2ƒ₁ (x)ƒ₂ (y)

4 ) 2ƒ₁ (x + y)ƒ₂ (x - y)

Question 23 :

Let f_k (x) = 1/k (sin ^k x + cos ^k xfor k = 1, 2, 3, ... Then for all x ∈ R, the value of f₄(x) - f₆(x) is equal to

1 ) 1/4

2 ) 5/12

3 ) -1/12

4 ) 1/12

Question 24 :

1 ) Both statements are true and statement 2 is the correct explanation of statement 1

2 ) Both statements are true and statement 2 is not the correct explanation of statement 1

3 ) statement 1 is true and statement 2 is false

4 ) statement-1 is false an statement -2 is true

Question 25 :

For real x, let f(x) = x³ + 5x + 1, then

1 ) f is one-one but not onto R

2 ) f is onto R but not one-one

3 ) f is one-one and onto R

4 ) f is neither one-one nor onto R

Question 26 :

Let f: N → N be a function defined as f(x) = 4x + 3 where Y = { y ∈ N, y = 4x + 3 for some x ∈ N }. Show that f is invertible and its inverse is

1 ) g(y) = (3y + 4)/4

2 ) 4 + (y+3)/4

3 ) (y+3)/4

4 ) (y-3)/4

Question 27 :

1 ) [- π /4 ,π/2)

2 ) [0,π /2)

3 ) [0, π]

4 ) (-π, π)

Question 28 :

A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given constant and f(0) = 1, f(2a - x) is equal to

1 ) - f(x)

2 ) f(x)

3 ) f(a) + f(a-x)

4 ) f(-x)

Question 29 :

The range of the function f(x) = ^7-xC_x-3 is

1 ) {1,2,3,4,5}

2 ) {1,2,3,4,5,6}

3 ) {1,2,3,4}

4 ) {1,2,3}

Question 30 :

The graph of the function y = f(x) is symmetrical about the line x = 2, then

1 ) f ( x ) = - f ( -x )

2 ) f ( 2 + x ) = f ( 2 -x )

3 ) f ( x ) = f ( -x )

4 ) f ( x + 2) = f ( x - 2 )

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Functions JEE(main) exam

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