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 ) ^{50}P_{17}
2 ) ^{50}P_{22}
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) = ax^{2} + 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 ) ^{50}P_{17}
2 ) ^{50}P_{22}
3 ) 33! × 17!
4 ) 25 × 49!
Question 9 :
Let f(x) = $\raisebox{1ex}{${\mathrm{x\; -\; 1}}^{}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{x+1}}^{}$}\right.$ ∈ 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 -3x^{2} -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) = x^{3} + 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) = ƒ_{1} (x) + ƒ_{2} (x), where ƒ_{1} (x) is an even function of ƒ_{2} (x) is an odd function. Then ƒ_{1} (x + y) + ƒ_{1} (x – y) equals
1 ) 2ƒ_{1} (x)ƒ_{1} (y)
2 ) 2ƒ_{1} (x + y)ƒ_{1} (x - y)
3 ) 2ƒ_{1} (x)ƒ_{2} (y)
4 ) 2ƒ_{1} (x + y)ƒ_{2} (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 f_{4}(x) - f_{6}(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) = x^{3} + 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-x}C_{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