CAT Question Bank  Functions

Q13) f(x) be a function for every real x, f(x+1)  f(x) = f(x2)  f(x1) and f(3) = 3 then find f(511) ?

Q14) A polynomial f(x) with positive integer coefficients satisfies f(1) = 12 and f(12) = 2080. Find the last three digits of f(10).

Q15) If we define a function f(x) on natural numbers as f(x + 1) = f(x) + x and f(1) = 1, what is the sum f(1) + f(2) + f(3) + ... + f(30)?

Q16) f(x) = ax^2 + bx + c where a is not equal to 0.
f(7) = 2f(5) and 5 is one of the root of f(x) = 0.
Find a + b + c

Q17) Function f(x) is defined for all positive integers that satisfy the condition:
f(1) + f(2) + f(3) + ... + f(p) = p^2 × f(p)
If f(1) = 2009, then what is the value of f(2008)?

Q18) f(x) + f (1/x) = f(x) f(1/x)
f(4) = 6
Find f(6).

Q19) Function f(x) is a continuous function defined for all real values of x, such that f(x) = 0 only for two distinct real values of x. It is also known that
f(6) + f(8) = 0
f(7).f(9) > 0
f(6).f(10) < 0
f(0) > 0 and f(1) < 0
How many of the following statements must be true?
I. f(1).f(2).f(3) < 0
II. f(3).f(5).f(7).f(9) > 0
III. f(7).f(8) < 0
IV. f(0) + f(1) + f(9) + f(10) > 0(a) 1
(b) 2
(c) 3
(d) 4

f(6) + f(8) = 0
indicates a root lies between 6 and 8
f(7).f(9) > 0
indicates f7 and f9 lie on the same side of the X axis
So, we can say
(a) root lies bw 6 and 7, and both 7 and 9 are either +ve or negative
f(6)f(10) 0 and f(1) < 0
indicates a root lies between 0 and 1
f(x) = 0 only for two distinct real values of x << Only two distinct roots. Both have been discovered.
f(0) > 0
hence it begins as a curve above the X axis, somewhere between (0,1) has a root, then becomes ve. The second root arrives between (6,7) and beyond this curve again becomes +veSo
I. f(1).f(2).f(3) < 0 << All three values are negative. Hence, true
II. f(3).f(5).f(7).f(9) > 0 << f3,f5 are ve whereas f7,f9 are +ve Hence true
III. f(7).f(8) < 0 << Both are +ve hence false
IV. f(0) + f(1) + f(9) + f(10) > 0 << f0,f9,f10 are +ve while f1 is ve
However it being a continuous curve with two roots, it cannot be predicted what will be the negative value. so not a "must be true" statement.So ans 2
[Credits : @hemant_malhotra]

Q20) Let f(x) be a function such that f(x).f(y)  f(xy) = 3(x+y+2). Then f(4)=?
a) can not be determined
b) 7
c) 8
d) either 7 or 8
e) none of these

Q21) f(x) = x + 1/x
[f(x)]^3 = f(x^3) + kf(1/x)
k = ?
a) 1
b) 3
c) 3
d) 1

Q22) f(x) is a 4 degree polynomial satisfying f(n) = 1/n for integers n = 1 to 5.
If f(0) = a/b, where a and b are coprime positive integers. What is the value of a + b ?

Q23) If f(f(x)) = x^4 + 4x^3 + 4x^2  2, then f(2) = ?
a. 2
c. 0
c. 7
d. 1
e. None of the above

Q24) If f(x) is cubic polynomial with coefficient of x^3 as 1. It is given that it has nonnegative real roots and f(0) = 64. Find the largest possible value of f(1).

Q25) f(x) is a cubic Polynomial with leading coefficient as 1.
If f(1) = 11
f(2) = 104
f(3) = 1027.
Find the value of f(0).
a) 4
b) 5
c) 6
d) 7

Q26) Range of f(x) = 1/√(x  [x]) ?

Q27) Let f(x) = 4x^3 + 21x^2 + 41x + 24 be identical to g(x) = a(x+2)^3 + b(x+2)^2 + c(x+2) + d
Find a  b + c  d

Q28) f(xy) = f(x) + f(y) ‐ 2. If f(2) = a and f(3) = b, then the value of f(72) is:
a) a + b ‐ 2
b) 2a + 2b ‐ 6
c) 2a + 3b ‐ 8
d) 3a + 2b ‐ 8

Q29) A function ƒ(x) satisfies ƒ(1) = 3600 and ƒ(1) + ƒ(2) + ... + ƒ(n) = n2 f(n), for all positive integers n > 1. What is the value of ƒ(9)?

Q30) f(xy) = f(x)f(y) for all real values of x and y, and f(2)=1/8.
Find the value of f(1) + f(1/2) + f(1/3)
a) 11/6
b) 1/6
c) 6
d) 36
