CAT Question Bank - Functions



  • Q2) A function f(x) is defined for all real values of x as 2f(x) + f(1 – x) = x^2. What is the value of f(5)?
    (a) 10
    (b) 17
    (c) 34/3
    (d) Cannot be determined



  • Q3) If F(x) = x^4 - 360x^2 + 400 (for any integral value of x) and if F(x) is a prime number, then what is the sum of all possible values of F(x)?



  • Q4) Evaluate f(x) if f(2x + 1) = 6x + 2



  • Q5) The graph of the function f(x) = a(x + 1)^3 passes through the point (1, 4). What is f(3)?



  • Q6) f(x)f(y) = f(x+y)
    If f(1) = 0.3333333...
    Find the value of f(1) + f(3) + f(5) + … up to infinity



  • Q7) If f(x) = 2x^2 + 6x - 1 and g(x) = |x + 5| then the value of f(g(f(g(-6))) is



  • Q8) If f(x) = log((1+x)/(1-x)), then f(x) + f(y) ?
    a) f(x + y)
    b) f(1 + xy)
    c) (x + y) f(1 + xy)
    d) f ((x +y) / (1+xy))



  • Q9) A function f(x) defined for all real values of x as 2f(x) + f(1-x) = x^2. Find f(5)?



  • Q10) Let f(x) be a function with the two properties:
    a) for any two real numbers x and y, f(x + y) = x + f(y)
    b) f(0) = 2
    What is the value of f(98) ?



  • Q11) F(x) is a Polynomial of degree 5 & leading coefficient is 2009.
    F(1) = 1
    F(2) = 3
    F(3) = 5
    F(4) = 7
    F(5) = 9
    Find F(6)



  • Q12) A function f(x) is said to be strictly increasing if f (x1) > f (x2) for x1 > x2. The number of strictly increasing functions that can be defined from set A with 4 positive integers to set B with 6 positive integers is?



  • Q13) f(x) be a function for every real x, f(x+1) - f(x) = f(x-2) - f(x-1) 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 +ve

    So
    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


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