Functions
Let f(x) = ax2 – b|x|, where a and b are constants. Then at x = 0, f(x) is
Let u = (log2x)2 – 6log2 x + 12 where x is a real number. Then the equation xu = 256, has
How many of the following products are necessarily zero for every x.
Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is the relation g(x+p) = g(x) necessarily true for every real x?
A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and female operator gets Rs. 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?
Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English, and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose?
A function f(x) satisfies f(1) = 3600 and f(1) + f(2) + ... + f(n) = n2 f(n), for all positive integers n > 1. What is the value of f(9)?
Directions for Questions 5 and 6:
Let f(x) = ax2 + bx + c, where a, b and c are certain constants and a ≠ 0 It is known that
f(5) = – 3f(2). and that 3 is a root of f(x) = 0.
What is the other root of f(x) = 0?
Let f(x) = ax2 + bx + c, where a, b and c are certain constants and a ≠ 0 It is known that
f(5) = – 3f(2). and that 3 is a root of f(x) = 0.
What is the value of a + b + c?
Let f(x) be a function satisfying f(x).f(y) = f(x.y) for all real x, y. If f(2) = 4, then what is the value of f(1/2)?
A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?
If y = f(x) and f(x) = (1−x) / (1 + x), which of the following is true?
Three machines, A, B and C can be used to produce a product. Machine A will take 60 hours to produce a million units. Machine B is twice as fast as Machine A. Machine C will take the same amount of time to produce a million units as A and B running together. How much time will be required to produce a million units if all the three machines are used simultaneously?
Q74 and 75 : A function f(x) is said to be even if f(x) = f(x), and odd if f(x) = f(x). Thus, for example, the function given by f(x) = x2 is even, while the function given by f(x) = x3 is odd. Using this definition, answer the following questions.
The function given by 
is
Q74 and 75 : A function f(x) is said to be even if f(x) = f(x), and odd if f(x) = f(x). Thus, for example, the function given by f(x) = x2 is even, while the function given by f(x) = x3 is odd. Using this definition, answer the following questions.
The sum of two odd functions
@ (A, B) = average of A and B
*(A, B) = product of A and B
/(A, B) = A divided by B
Answer these questions with the above data.
The sum of A and B is given by
@ (A, B) = average of A and B
*(A, B) = product of A and B
/(A, B) = A divided by B
Answer these questions with the above data.
The sum of A, B, and C is given by
Q63 64 : are based on the following information:
Q63 64 : are based on the following information:
Q87 90 : are based on the following information:
fog(x) is equal to
Q87 90 : are based on the following information:
For what value of x; f (x) = g(x −3)?
Q87 90 : are based on the following information:
What is the value of (gofofogogof) (x) (fogofog)(x)?
E.g. fogofog(x) = fog(x) = x. Since both the brackets have the functions repeated for even number of times, each of their value will be x and their product will be x2.
Q87 90 : are based on the following information:
What is the value of fo(fog)o(gof)(x)?
fo(fog)o(gof)(x) = fo(fog)(x) = f(x) = 2x + 3.
Direction for questions 88 to 91: Answer the questions based on the following information.
le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)
Find the value of me(a + mo(le(a, b)); mo(a + me(mo(a), mo(b))), at a = 2 and b = 3.
= me(2 + mo(3), mo(2 + me(2, 3)))
= me(2 + 3, mo(2 + 3))
= me(1, mo(1)) = me(1, 1) = 1.
Direction for questions 88 to 91: Answer the questions based on the following information.
Which of the following must always be correct for a, b > 0?
In this case, option (a) LHS = mo(le(2, 3)) = mo(2) = 2.
RHS = (me(mo(2), mo(3)) = (me(2, 3)) = 3.
Hence, LHS < RHS.
(b) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (me(mo(2), mo(3)) = (me(2, 3)) = 3.
Hence, LHS < RHS.
(c) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (le(mo(2), mo(3)) = le(2, 3) = 2.
Hence, LHS = RHS.
(d) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (le(mo(2), mo(3)) = le(2, 3) = 2.
Hence, LHS = RHS.
Direction for questions 88 to 91: Answer the questions based on the following information.
For what values of a is me(a2 3a, a 3) < 0?
Direction for questions 88 to 91: Answer the questions based on the following information.
For what values of a is le(a2 3a, a 3) < 0?
Direction for questions 115 and 116: Answer the questions based on the following information.
A, S, M and D are functions of x and y, and they are defined as follows.
What is the value of M(M(A(M(x, y), S(y, x)), x), A(y, x)) for x = 2, y = 3?
Direction for questions 115 and 116: Answer the questions based on the following information.
A, S, M and D are functions of x and y, and they are defined as follows.
What is the value of S[M(D(A(a, b), 2), D(A(a, b), 2)), M(D(S(a, b), 2), D(S(a, b), 2))]?
A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition
connected to each of the other 11 points, giving a total
of 11 lines. One can move from any point to any other
point via the common point.
The maximum edges will be when a line exists between
any two points. Two points can be selected from 12
points in 12C2 i.e. 66 lines.
If f1 (x) = x2 + 11x + n and f2 (x) = x, then the largest positive integer n for which the equation f1 (x) = f2 (x) has two distinct real roots, is
Let f(x) = x2 and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is
If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is
If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals
Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers, and f(1) = 2. If f(a + 1) + f(a + 2) + ... + f(a + n) = 16(2n – 1) then a is equal to
For any positive integer n, let f(n) = n(n + 1) if n is
even, and f(n) = n + 3 if n is odd.
If m is a positive integer such that 8f(m + 1) – f(m) =
2, then m equals
Let f be a function such that f (mn) = f (m) f (n) for every positive integers m and n. If f (1), f (2) and f (3) are positive integers, f (1) < f (2), and f (24) = 54, then f (18) equals
If f(5 + x) = f(5 – x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these roots is
If f(x + y) = f(x)f(y) and f(5) = 4, then f(10) – f(–10) is equal to
Let a, b, c be non-zero real numbers such that b2 < 4ac, and f(x) = ax2 + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be
Suppose for all integers x, there are two functions f and g such that f(x) + f(x – 1) – 1 = 0 and g(x) = x2. If f(x2 – x) = 5, then the value of the sum f(g(5)) + g(f(5)) is
Let f(x) be a quadratic polynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f(4) = 6, then f(–2) is equal to
f(r) = 2r – r = r
So f(f(r)) = f(r) = r
For x < r,
f(x) = r
f(f(x)) = f(r) = r
For x > r,
Let x = kr, where k is some number greater than 1.
f(x) = f(kr) = 2kr – r = (2k – 1)r
Since k > 0, 2k – 1 > k
So for any x > r, f(f(x)) ≠ f(f(r))
Hence, for f(f(x)) = f(x) to be satisfied, the necessary condition is, x ≤ r.
Suppose f(x, y) is a real-valued function such that f(3x + 2y, 2x – 5y) = 19x, for all real numbers x and y. The value of x for which f(x, 2x) = 27, is
E.g. fogofog(x) = fog(x) = x. Since both the brackets have the functions repeated for even number of times, each of their value will be x and their product will be x2.
fo(fog)o(gof)(x) = fo(fog)(x) = f(x) = 2x + 3.
= me(2 + mo(3), mo(2 + me(2, 3)))
= me(2 + 3, mo(2 + 3))
= me(1, mo(1)) = me(1, 1) = 1.
In this case, option (a) LHS = mo(le(2, 3)) = mo(2) = 2.
RHS = (me(mo(2), mo(3)) = (me(2, 3)) = 3.
Hence, LHS < RHS.
(b) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (me(mo(2), mo(3)) = (me(2, 3)) = 3.
Hence, LHS < RHS.
(c) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (le(mo(2), mo(3)) = le(2, 3) = 2.
Hence, LHS = RHS.
(d) LHS = mo(le(2, 3)) = mo(2) = 2. RHS = (le(mo(2), mo(3)) = le(2, 3) = 2.
Hence, LHS = RHS.
connected to each of the other 11 points, giving a total
of 11 lines. One can move from any point to any other
point via the common point.
The maximum edges will be when a line exists between
any two points. Two points can be selected from 12
points in 12C2 i.e. 66 lines.
f(r) = 2r – r = r
So f(f(r)) = f(r) = r
For x < r,
f(x) = r
f(f(x)) = f(r) = r
For x > r,
Let x = kr, where k is some number greater than 1.
f(x) = f(kr) = 2kr – r = (2k – 1)r
Since k > 0, 2k – 1 > k
So for any x > r, f(f(x)) ≠ f(f(r))
Hence, for f(f(x)) = f(x) to be satisfied, the necessary condition is, x ≤ r.
