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QUESTION: 1

Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________

Solution:

The composition of f and g is given by f(g(x)) which is equal to 2(3x + 4) + 1.

QUESTION: 2

A certain function always obeys the rule: If f (x.y) = f(x). f(y) where x and y are positive realnumbers. A certain Mr. Mogambo found that the value of f (128) = 4, then find the value of thevariable M = f (0.5). f (1). f (2). f (4). f (8). f (16). f (32). f (64). f (128). f (256)

Solution:

Since f (128) = 4, we can see that the product of f (256). f (0.5) = f (256 × 0.5) = f (128) = 4.

Similarly, the products f (1). f (128) = f (2). f (64)

= f (4). f (32) = f (8). f (16) = 4.

Thus, M = 4 × 4 × 4 × 4 × 4 = 1024.

Option (d) is the correct answer.

QUESTION: 3

If f(t) = 2^{t}, then f(0), f(1), f(2) are in

Solution:

f(0) = 1, f(1) = 2 and f(2) = 4

Hence, they are in G.P.

QUESTION: 4

The graph of y = (x + 3)^{3} + 1 is the graph of y = x^{3} shifted

Solution:

(x + 3)^{3} would be shifted 3 units to the left and hence (x + 3)^{3} + 1 would shift 3 units to the left

and 1 unit up. Option (c) is correct.

QUESTION: 5

If f(x) = x^{2} and g(x) = log_{e}x, then f(x) + g(x) will be

Solution:

(x^{2} + loge x) would be neither odd nor even since it obeys neither of the rules for even function

(f(x) = f(–x)) nor for odd functions (f(x) = –f(–x)).

QUESTION: 6

f(x) is any function and f^{–1}(x) is known as inverse of f(x), then f–1(x) of f(x) = e^{x} is

Solution:

y = e^{x}

fi log_{e} y = x.

fi f ^{–1}(x) = log_{e} x.

QUESTION: 7

Which of the following functions will have a minimum value at x = –3?

Solution:

If you differentiate each function with respect to x, and equate it to 0 you would see that for none

of the three options will get you a value of x = –3 as its solution. Thus, option (d) viz. None of

these is correct.

QUESTION: 8

Define the following functions:

f(x, y, z) = xy + yz + zx

g(x, y, z) = x^{2}y + y^{2}z + z^{2}x and

h(x, y, z) = 3 xyz

Q.

Find the value of the following expressions:37. h[f(2, 3, 1), g(3, 4, 2), h(1/3, 1/3, 3)]

Solution:

The given function would become h[ 11, 80, 1] = 2640.

QUESTION: 9

Define the following functions:

f(x, y, z) = xy + yz + zx

g(x, y, z) = x^{2}y + y^{2}z + z^{2}x and

h(x, y, z) = 3 xyz

Find the value of the following expressions:

Q.

f[ f (1, 1, 1), g(1, 1, 1), h(1, 1, 1)]

Solution:

The given function would become f[3, 3, 3] = 27.

QUESTION: 10

If f(x) = 1/ g(x), then which of the following is correct?

Solution:

The number of g’s and f’s should be equal on the LHS and RHS since both these functions are

essentially inverse of each other.

Option (c) is correct.

QUESTION: 11

If R(a/b) = Remainder when a is divided by b;

Q(a/b) = Quotient obtained when a is divided by b;

SQ(a) = Smallest integer just bigger than square root of a.

Q.

If a = 12, b = 5, then find the value of SQ[R {(a + b)/b}].

Solution:

SQ [R[(a + b)/b]] = SQ [R[17/5]] fi SQ [2] = 2.

QUESTION: 12

If R(a/b) = Remainder when a is divided by b;

Q(a/b) = Quotient obtained when a is divided by b;

SQ(a) = Smallest integer just bigger than square root of a.

Q.

If a =18, b = 2 and c = 7, then find the value of Q [{SQ(ab) + R(a/c)}/b].

Solution:

Q [[SA (36) + R (18/7)]/2] = Q [(7 + 4)/2] = Q [11/2] = 5.

QUESTION: 13

Read the following passage and try to answer questions based on

them.

[x] = Greatest integer less than or equal to x

{x} = Smallest integer greater than or equal to x.

Q.

If x is not an integer, then ({x} + [x]) is

Solution:

[x] + {x} will always be odd as the values are consecutive integers.

QUESTION: 14

If f(t) = t^{2} + 2 and g(t) = (1/t) + 2, then for t = 2, f [g(t)] – g[f(t)] = ?

Solution:

f(g(t)) – g(f(t)) = f(2.5) – g(6) = 8.25 – 2.166 = 6.0833.

QUESTION: 15

Let F(x) be a function such that F(x) F(x + 1) = – F(x – 1)F(x–2)F(x–3)F(x–4) for all x ≥ 0.Given the values of If F (83) = 81 and F(77) = 9, then the value of F(81) equals to

Solution:

When the value of x = 81 and 82 is substituted in the given expression, we get,

F (81) F (82) = – F (80) F (79) F(78) F(77)

F (82) F (83) = – F (81) F (80) F(79) F(78)

On dividing (i) by (a), we get

Option (a) is the correct answer.

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