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QUESTION: 1

The minimum value of ax^{2} + bx + c is 7/8 at x = 5/4. Find the value of the expression at x = 5, ifthe value of the expression at x = 1 is 1.

Solution:

**Correct Answer :- B**

**Explanation : **ax^{2} + bx + c

a[x^{2} + b/a x + c/a]

a[x^{2} + 2b/2a x + c/a + (b/2a)^{2} - (b/2a)^{2}]

a[(x+b/2a)^{2} + c/a - (b/2a)^{2}]

Minimum value occurs when

x = - b/2a

=> -b/2a = 5/4

=> b/a = -5/2

And, minimum value is a[(c/a - (b/2a)^{2}]

a[(c/a - (b/2a)^{2}] = 7/8

=> c - 25s/16 = 7/8

=> 16c - 25a = 14.....(1)

Again value of the expression at x = 1 is 1

a * 1^{2} + b * 1 + c = 1

a - 5a/2 + c = 1

=> 2a - 5a + 2c = 2

=> - 3a + 2c = 2.......(2)

Multiplying eq (2) by 8 and subtracting it from eq (1)

-a = -2

or a = 2

Thus, b = -5/2 × 2 = -5

Putting the value of a in eq (2)

-6 + 2c = 2

2c = 8

c = 4

Thus, the expressions is 2x^{2} - 5x + 4

Value of the expression at x = 5 is

2 * 5^{2 }- 5 * 5 + 4

= 50 - 25 + 4

= 29

QUESTION: 2

A function a(x) is defined for x as 3a(x) + 2a (2 – x) = (x + 3)^{2}. What is the value of [G (–5)]

where [x] represents the greatest integer less than or equal to x?

Solution:

The equation given in the question is: 3a(x) + 2a (2–x) = (x + 3)^{2}

Replacing x by (2–x) in the above equation, we get

3a(2–x) + 2a(x) = (5–x)^{2}

Solving the above pairs of equation, we get

a(x) = (x2 + 38 – 23)/5

Thus, G(–5) = –188/5 = –37.6. The value of [–37.6] = –37. Hence, Option (c) is the correct

answer.

QUESTION: 3

Find the domain of the definition of the functiony = [(x – 3)/(x + 3)]^{1/2} + [(1 – x)/(1 + x)]1/2.

Solution:

Both the brackets should be non-negative and neither (x + 3) nor (1+ x) should be 0.

For (x – 3)/(x + 3) to be non negative we have x>3 or x< – 3.

Also for (1– x)/(1+ x) to be non-negative –1 < x < 1. Since there is no interference in the two

ranges, Option (d) would be correct.

QUESTION: 4

If f(x) = e^{x} and g(x) = log_{e}x then value of fog will be

Solution:

fog = f (log_{e}x) = e^{logex} = x.

QUESTION: 5

The function y = 1/x shifted 1 unit down and 1 unit right is given by

Solution:

Looking at the options, one unit right means x is replaced by (x – 1). Also, 1 unit down means –1

on the RHS.

Thus, (y + 1) = 1/(x – 1)

QUESTION: 6

If f(x) = |x – 2| , then which of the following is always true?

Solution:

Take different values of n to check each option. Each of Options (a), (b) and (c) can be ruled out.

Hence, Option (d) is correct.

QUESTION: 7

Read the instructions below and solve:

f(x) = f(x – 2) – f(x – 1), x is a natural number

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

Q.

The value of f[f(6)] is

Solution:

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

f(3) = f(1) – f(2) = –1

f(4) = f(2) – f(3) = 2

f(5) = f(3) – f(4) = –3

f(6) = f(4) – f(5) = 5

f(7) = f(5) – f(6) = –8

f(8) = f(6) – f(7) = 13

Ans:-

f(f(6) = f(5) = –3.

QUESTION: 8

If f(x) is a function satisfying f(x). f(1/x) = f(x) + f(1/x) and f(4) = 65, what will be the value off(6)?

Solution:

We have f(x) ◊ f(1/x) = f(x) + f(1/x)

fi f(1/x) [f(x) – 1] = f(x)

For x = 4, we have f(1/4) [f(4) – 1] = f (4)

fi f(1/4) [64] = 65

fi f(1/4) = 65/64 = 1/64 + 1

This means f(x) = x3 + 1

For f(6) we have f(6) = 216 + 1 = 217.

QUESTION: 9

Define the functions:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Q.

For what condition will A(x, y, z) be equal to Max (x, y, z)?

Solution:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Ans:-

When either x or y is maximum.

QUESTION: 10

Define the functions:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Q.

For what condition will A(x, y, z) not be equal to B (x, y, z)?

Solution:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Ans:-

When z is maximum, A and B would give different values. Thus, option (c) is correct.

QUESTION: 11

Define the functions:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Q.

The highest value amongst the following will be

Solution:

A(x, y, z) = Max (max (x, y), min (y, z) min (x, z))

B(x, y, z) = Max (max (x, y), min (y, z) max (x, z))

C(x, y, z) = Max (min (x, y), min (y, z) min (x, z))

D(x, y, z) = Min (max (x, y), max (y, z) max (x, z))

Max (x, y, z) = Maximum of x, y and z.

Min (x, y, z) = Minimum of x, y and z.

Assume that x, y and z are distinct integers.

Ans:-

We cannot determine this because it would depend on whether the integers x, y, and z are positive

or negative.

QUESTION: 12

A0, A1, A2,...... is a sequence of numbers withA0 = 1, A1 = 3, and At = (t +1) At–1 – t At–2 = 2, 3, 4,....Conclusion I. A8 = 77Conclusion II. A10 = 121Conclusion III. A12 = 145

Solution:

This question is based on the logic of a chain function. Given the relationship

At = (t + 1) At – 1 – t At – 2

We can clearly see that the value of A2 would depend on the values of A_{0} and A_{1}. Putting t = 2 in

the expression, we get:

A_{2} = 3A1 – 2A_{0} = 7; A_{3} = 19; A_{4} = 67 and A_{5} = 307. Clearly, A6 onwards will be larger than 307

and hence none of the three conclusions are true. Option (e) is the correct answer.

QUESTION: 13

The figure below shows the graph of a function f (x). How many solutions does the equation f ( f(x)) = 15 have?

Solution:

f ( f (x) = 15 when f (x) = 4 or f (x) = 12 in the given function. The graph given in the figure

becomes equal to 4 at 4 points and it becomes equal to 12 at 2 points in the figure. This gives us 6

points in the given figure when f (f (x) =15. However, the given function is continuous beyond the

part of it which is shown between –10 and +13 in the figure. Hence, we do not know how many

more solutions to f (f (x) = 15 would be there. Hence, Option (e) is the correct answer.

QUESTION: 14

For all real numbers x, except x = 0 and x = 1, the function F is defined by

If 0 < a < 90° then F((cosec a)2) =

Solution:

QUESTION: 15

where [x] is defined as integral part of x and f is a fraction, then x(1 – f) equals–

Solution:

retained the even terms which are integral. Hence, the value of x+y is an integer.

Further, x+y = [x] + f and hence, if x+y is an integer, [x] + f + y would also be an integer. This

automatically means that f+y must be an integer (as [x] is an integer).

Now, the value of y is between 0 to 1 and hence when we add the fractional part of x i.e. ‘f ’ to y,

and we need to make it an integer, the only possible integer that f + y can be equal to is 1.

Thus, if f + y = 1 Æ y = (1 – f ).

In order to find the value of x(1 – f ) we can find the value of x × y.

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