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**Question 12****:** The value of f∘g∘h(9) could be, if

f(x) **=**** 1/x****g(x) = 1/(x-2)**

h(x) **= √**x

A. 3

B. 1/3

C. -5

D. None of these

**Answer. None of these **

**Explanation.**

**fogoh(9) means f(g(h(9)))Start by solving h(9) = √9 = 3 and not (3, -3) as square root of a number is always positive.Taking the value to be 3, g(3) = **1/(3-2) = 1 ⇒ f(1) = 1

The question is **"What is value of f∘g∘h(9) could be?"**

Choice D is the correct answer.

**Question 13****:** For this question, assume the following operators:

A*B = A^{2} - B^{2}

A-B = **A/B**

A+B = A * B

**A/B **= A+B

Which of the following expression would yield the result as x subtracted by y?

A. (x*y)-(x+5)

B. (x/y)*(x-y)

C. (x*y)-(x/y)

D. (x+y)*(x-y)

**Answer. (x*y)-(x/y)**

**Explanation.**

**Never solve each expression. Solve by assigning values of x and yLet x = 6, y = 2 Therefore, required x – y = 4x*y = 6**

**x-y = 6/2 = 3**

**x+y = 6 * 2 = 12 **

**x/y = 6+2 = 8**

**a) 32-12 = 32/12 not equal to 4 **

**b) 8*3 = 8 ^{2} - 3^{2} = 55 not equal to 4 **

**c) 32 - 8 = 32/8 = 4**

**d) 12*3 = 12 ^{2} - 3^{2} not equal to 4 **

**The question is "Which of the following expression would yield the result as x subtracted by y?" **

Choice C is the correct answer.

**Question 14****:** Find the domain of:

A. (-∞,9)

B. [-1,9)

C. [-1,9) excluding 0

D. (-1,9)

**Answer. (-1,9)**

**Explanation.**

In the expression,

9-x > 0

⇒ x < 9

Also, 1-log ( 9-x) ≠ 0

⇒ log (9 – x) ≠ 1

⇒ 9-x ≠ 10

⇒ x ≠ -1

And, x + 1 > 0

⇒ x > -1

The question is **"Find the domain of: **

Choice D is the correct answer.

**Question 15****:** If [X] – Greatest integer less than or equal to x. Find the value of

[√1] + [√2] + [√3] +……………………………………………………+ [√100]

A. 615

B. 625

C. 5050

D. 505

**Answer. 625**

**Explanation.**

**As can be seen, Thus, the series becomes:3 * 1 + 5 * 2 + ……………………………………+ 19 * 9 + 10⇒ ∑(2n+1)n + 10 (where, n = 1 to 9)⇒ ∑2n**

**⇒ **

**⇒ **

**⇒ **570 + 45 + 10

The question is **"Find the value of [√1] + [√2] + [√3] +……………………………………………………+ [√100]?"**

Choice B is the correct answer.

**Question 16****:** Find the value of x for which x[x] = 39?

A. 6.244

B. 6.2

C. 6.3

D. 6.5

**Answer. 6.5**

**Explanation.**

When x = 7,

x[x] = 7 * 7 = 49

When x = 6,

x[x] = 6 * 6 = 36

Therefore, x must lie in between 6 and 7 ⇒ [x] = 6

⇒

The question is **"Find the value of x for which x[x] = 39?"**

Choice D is the correct answer.

**Question 17****:** Find the value of x for which x[x] = 15?

A. 3.5

B. 5

C. 6.1

D. None of these

**Answer. None of these**

**Explanation.**

Using similar approach as previous question, [x] = 3

⇒ x = 15/3 ⇒ x = 5 which is not possible since 3 < x < 4

The question is **"Find the value of x for which x[x] = 15?"**

Choice D is the correct answer.

**Question 18****:** If f(x) = **1/g(x),** then which of the following is correct?

A.** **f(f(g(g(f(x))))) = g(f(g(g(g(x)))))

B. f(f(f(g(g(g(f(g(x)))))))) = g(g(g(g(f(g(f(f(x))))))))

C. f(f(g(f(x)))) = g(g(f(g(x))))

D. f(g(f(f(g(f(g(g(x)))))))) = g(g(g(g(f(f(f(f(x))))))))

**Answer. f(g(f(f(g(f(g(g(x)))))))) = g(g(g(g(f(f(f(f(x))))))))**

**Explanation.**

**Given, f(x) = **1/g(x) ⇒ f(x).g(x) = 1** which implies that f(x) and g(x) are essentially inverse of one another.So, one just has to look for an option which has equal number of fs and gs on both side of the equation. **

**The question is ****"**** If f(x) =** 1/g(x),

Choice D is the correct answer.

**Question 19****:** If f(x) = Find the value of x for which f(x) = f^{-1}(x)?

A. -3

B. 2

C. Both A and B

D. None of these

**Answer. Both A and B**

**Explanation.**

**f(x) = f ^{-1}(x) when f(x) = x **

**⇒ **

**⇒**** x + 6 = x ^{2} + 2x⇒ x^{2} + x – 6 = 0⇒ (x+3) (x-2) = 0⇒ x = -3, 2 **

The question is **"Find the value of x for which f(x) = f ^{-1}(x)?"**

Choice C is the correct answer.

**Question 20****:** If f(x) = |x| + |x+3| + |x+6| + ……………………………..+ |x+3t|, where x is an integer and t is a positive integer, find the minimum value of f(x) when t = 6?

A. 63

B. 36

C. 30

D. 25

**Answer. 36**

**Explanation.**

When t = 3,

f(x) = |x| + |x+3| + |x+6| + ……………………………..+ |x+18|

Minimum value of this function will occur when x = -9 i.e. the middle term is at its minimum which is 0.

Therefore, f(-9) = 9 + 6 + 3 + 0 + 3 + 6 + 9

= 2 * 18

= 36

The question is **"find the minimum value of f(x) when t = 6?"**

Choice B is the correct answer.

**Question 21****:** In the previous question if t = 7, for how many values of x, f(x) will be minimum?

A. 1

B. 2

C. 4

D. 8

**Answer. 4**

**Explanation.**

When t = 7,

f(x) = |x| + |x+3| + |x+6| + ……………………………..+ |x+21|

This expression has two middle terms: |x+9|, |x+12|

The value of f(x) will be minimized when the sum of the two middle terms are minimized

=) |x+9|+ |x+12| should be minimum

This happens when -12 ≤ x ≤ -9, Note that for x = -12, -11, -10, -9 the value of the sum of the middle terms = 3.

Therefore, for all 4 values of x, f(x) will have minimum value.

The question is **"In the previous question if t = 7, for how many values of x, f(x) will be minimum?"**

Choice C is the correct answer.

**Question 22****: **If f(x^{2} – 1) = x^{4} – 7x^{2} + k1 and f(x^{3} – 2) = x^{6} – 9x^{3} + k_{2} then the value of (k_{2} – k_{1}) is

A. 6

B. 7

C. 8

D. 9

E. None of the above**Answer. 8**

**Explanation. **

Given Data

f(x^{2} – 1) = x^{4} – 7x^{2} + k_{1}

f(x^{3} – 2) = x^{6} – 9x^{3} + k_{2}

When x^{2} = 1, f(x^{2} – 1) = f(1 - 1) = f(0) =(1)^{2} - 7(1) + k_{1}

f(0) = - 6 + k_{1} ..........(1)

Essentially, we have replaced all x^{2} with 1.

When x^{3} = 2,f(x^{3} – 2) = f(2 - 2) = f(0) =(2)^{2} - 9(2) + k_{2}

f(0) = - 14 + k_{2} ..........(2)

Essentially, we have replaced all x^{3} with 2.

Equating f(0) in equations (1) and (2)

(-6 + k_{1}) = (-14 + k_{2})

or k_{2} - k_{1} = 8

**Question 23****: **Which of the following is not an odd function?

(1) f(x) = -x^{3}

(2) f(x) = x^{5}

(3) f(x) = x^{2} – x

(4) f(x) = |x|^{3}

**Answer. f(x) = |x| ^{3}**

**Explanation. **

An odd function is a function whose value reverses in sign for a reversal in sign of its argument. i.e. f(x) = -f(-x).

Except f(x) = |x|^{3} all other functions mentioned in the choices change values.

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