# Exercise 8

1. Which of the following expressions are different in value?

(A) (2x + 3y)2

(B) (2x + y)2 + 8y(x + y)

(C) (2x – y)2 – 8y(x + y)

(D) 22(x + y)2 + 4xy + 5y2

(a) A and B
(b) B and C only
(c) A, B and D only
(d) B and D only
(e) All are different

#### View Ans & Explanation

Ans.b

All others are equal except (C).

2. Which of the following values of x will satisfy the in equality 2x2 – 7x < 15?
(a) -32 < x < 5
(b) x > 5 or x < -32
(c) x< 5 and x < -32
(d) x > -32 and x > 5
(e) None of these

#### View Ans & Explanation

Ans.a

2x2 – 7x < 15

or, 2x2 – 7x – 15 < 0

or, 2x2 – 10x + 3x – 15 < 0

or, 2x(x – 5) + 3(x – 5) < 0

or, (x – 5)(2x + 3) < 0

----------|--------------------|--------

+ve -32 –ve 5 +ve

-32 < x < 5

3. Which values of ‘x’ satisfies the inequality
x2 - 3x + 2 > 2x - 4 ?
(a) 2< x < 3
(b) x > 3 or x < 2
(c) x ³ 3
(d) x £ 2
(e) None of these

#### View Ans & Explanation

Ans.b

Given expression is

x2 – 3x + 2 – 2x + 4 > 0

or, x2 – 5x + 6 > 0

or, x2 – 3x – 2x + 6 > 0

or, x(x – 3) – 2(x – 3) > 0

or, (x – 3)(x – 2) > 0

or, x > 3 or, x < 2

4. Which of the following values of x satisfies the inequality 2x2 – 3x + 1 > 0?
(a) -1 < x < -12
(b) 12 < x < 1
(c) x > 1 or x < 12
(d) -12 < x < 1
(e) None of these

#### View Ans & Explanation

Ans.c

2x2 – 3x + 1 = 0.

or (2x – 1)(x – 1) = 0

Hence, α = 12 and β = 1

Now, the given inequality is 2x2 – 3x + 1 > 0,

Hence, sign scheme will be as follows:

--------|---------------------|----------------

+ve 12 –ve 1 +ve

∴ x > 1 or x < 12

[Note:If the inequality were 2x2 – 3x + 1 < 0; the answer will be 12 < x < 1]
5. If 3x – 5y = 5 and $\frac{x}{x + y} = \frac{5}{7}$, then what is the value of x – y?
(a) 9
(b) 6
(c) 4
(d) 3
(e) None of these

#### View Ans & Explanation

Ans.d

3x – 5y = 5...(i)

And $\frac{x}{x + y} = \frac{5}{7}$

⇒ 7x = 5x+ 5y

⇒ 2x = 5y... (ii)

From (i) and (ii), x = 5 and y = 2

∴ x – y = 3

6. Which of the following values of x will satisfy the inequality x2 – x – 6 > 0 ?
(a) x < – 2 or x > 3
(b) –2 < x < 3
(c) –3 < x < 2
(d) x > – 2 or x < 3
(e) None of these

#### View Ans & Explanation

Ans.a

x2 – x – 6 > 0

or, x2 – 3x + 2x – 6 > 0

or, x(x – 3) + 2(x – 3) > 0

or, (x – 3)(x + 2) > 0

or, x = 3 or – 2

∴ x < – 2 or x > 3

7. 57 of 415 of a number is 8 more than 25 of 49 of the same number. What is half of that number?
(a) 630
(b) 315
(c) 210
(d) 105
(e) None of these

#### View Ans & Explanation

Ans.d

Let the number be x.

57 × 415 × x - 25 × 49 × x = 8

or, x = $\frac{8 \times 315}{12} = 210$

∴ Half of the number = 105

8. The difference between a two-digit number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
(a) 4
(b) 9
(c) 3
(d) Cannot be determined
(e) None of theses

#### View Ans & Explanation

Ans.a

Let the two-digit number be 10x + y.

Then, (10x + y) – (10y + x) = 36

or x – y = 4

9. By the how much is two-fifth of 200 greater than three -fifths of 125?
(a) 15
(b) 3
(c) 5
(d) 30
(e) None of these

#### View Ans & Explanation

Ans.c

Reqd no. = 25 × 200 - 35 × 125

= 80 - 75 = 5

10. If $\frac{x^{2} - 1}{x + 1}$ = 2, then, x = ?
(a) 1
(b) 0
(c) 2
(d) Can’t be determined
(e) None of these

#### View Ans & Explanation

Ans.e

(x – 1) = 2 ⇒ x = 3