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
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
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
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
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
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
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
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
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
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
Ans.e

(x – 1) = 2 ⇒ x = 3