Exercise 9


1. Which value of x does statisfy the inequality 2x2 + x – 3 < 0?
(a) -32 < x < 1
(b) -1 < x < 32
(c) x > 1
(d) x < -25
(e) None of these
Ans.a

2x2 + x – 3 < 0

⇒ (x – 1)(2x + 3) < 0 or, -32 < x < 1

2. The difference between a number and its one-third is double of its one-third. What is the number?
(a) 60
(b) 18
(c) 30
(d) Cannot be determined
(e) None of these
Ans.d

Let the no. be x.

Then, x - x3 = 23x

or, 23x = 23x

So, can’t be determined is the correct choice.

3. Two pens and three pencils cost 86. Four pens and a pencil cost 112. What is the difference between the cost of a pen and that of a pencil?
(a) 25
(b) 13
(c) 19
(d) Cannot be determined
(e) None of these
Ans.b

Let the cost of a pen and a pencil be ‘x’ and ‘y’ respectively. We have to find (x – y).

From the question,

2x + 3y = 86 ..... (i)

4x + y = 112 ......(ii)

Subtracting (i) from (ii), we get

2x – 2y = 26 or, x – y = 13

4. The difference between a two-digit number and the number after interchanging the position of the two digits is 36. What is the difference between the two digits of the number?
(a) 4
(b) 6
(c) 3
(d) Cannot be determined
(e) None of these
Ans.a

Let the two-digit no. be l0x + y.

Then, (10x + y) – (10y + x) = 36

or, 9(x – y) = 36

or, x – y = 4

5. If the digit in the unit’s place of a two-digit number is halved and the digit in the ten’s place is doubled, the number thus, obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true?
(a) Digits in the unit’s place and the ten’s place are equal.
(b) Sum of the digits is a two-digit number.
(c) Digit in the unit’s place is half of the digit in the ten’s place.
(d) Digit in the unit’s place is twice the digit in the ten’s place.
(e) None of these
Ans.d

Suppose the two-digit number is

10 x + y

Then, 10 y + x = 20x + y2

or 20y + 2x = 40x + y or, y = 2x

6. If A and B are positive integers such that 9A2 = 12A + 96 and B2 = 2B + 3, then which of the following is the value of 5A + 7B ?
(a) 31
(b) 41
(c) 36
(d) 43
(e) 27
Ans.b

9A2 = 12A + 96 ⇒ 3A2 – 4A – 32 = 0

\ A = \(\frac{4 \pm \sqrt{16 + 384}}{6}\) = 4 , -83

B2 = 2B + 3 ⇒ B2 – 2B – 3 = 0

\ B = \(\frac{2 \pm \sqrt{4 + 12}}{2}\) = 3 , -1

\ 5A + 7B = 5 × 4 + 7 × 3 = 20 + 21 = 41

7. On Children’s Day, sweets were to be equally distributed among 175 children in a school. Actually on the Children’s Day 35 children were absent and therefore, each child got 4 sweets extra. How many sweets were available in all for distribution?
(a) 2480
(b) 2680
(c) 2750
(d) 2400
(e) None of these
Ans.e

Let the original number of sweets be x.

According to the question,

x140 - x175 = 4

or, 175x – 140x = 4 × 140 × 175

or, x = \(\frac{4 \times 140 \times 175}{35} = 2800\)

8. A two-digit number is seven times the sum of its digits. If each digit is increased by 2, the number thus obtained is 4 more than six times the sum of its digits. Find the number.
(a) 42
(b) 24
(c) 48
(d) Data inadequate
(e) None of these
Ans.a

Let the two-digit number be l0 x + y.

10x + y = 7(x + y) Þ x = 2y...(i)

10(x + 2) + y + 2 = 6(x + y + 4) +4

or 10x + y + 22 = 6x + 6y + 28 Þ 4x – 5y = 6 ... (ii)

Solving equations (i) and (ii), we get x = 4 and y = 2

9. One-third of Ramani’s savings in National Savings Certificate is equal to one-half of his savings in Public Provident Fund. If he has 150000 as total savings, how much he saved in Public Provident Fund?
(a) 60000
(b) 50000
(c) 90000
(d) 30000
(e) None of these
Ans.a

Ratio of Ramani’s savings in NSC and PPF = 3: 2

His savings in PPF = 25 × 150000 = 60000

10. 15 of a number is equal to 58 of the second number. If 35 is added to the first number then it becomes 4 times of second number. What is the value of the second number?
(a) 125
(b) 70
(c) 40
(d) 25
(e) None of these
Ans.c

Let x be the first number and y be the second number.

15x = 58y

\ xy = 258......(i)

x + 35 = 4y

or, 258y + 35 = 4y

\ y = 40