Question 1

If √ 2 = 1.4142, then the value of 7 / (4 + √ 2) is?

Accepted Answer

√7 / (4 + √2) = [7 / (4 + √2)] x [(4 - √2 ) / (4 - √2] = 7(4 - √2) / (16 - 2) [∵ (a + b)(a - b) = a2 - b2] = 7(4 - √2 )/14 = (4 - √2)/2 = (4 - 1.4142)/2 = 2.5858/2 = 1.2929

Question 2

The value of (0.96) 3 - (0.1) 3/ [(0.96) 2 + (0.096) + 0.01] is?

Accepted Answer

(0.96)3 - (0.1)3/ [(0.96)3 + 0.096 + 0.01] = (0.96)3 - (0.1)3 = (0.96 - 0.1) [(0.96)2 + 0.96 x 0.1 + (0.1)2] [∵ a3 - b3 = (a - b) (a2 + ab + b2)] = (0.96 - 0.1) [(0.96)2 + 0.096 + 0.01]/(0.96)2 + 0.096 + 0.01 = 0.96 - 0.1 = 0.86

Question 3

3 2/ 3 + 2 3/ 4 + 1 1/ 2 =?

Accepted Answer

? = 32/3 + 23/4 + 11/2 = 3 + 2 + 1 + (2/3 + 3/4 + 1/2) = 6 + (8 + 9 + 6)/12 = 6 + (23/12) = 6 + (111/12) = 711/12

Question 4

(5568 ÷ 87) 1/3 + (72 x 2) 1/2 = (?) 1/2

Accepted Answer

(?)1/2 = (5568 ÷ 87)1/3 + (72 x 2)1/2 = (64)1/3 + (144)1/2 ∴ ? = (4 + 12)2 = 256

Question 5

(0.49) 4 x (0.343) 4 ÷ (0.2401) 4 = (70 ÷ 100) ? + 3

Accepted Answer

(70 ÷ 100) ? + 3 = (0.49)4 x (0.343)4 ÷ (0.2401)4 (0.7) ? + 3 = (0.49 x 0.343/0.2401)4 (0.7)? + 3 = (0.7)4 On comparing the exponents both sides, we get ? + 3 = 4 ∴ ? = 4 - 3 = 1

Question 6

[0.5 x 0.5 x 0.5 + 0.2 x 0.2 x 0.2 + 0.3 x 0.3 x 0.3 - 3 x 0.5 x 0.3 x 0.2] / [0.5 x 0.5 + 0.2 x 0.2 + 0.3 x 0.3 - 0.5 x 0.2 - 0.2 x 0.3 - 0.5 x 0.3]

Accepted Answer

Given expression = a3+ b3 + c3 - 3abc/a2 + b2 + c2 - ab - bc - ca = a + b + c = 0.5 + 0.2 + 0.3 where, a = 0.5, b = 0.2, c = 0.3

Question 7

What is the value of [725 x 725 x 725 + 371 x 371 x 371] / [725 x 725 - 725 x 371 + 371 x 371] =?

Accepted Answer

[725 x 725 x 725 + 371 x 371 x 371] / [725 x 725 - 725 x 371 + 371 x 371] = 725 + 371 = 1096 [∵ (a3 + b3) / (a2 - ab + b2) = (a + b) (a2 - ab + b2)/(a2 - ab + b2) = a + b ]

Question 8

If a 2 + b 2 = 234 and ab = 108, find the value of (a + b) / (a - b).

Accepted Answer

We know that, (a + b)2 = a2 + b2 + 2ab = 234 + 2 x 108 = 450 (a - b)2 = a2 + b2 - 2ab = 234 - 2 x 108 = 18 ∴ (a + b)2/(a - b)2 = 450/18 = 25 ⇒ [(a + b)/(a - b)]2 = 25 ∴ (a + b)/(a - b) = √25 = 5

Question 9

If x + 1/x = 2, then 2x 2 + 2/3x 2 + 5x + 3 =?

Accepted Answer

Given expression = 2x2 + 2/3x2 + 3 + 5x On dividing numerator and denominator by x, we get = 2(x + 1/x)/3(x + 1/x) + 5 = 2 x 2/3 x 2 + 5 = 4/11 [given x + 1/x = 2]

Question 10

If a, b and c are non-zero, a + (1/b) = 1 and b + (1/c) = 1, then the value of abc is

Accepted Answer

∵ a + (1/b) = 1 ab + 1 = b ......(i) Also, b + (1/c) = 1 ⇒ b = 1 - (1/c) ....(ii) From Eqs. (i) and (ii), we get ab + 1 = 1 - (1/c) ⇒ ab = -1/c ⇒ abc = -1

Question 11

If a + b + c = 14 and a 2 + b 2 + c 2 = 96, then (ab + bc + ca) =?

Accepted Answer

We know that, (a + b + c)2 = (a2 + b2 + c2) + 2(ab + bc + ca) ⇒ 196 = 96 + 2(ab + bc + ca) ⇒ 2(ab + bc + ca) = 196 - 96 = 100 ∴ (ab + bc + ca) = 100/2 = 50

Question 12

The highest score in an inning was 3/11 of the total score and the next highest was 3/11 of the remainder. If the score differ by 18, find the total score.

Accepted Answer

Let total score = N. Then, the highest score = 3N/11. Remainder = (N - 3N/11) = 8N/11 Next highest score = 3/11 of 8N/11 = 24N/121 According to the question, (3N/11) - (24N/121) = 18 ⇒ 33N - 24N = 18 x 121 ⇒ 9N = 18 x 121 ∴ N = (18 x 121)/9 = 242

Question 13

If a + b + c = 0, then the value of [(a + b)/c + (b + c)/a + (c + a)/b)] x [a/(b + c) + b/(c + a) + c/(a + b)] is?

Accepted Answer

Since, a + b + c = 0 Then, a + b = - c .....(i) a + c = - b ........(ii) b + c = - a ......(iii) Now,[(a + b)/c + (b + c)/a + (c + a)/b)] x [a/(b + c) + b/(c + a) + c/(a + b)] Now, putting the value of a + b, b + c and c + a from Eqs (i), (ii) and (iii), we get Given expression = [(-c)/c + (-a)/a + (-b)/b] [a/-a + b/-b + c/-c] [(-1) + (-1) + (-1)] [(-1) + (-1) + (-1)] = (-3) x (-3) = 9

Question 14

(2 - 1/3) (2 - 3/5) (2 - 5/7) ... (2 - 997/999) is equal to?

Accepted Answer

Given expression = (2 - 1/3) (2 - 3/5) (2 - 5/7) ... (2 - 997/999) = ((6 - 1)/3) ((10 - 3)/5) ((14 - 5)/7) ... ((1998 - 997)/999) = 5/3 x 7/5 x 9/7 x ... x 1001/999 Multiply and solve by using the law of algebra, = 5/3 x 7/5 x 9/7 x ... x 1001/999 = 1001/3

Question 15

In a certain office (1/3) of the worker are women (1/2) of the women are married and (1/3) of the married women have children. If (3/4) of the man are married and (2/3) of the married men have children. What part of workers are without children?

Accepted Answer

Let, total number of workers be N . Then number of women = N/3 Number of men = 2N/3 Number of women having children = 1/3 of 1/2 of N/3 = N/18 Number of men having children = 2/3 of 3/4 of 2N/3 = N/3 Number of workers having children = N/18 + N/3 = 7N/18 Number of workers having no children = N - 7N/18 = 11N/18 =11/18 of all workers

Question 16

If x = √3 + √2 , then the value of (x + 1/x) is

Accepted Answer

Given , X = √3 + √2 ∴ 1/x = 1/(√3 + √2) Now Multiply and Divide by (?3 - ?2) in 1/x. After multiply and divide, we will get 1/x = (1/(?3 + ?2)) X (?3 - ?2) /(?3 - ?2) [rationalising] ⇒ 1/x = √3 - √2/3 - 2 ∴ x + 1/x = √3 + √2 + √3 - √2 = 2√3

Question 17

Find the value of a 3 + b 3 + c 3 - 3abc when a = 225, b = 226 , c = 227

Accepted Answer

We know that, a3 + b3 + c3 - 3abc = (a + b + c) 1/2 [(a - b) 2 + (b - c) 2 + (c - a) 2] = (225 + 226 + 227) 1/2 [1 + 1 + 4] = 678 x 3 = 2034

Question 18

Simplify 6 - [9 - {18 - (15 -12 - 9)}]

Accepted Answer

Given expression = 6 - [9 - {18 - (15 -12 - 9)}] = 6 - [9 - {18 - (15 - 12 + 9)}] = 6 - [9 - {18 - 12}] = 6 - [9 - 6] = 6 - 3 = 3

Question 19

10.8 x 5.5 x 8.4 =?

Accepted Answer

Multiply the given in question. The answer would be ? = 10.8 x 5.5 x 8.4 = 498.96

Question 20

8648 - 7652 =? x 40

Accepted Answer

8648 - 7652 = ? x 40 996 = ? x 40 ⇒ ? = 996/40 = 24.9

Simplification Questions