Simplification problems
- 1. If a + b + c = 14 and a 2 + b 2 + c 2 = 96, then (ab + bc + ca) =?
Options- A. 51
- B. 55
- C. 50
- D. 65 Discuss
Correct Answer: 50
Explanation:
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
- 2. 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.
Options- A. 244
- B. 242
- C. 294
- D. 255 Discuss
Correct Answer: 242
Explanation:
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
- 3. 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?
Options- A. 9
- B. 0
- C. 8
- D. -3 Discuss
Correct Answer: 9
Explanation:
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
- 4. (2 - 1/3) (2 - 3/5) (2 - 5/7) ... (2 - 997/999) is equal to?
Options- A. 1001/999
- B. 999/1001
- C. 1001/3
- D. 5/1001 Discuss
Correct Answer: 1001/3
Explanation:
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
- 5. 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?
Options- A. 5/11
- B. 4/9
- C. 11/18
- D. 17/36 Discuss
Correct Answer: 11/18
Explanation:
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
- 6. If x = ?3 + ?2 , then the value of (x + 1/x) is
Options- A. 2
- B. 3
- C. 2?2
- D. 2?3 Discuss
Correct Answer: 2?3
Explanation:
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
- 7. Find the value of a 3 + b 3 + c 3 - 3abc when a = 225, b = 226 , c = 227
Options- A. 2304
- B. 2430
- C. 2034
- D. 2340 Discuss
Correct Answer: 2034
Explanation:
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
- 8. Simplify 6 - [9 - {18 - (15 -12 - 9)}]
Options- A. 1
- B. 4
- C. 5
- D. 3 Discuss
Correct Answer: 3
Explanation:
Given expression
= 6 - [9 - {18 - (15 -12 - 9)}]
= 6 - [9 - {18 - (15 - 12 + 9)}]
= 6 - [9 - {18 - 12}]
= 6 - [9 - 6] = 6 - 3 = 3
- 9. 10.8 x 5.5 x 8.4 =?
Options- A. 458.69
- B. 489.96
- C. 498.96
- D. 485.69 Discuss
Correct Answer: 498.96
Explanation:
Multiply the given in question.
The answer would be
? = 10.8 x 5.5 x 8.4 = 498.96
- 10. 8648 - 7652 =? x 40
Options- A. 24.7
- B. 28.9
- C. 27.4
- D. 25.9 Discuss
Correct Answer:
Explanation:
8648 - 7652 = ? x 40
996 = ? x 40 ? ? = 996/40 = 24.9
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