Elementary Algebra problems
- 1. If a, b, c are real and
a3 + b3 + c3 = 3abc and
a + b + c ≠ 0 , then the relation between a, b, c will be
Options- A. a + b = c
- B. a + c = b
- C. a = b = c
- D. b + c = a Discuss
Correct Answer: a = b = c
Explanation:
Given that :-
Now ,a3 + b3 + c3 = 3abc
( a+b+c )( a2 + b2 + c2 - ab - bc - ca ) + 3abc = 3abc
( a+b+c )( a2 + b2 + c2 - ab - bc - ca ) = 0
∴ a² + b² + c² - ab - bc - ca = 0 or a + b + c ≠ 0
Where ,
a = b = c
- 2. The line passing through the points (? 2, 8) and (5, 7)
Options- A. does not cut any axes
- B. cuts x- axis only
- C. cuts y - axis only
- D. cuts both the axes Discuss
Correct Answer: cuts y - axis only
Explanation:
As indicated in the graph, the line passing through the points cuts Y-axis only.
- 3. If the difference of two numbers is 3 and the difference of their squares is 39; then the larger number is:
Options- A. 9
- B. 12
- C. 13
- D. 8 Discuss
Correct Answer: 8
Explanation:
Let the numbers are p and q.
p ? q = 3 ? (1)
p2 ? q2 = 39
(p ? q) (p + q) = 39 ? x + y = 13 ? (2)
Adding eqn (1) and (2)
p + q + p ? q = 16 ? p = 8
? q = 3
Hence, 8 is the larger number.
- 4. Let a = ?6 - ?5, b = ?5 - 2, c = 2 - ?3 Then point out the correct alternative among the four alternatives given below.
Options- A. a < b < c
- B. b < a < c
- C. a < c < b
- D. b < c < a; Discuss
Correct Answer: a < b < c
Explanation:
We know that , ?6 = 2.44, ?5 = 2.23, ?3 = 1.73
a = ?6 - ?5 = 2.44 - 2.23 = 0.21
b = ?5 - 2 = 2.23 - 2 = 0.23
c = 2 - ?3 = 2 - 1.73 = 0.27
- 5. If a2 + b2 + c2 = 2 (a ? b ? c) ? 3, then the value of 2a ? 3b + 4c is
Options- A. 1
- B. 7
- C. 2
- D. 3 Discuss
Correct Answer: 1
Explanation:
Given that :- a2 + b2 + c2 = 2 (a ? b ? c) ? 3
? a2 + b2 + c2 - 2 (a ? b ? c) + 3 = 0
? a2 + b2 + c2 - 2a + 2b + 2c + 3 = 0
? ( a2 + 1 - 2a ) + ( b2 + 1 + 2b ) + ( c2 + 1 + 2c ) = 0
? ( a - 1 )2 + ( b + 1 )2 + ( c + 1 )2 = 0
This is possible when ( a - 1 )2 = 0, ( b + 1 )2 = 0, and ( c + 1 )2 = 0
? a = 1, b = ?1, c = ?1
Thus, 2a ? 3b + 4c = 2 (1) ? 3 (?1) + 4 (?1)
2a ? 3b + 4c = 2 + 3 ? 4 = 1.
- 6. If the square of the sum of two numbers is equal to 4 times of their product. then the ratio of these numbers is:
Options- A. 2 : 1
- B. 1 : 3
- C. 1 : 1
- D. 1 : 2 Discuss
Correct Answer: 1 : 1
Explanation:
Let the number be p and q.
According to question,
(p + q)2 = 4pq
? p2 + q2 + 2pq ? 4pq = 0
? (p ? q)2 = 0
? p = q
Hence , required ratio is 1 : 1 .
- 7. A man buys 3 cows and 18 goats in ? 47,200. Instead if he would have bought 8 cows and 3 goats, he had to pay ? 53,000 more. Cost of one cow is:
Options- A. ? 10,000
- B. ? 11,000
- C. ? 12,000
- D. ? 13,000 Discuss
Correct Answer: ? 12,000
Explanation:
- 8. In a two-digit number, the digit at the unit's place is 1 less than twice the digit at the ten's place. If the digits at unit's and ten's place are interchanged, the difference between the new and the original number is less than the original number by 20. The original number is
Options- A. 47
- B. 59
- C. 23
- D. 35 Discuss
Correct Answer: 47
Explanation:
Since two digit number = 10p + q
According to question , ? q = 2p ? 1.. (i)
When digits are interchanged then new number = 10q + p
then original number ? [new number ? original number] = 20
? 10p + q ? [10q + p ? (10p + q)] = 20
? 10p + q ? 10q ? p + 10p + q = 20
19p ? 8q = 20
19p ? 8 (2p ? 1) = 20 (Using eq. (i))
19p ? 16p + 8 = 20
3p = 12 ? p = 4
From (i) q = 2 × 4 ? 1 ? q = 7
? original number = 10p + q = 10 × 4 + 7 = 47
- 9. The linear equation such that each point on its graph has an ordinate four times its abscissa is:
Options- A. y + 4x = 0
- B. y = 4x
- C. x = 4y
- D. x + 4y = 0 Discuss
Correct Answer: y = 4x
Explanation:
The linear equation is y = 4x,
When, x = 1, y = 4
- 10. A sum of Rs. 13.50 is made by 38 coins which are either 50 paise or 25 paise. Find the number of 25 paise coins?
Options- A. 16
- B. 24
- C. 18
- D. 22 Discuss
Correct Answer: 22
Explanation:
Let the number of 25 paise coins and 50 paise cons be x and y,respectively.
According to the question,
x + y = 38
Also, 25x + 50y = 1350
On multiplying Eq.(i) by 25 and substracting from Eq.(ii), we get y =16
On substituting the value of y in Eq.(i),we get y=22
So,the number of 25 paise coins in 22.
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