Quadratic Equation problems
- 1. If ?, ? are the roots of the equation x2 + kx + 12 = 0 such that ? ? ? = 1, the value of k is
Options- A. 0
- B. ± 5
- C. ± 1
- D. ± 7 Discuss
Correct Answer: ± 7
Explanation:
As per the given above question , we know that
Let, ?, ? be the roots of the equation x2 + kx + 12 = 0
? ? + ? = ?k and ?? = 12
Now ( ? - ? )2 = ( ? + ? )2 - 4??
( 1 )2 = k2 - 4(12)
? k2 = 49 ? k = ± 7.
- 2. Determine k such that the quadratic equation x2 - 2( 1 + 3k ) x + 7 ( 3 + 2k ) = 0 has equal roots.
Options- A. 2, - 10 9
- B. 2, 10 9
- C. - 2, 10 9
- D. - 2, -10 9 Discuss
Correct Answer: 2, - 10 9
Explanation:
- 3. If ? and ? are the roots of the equation x2 - 3?x + ?2 = 0,
find ? if ?2 + ?2 = 7 4
Options- A. ± 1 2
- B. ± ?7 2
- C. ± ?3 2
- D. None of these Discuss
Correct Answer: ± 1 2
Explanation:
- 4. If a and b are the roots of the equation x 2 - 6x + 6 = 0, find the value of 2(a 2 + b 2).
Options- A. 40
- B. 42
- C. 48
- D. 46 Discuss
Correct Answer: 48
Explanation:
a and b are the roots of the equation
x2 - 6x + 6 = 0
? a + b = -B/A = 6 and ab = C/A = 6
We know that,
a2 + b2 = (a + b)2 - 2ab
= (6)2 - 2 x 6 = 36 - 12 = 24
? 2(a2 + b2) = 2 x 24 = 48
- 5. If ? and ? be the roots of the equation ax 2 + bx + c = 0, find the value of ? 2/? + ? 2/? .
Options- A. ab -b2c/2b2c
- B. 3bc - a3/b2c
- C. 3ac - b2/a3c
- D. 3abc - b3/a2c Discuss
Correct Answer: 3abc - b3/a2c
Explanation:
Given, ? and ? are the roots of the equation ax2 + bx + c = 0
? Sum of two roots = -b/a
? + ? = -b/a
and product of two roots = c/a
? ? = c/a
? ?2/? + ?2/? = ( ?3 + ?3) / ? ?
= [(? + ?) 3 - 3 ? ? (? + ?)] / ??
[? (a + b)3 = a3 + b3 + 3ab(a + b) ]
= (-b/a)3 - [3c/a(-b/a)/c/a = -b3/a3 + 3bc/a2]/(c/a)
= 3abc - b3/a2c
- 6. If ? and ? are the roots of the equation x 2 - 11x + 24 = 0, find the equation having the roots ? + 2 and ? + 2
Options- A. x2 + 15x + 24 = 0
- B. x2 - 15x + 24 = 0
- C. x2 + 15x - 50 = 0
- D. x2 + 15x - 60 = 0 Discuss
Correct Answer:
Explanation:
Given equation is
x2 - 11x + 24 = 0
Then, required equation is
(x - 2)2 - 11(x - 2) + 24 = 0
? x2 - 4x + 4 - 11x + 22 + 24 = 0
? x2 - 15x + 50 = 0
- 7. If x 2 = 6 + ? 6 + ?6 + ?6 + .... ? , then what is one of the value of x equal to?
Options- A. 6
- B. 5
- C. 4
- D. 3 Discuss
Correct Answer: 3
Explanation:
Here, x2 = 6 + ?6 + ?6 + ?6 + .... ? ,
So, x2 = 6 + ?x2
? x2 = 6 + x
? x2 - x - 6 = 0
? x2 + 2x - 3x - 6 = 0
? x(x + 2) - 3(x + 2) = 0
? (x - 3) (x + 2) = 0
? x = 3
- 8. If a = p / (p + q) and b = q / (p - q), then and ab/a + b is equal to
Options- A. pq / (p2 + q2)
- B. (p2 + q2) / pq
- C. p / (p + q)
- D. [p / (p + q)]2 Discuss
Correct Answer: pq / (p2 + q2)
Explanation:
ab/(a + b) = [p/(p + q)] x [q/(p - q)] / [p/(p + q) + q/(p - q)]
= pq/p2 - pq + pq + q2
= pq/(p2 + q2)
- 9. The sum of the roots of the equation x 2 + px + q = 0 is equal to the sum of their squares, then
Options- A. p2 - q2 = 0
- B. p2 + q2 = 2q
- C. p2 + p = 2q
- D. q2 + q2 = 2p Discuss
Correct Answer: p2 + p = 2q
Explanation:
Let ? and ? be the roots of the equation
x2 + px + q = 0
Then, ? + ? = -p, ? ? = q
According to the question,
? + ? = ?2 + ?2
? ? + ? = (? + ?)2 - 2? ?
? -p = p2 - 2q ? p2 + p = 2q
- 10. If ? and ? are the roots of the equation 8x 2 - 3x + 27 = 0, find the value of (? 2/?) 1/3 + (? 2/?) 1/3
Options- A. 1/3
- B. 1/4
- C. 7/2
- D. 4 Discuss
Correct Answer: 1/4
Explanation:
Since, ? and ? are the roots of the equation
8x2 - 3x + 27 = 0
? ? + ? = 3/8 and ? ? = 27/8
? (?2/?)1/3 + (?2/? )1/8
= (?3)1/3 + (?3)1/3/(??)3
= ? + ? / (??)1/3 = 3/8/(27/8)1/3
= 3/8 x 2/3 = 1/4
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