| 1. For what value of k, are the roots of the equation x2+56-k(4x-20) = 0 equal? | Easy |
| A. -2, 7 B. 3, 5 C. 3, 6 D. None of these |
View Answer
Answer: Option A
Explanation:
Rearranging the terms of the given equation, we get
x2-4kx+(20k+56) = 0
If the roots of the given equation are equal, then it’s discriminant = 0
i.e. (-4k)2-4(20k+56) = 0 or k2-5k-14 = 0 or k = -2, 7
| 2. When the coefficient of x in the quadratic equation x2+px+q = 0 was taken as -11 in place of -9, and the value of q was taken as 2q, its roots were found to be 4 and 7. Find the roots of the original equation. | Medium |
| A. -2, 7 B. 2, 7 C. -2, -7 D. None of these |
View Answer
Answer: Option B
Explanation:
Given roots of the equation x2+px+q = 0 are 4, 7 if p = -11 and the constant is twice of q.
-> 4, 7 will satisfy x2 – 11x+r = 0 (where r = 2q)
i.e. (4)2 – 11(4)+ r = 0 , if x = 4
or r = 28
Therefore, putting p =-9 (actual value) and q = r/2 = 14 in x2+px+q = 0, the original equation is x2 -9x+14 = 0 or (x-2)(x-7) = 0 which gives the required roots as 2, 7
| 3. The quadratic equation in x was solved by two students. The first student made an error in the constant term, resulting in the roots being -1 and -4. The second student made a mistake in the coefficient of x, which led to the roots being 2 and 3. Determine the accurate roots of the quadratic equation. | Difficult |
| A. 2, 3 B. -2, -3 C. 2, -3 D. None of these |
View Answer
Answer: Option B
Explanation:
Let the correct equation be x2+px+q = 0 —– ❶
The roots found by the first student are -1 and -4. Their sum = -5 and product = 4
So, ❶ reduces to x2 – (sum of the roots)x+ (product of the roots) = 0 -> x2 + 5x+ 4 = 0 —– ❷
But he has committed a mistake only in the constant term, i.e., in q, so the coefficient of x remains equal to 5.
Now roots found by the second student are 2 and 3. Their sum = = 5 and product = 6
So ❶ reduces to x2 – (sum of the roots)x+ (product of the roots) = 0 à x2 – 5x+ 6 = 0 —– ❸
But he has committed a mistake only in the coefficient of x, i.e., the constant remains equal to 6.
Hence the correct equation from ❶ is x2 + 5x+6 = 0
-> (x+2)(x+3) = 0 or x = -2, -3 are the required roots.
| 4. Determine the values of k for which the equation 5x2-4x+2+k(4x2-2x-1) = 0 has sum of roots as 6. | Easy |
| A. -13/11 B. 13/11 C. -11/13 D. None of these |
View Answer
Answer: Option A
Explanation:
Given equation can be written as
(5+4k)x2-(4+2k)x+(2-k) = 0
Sum of its roots = (4+2k)/(5+4k) = 6 (given)
-> 2(2+k) = 6(5+4k)
-> 2+k = 15+12k or k = -13/11
| 5. If x is real, then find the minimum value of x2-8x+17. | Easy |
| A. 0 B. 1 C. -1 D. Can’t be determined |
View Answer
Answer: Option B
Explanation:
x2-8x+17 = (x2-8x+16)+ 1 = (x-4)2+1
Since x is real, (x-4)2 is always positive and its least value is 0 and so the minimum value of given expression is 1.
| 6. If the difference of the squares of the roots of the equation x2-6x+q = 0 is 24, find the value of q. | Medium |
| A. 4 B. 5 C. 6 D. None of these |
View Answer
Answer: Option B
Explanation:
Let α,β be the roots of the given equation x2-6x+q = 0
Then α + β= sum of roots = 6
and αβ = product of roots = q
Therefore, (α-β)2 = (α+β)2 -4αβ = 36 – 4q
According to the problem, α2 – β2 = 24
or (α – β)(α + β) = 24
or (α – β)2(α + β)2 = (24)2
or (36-4q)(6)2 = (24)2
or 36-4q = 16
or 4q = 20 or q = 5
| 7. If α,β are roots of the equation 4x2+3x+7 = 0, then find the value of (1/β) + (1/α). | Easy |
| A. -3/7 B. -2/7 C. -4/7 D. None of these |
View Answer
Answer: Option A
Explanation:
Here α+β = -3/4; αβ = 7/4
Therefore, 1/β + 1/α = (α+β)/αβ = (-3/4)/(7/4) = -3/7
| 8. If α,β are the roots of the equation ax2+2bx+c = 0, then express (α/β)+(β/α) in terms of a, b and c. | Medium |
| A. (4b2-2ac)/ac B. (4b3-2ac)/ac C. (4b2-2ac)/a D. None of these |
View Answer
Answer: Option A
Explanation:
Here α+β = -2b/a, αβ = c/a
α/β + β/α = (α2 + β2)/ αβ = [( α+β)2-2αβ]/αβ
= [(-2b/a)2-2(c/a)]/(c/a)
= (4b2-2ac)/ac
| 9. If the roots of a quadratic equation are 3 and -4, then what is the equation? | Easy |
| A. x2-x-12 = 0 B. x2+x-12 = 0 C. x2+x+12 = 0 D. None of these |
View Answer
Answer: Option B
Explanation:
Since 3 and -4 are the roots, the equation can be expressed as:
(x-3)(x-(-4)) = 0
-> (x-3)(x+4) = 0
-> x2+4x-3x-12 = 0
-> x2+x-12 = 0
| 10. If x is real, then find the maximum value of the expression 5+4x-4x2 and also find the value of x for which this is maximum. | Difficult |
| A. 6, x = ½ B. 5, x = 0 C. 5, x = 1 D. None of these |
View Answer
Answer: Option A
Explanation:
5+4x-4x2 = 5-4(x2-x)
= 5-4[(x-1/2)2 – 1/4]
=5-4(x-1/2)2+1 = 6-4(x-1/2)2
Since x is real, (x-1/2)2 is always positive and is least when (x-1/2) = 0 i.e., when x = ½, the value of the given expression is maximum, and is equal to 6-0 i.e., 6
Therefore, required maximum value of the given expression is 6, when x = 1/2