Algebra -2 Quadratic Equations HA-10
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Texas A&M University *
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601
Subject
Mathematics
Date
Feb 20, 2024
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Quadratic Equations Home Assignment
Level A
1. Factorise: 12x
2
+ 23x + 5
(A) (3x + 1)(2x + 5) (B) (4x + 1)(3x + 5)
(C) (6x + 1)(x + 5) (D) (2x + 1)(6x + 5)
2. (a) Find the roots of the quadratic equation: x
2
- x - 20 = 0.
(A) -5, 4 (B) -5, -4 (C) 5, -4 (D) 5, 4
(b) What are the roots of the quadratic equation: 2x
2
- 5x - 3 = 0?
(A) -1/2, –3 (B) 1/2, -3
(C) 1/2, 3 (D) -1/2, 3
3. Find a quadratic equation whose roots are 3 and 4.
(A) x
2
+ 7x + 12 = 0 (B) x
2
- 7x + 12 = 0
(C) x
2
- 7x - 12 = 0 (D) x
2
+ 7x - 12 = 0
4. Find the roots of the quadratic equation : x
2
– 12x + 13 = 0.
(A) 1, 13 (B) –1, –13
(C) 6 + √23 , 6 – √23 (D) None of these
5.
If the sum of the roots and the product of the roots of a quadratic equation are 13 and 30
respectively, find its roots.
(A) 10, 3 (B) –10, –3 (C) 10, –3 (D) –10, 3
6. The value of the discriminant of the equation : 3x
2
+ 7x + 2 = 0 is
7. (a) Construct a quadratic equation whose roots are 2 more than the roots of the equation
x
2
+ 9x + 10 = 0.
(A) x
2
+ 5x – 4 = 0 (B) x
2
+ 13x + 32 = 0
(C) x
2
– 5x – 4 = 0 (D) x
2
– 13x + 32 = 0
(b) Construct a quadratic equation whose roots are reciprocals of the roots of the equation
2x
2
+ 8x + 5 = 0.
(A) 5x
2
+ 8x + 2 = 0
(B) 8x
2
+ 5x + 2 = 0
(C) 2x
2
+ 5x + 8 = 0
(D) 8x
2
+ 2x + 5 = 0
8. The sum of the roots of a quadratic equation is 33 and the product of its roots is 90, the
sum of the squares of its roots is
9. A quadratic equation in x has the sum of its roots as 19 and the product of its roots as 90,
the difference of its roots is
10. If one root of the quadratic equation 4x
2
- 8x + k = 0, is three times the other root, the
value of k is
11. The square of the sum of the roots of a quadratic equation E is 8 times the product of its
roots. The value of the square of the sum of the roots divided by the product of the roots of
the equation whose roots are reciprocals of those of E is
12. (a) The expression (4ac - b
2
)/4a represents the maximum/minimum value of the quadratic
expression ax2 + bx + c. Which of the following is true?
(A) It represents the maximum value when a > 0.
(B) It represents the minimum value when a < 0.
(C) Both (A) and (B)
(D) Neither (A) nor (B)
(b) The quadratic expression ax
2
+ bx + c has its maximum/minimum value at x = ______.
(A) –b/2a
(B) 2a/b
(C)b/2a
(D) 2b/a
(c) Find the maximum value of the quadratic expression –3x
2
+ 4x + 5.
(A)19/3 (B) 31/12 (C) 3/19 (D) - 19/3
Q13. Q14. Find the value of p in x
2
+ px + 8 = 0 for each of the following two cases:
i. If one root is square of the other root.
ii. If both the roots are equal
iii. If the difference of the two roots is 2.
Q15.
If p and q are the roots of x
2
– 7
x – 6 = 0, find the value of p/q + q/p.
Q16. If 3
p + 1 and 3
q + 1 are the roots of the equation x
2
– 7
x + 10 = 0, find the equation whose roots are p and q
.
Q17.
Q18.
Find the value of p if the equation 3
x
2
– 2
x + p = 0 and 6
x
2
– 17
x + 12 = 0 have a common root.
Level B 1.
Find the equation whose roots are thrice the roots of the equation 2x
2
- 15x + 18 = 0.
(A) x
2
+ 45x + 324 = 0 (B) 2x
2
- 45x + 81 = 0
(C) x
2
+ 45x - 324 = 0 (D) 2x
2
- 45x + 162 = 0
2.
A person bought a certain number of oranges for Rs.70. If the price of each orange was Rs.2
less, he would have bought 4 more oranges for the same amount, the number of oranges he
bought originally is
3.
If a positive number is increased by three and then squared, the result is 23 more than the
original number. The original number is
4.
Find the value of R, so that one of the roots of x
2
+ 6Rx + 64 = 0 is the square of the other
root.
(A) –10/3 (B) 8/3 (C) 5/3 (D) 7/3
5.
Find the values of k for which the roots of x
2
+ x(14 - k) - 14k + 1 = 0 are equal integers.
(A) -11, -13 (B) -12, -16
(C) -13, -15 (D) –11, –12
6.
If the roots of 2x
2
+ (4m + 1)x + 2(2m - 1) = 0 are reciprocals of each other, m =
7.
If the roots of 2mx
2
+ 8x + 64m = 0 are real and equal, find m.
(A) 2/3 (B) 1/2 (C) 7/4 (D) 4/7
8.
If the roots of the equation ax2 + bx + c = 0 are x and y, find the value of : x/y +y/x – 2(1/x+1/y) +2xy
(A) (b
2
+ 2ac)/ac (B) (b
2
- 2ac)/ac
(C) (b
2
+ 4ac)/ac (D) None of these
9.
If the price of a book goes down by Rs.20 per dozen, a person can purchase 50 dozen books
more for Rs.30,000. The original price (in Rs.) of each book is
10. If 31 is split up into two parts such that the sum of the squares of the two parts is 481, the
difference between the two parts is
11.
Find the equation whose roots are twice the roots of the equation 3x
2
– 7x + 4 = 0.
(A) 3x
2
– 14 x + 8 = 0 (B) 3x
2
+ 14x + 16 = 0
(C) 3x
2
+ 14 x – 16 = 0 (D) 3x
2
– 14x + 16 = 0
12. The roots of the equation x
2
– 3x – 108 = 0 are a and b, where |a| > |b| . Which of the
following holds true?
(A) a – b = 3 (B) a – b = –3
(C) a – b = –21 (D) a – b = 21
13. P and Q were trying to solve a quadratic equation. P copied the coefficient of x wrong and
obtained 12 and 9 as the roots. Q copied the constant term wrong and obtained 8 and 16 as the
roots. Find the roots of the equation.
(A) 18, –6 (B) –18, 6 (C) 18, 6 (D) –18, –6
(B)
14.
Find the respective values of a and k if the roots of the quadratic equation 27
x
2
– 87
x + k = 0,
are a and 8/3.
(A) 5/9, 40 (B) 4/9, 32
(C) 7/9, 72 (D) 2/9, 16
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15.
Find the quadratic equation whose roots are the reciprocals of the roots of 3
x
² - 8
x + 4 = 0.
(A) –4
x
² + 8
x + 3 = 0 (B) -8
x ² + 4
x + 3 = 0
(C) 4
x
² - 8
x + 3 = 0 (D) 8
x
² + 3
x - 4 = 0
Answers : Level A : Q1 – B
Q7 – A , A
Q13 – 216 , 8
Q2 – C ; D
Q8 – 909 Q14 – 2 ; -1 ; 3 Q3 – B
Q9 – 1
Q15 – 2
Q4 – C
Q10 – 3
Q16 – 4
Q5 - A
Q11 – 8 Q17 – 4
Q6 - 25
Q12 – D , A , A
Q18 – 4
Level B
Q1 – D Q6 – 1
Q11 – D
Q2 – 10
Q7 – D Q12 – D
Q3 – 2
Q8 – D
Q13 – C
Q4 – A
Q9 – 10
Q14 – A
Q5 - B
Q10 – 1 Q15 – C