Question Bank  Algebra  Hemant Malhotra

Q25) f(x) and g(x) are two quadratic equations with real roots
f(x). g(x) > 0
Then, which of the following is true wrt solution of the above inequality?
a) solution is same as union of solution for f(x) > 0 and g(x) > 0
b) solution is same as union of solution for f(x) < 0 and g(x) > 0
c) solution is same as union of solution for f(x) < 0 and g(x) < 0
d) solution is same as union of solution for f(x) > 0 and g(x) < 0
e) solution CBD

Q26) The quadratic polynomial f(x) = ax^2 + bx + c has integer coefficients such that f(1), f(2), f(3), and f(4) are all perfect squares of integers but f(5) is not. What is the value of a, b and c?

Q27) Suppose p(x) = ax^2 + bx + c is a quadratic polynomial with real coefficients and p(x) ≤ 1 for 0 ≤ x ≤ 1. Find the largest possible value of a + b + c.

Q28) Find integral value of a for which x^2  2(4a1)x + 15a^2  2a  7 > 0 is valid for any x
a) 2
b) 3
c) 4
d) none of the above

Q29) If the sum of the reciprocals of the roots of the quadratic equation x^2 − ax + b = 0 is 2, what is the sum of the reciprocals of the roots of the quadratic equation x^2 − bx + a = 0?

Q30) If a^2 = 5a  3 and b^2 = 5b  3 then find the quadratic eqn whose roots are a/b and b/a

Q31) if the roots of ax^2 + bx + 10 are not real and distinct where a,b real . and p and q are the values of a and b for which 5a + b is minimum . then family of line p(4x + 2y + 3) + q(x  y  1) = 0 are concurrent at
a) (1,1)
b) (1,1)
c) (1/6,1)
d) none of the above

Q32) find the number of quadratic polynomials (ax^2 + bx + c ) such that a,b,c are distinct natural numbers € (1,2,3,4,.......,1999) and polynomial is divisible by (x + 1).

Q33) A and B are real numbers such that the two quadratic equations 13x^2 + 3x +2 = 0 and Ax^2 + Bx + 5 = 0 have a common root. What is the value of A + B?

Q34) The roots of the quadratic equation ax^2 + bx + c = 0 are integers and the product of the roots is 36. How many values are possible for the sum of the roots?
a) 4
b) 5
c) 8
d) 10

Q35) The equation ax^2 + bx + c = 0 (a, b, c are real numbers) has one root greater than 2 and the other root less than zero. Which of the following is necessarily true?
a) a(a + b + c) > 0
b) a(a + b + c) < 0
c) a + b + c > 0
d) a + b + c < 0

@hemant_malhotra
Q29. 1/r + 1/s = 2
(r+s) / rs = 2
a/b =2
1/r + 1/s = (r+s) / rs = b/a = 1/2 :)

@hemant_malhotra
a^2b^2 + a^2c^2 + b^2c^2 = (ab + bc + ac)^2  2abc (a + b + c) = (1/3)^2  2(1)(2/3)
= 1/9  4/3 = 11/9
= 11/9 / (1)^2 = 11/9

Q4. # of Positive Roots = 0 ( no variations)
# of Negative roots = 1 (Substitute f(x) then check variations)Thus only 1 real root :)

Q16: We get the x and y  intercepts .
3x = +/12, x = +/ 4
4y = +/ 12, y = +/3This is a rhombus with diagonals 6 and 8 thus A = 1/2 (8)(6) = 24 square units :)

Q21: p/q + q/p = (p^2 + q^2) / pq = ((p+q)^2  2pq)/pq
By Vieta's theorem, p + q = 6/2 = 3, pq = b/2
9/(b/2)  2 = 18/b  2
Max value of the above equation if b =  infinityThus it will be 18/(inf)  2 = 0 = 2 = 2

Q24: Discriminant should be a perfect square
b^2  4ac = a^2  24a should be a perfect square
a(a24) is a perfect square
a = 0 obviously
a = k^2
a  24 = m^2
k^2  m^2 = 24
k + m = 12 or km = 2
k + m = 6 or k  m = 4
k = 7 or k = 5 for both cases
Thus the only possible values of a are 0, 25 and 49. Therefore, 3 values of a :)


@HarikrishnaShenoy Dear Hari, Kindly provide detailed methods. So that others can learn from your methods and mentors can suggest improvements if possible :)

Q36) Number of real roots of the equation x^4 + 10 * x^3 + 35 * x^2 + 50 * x + 24 = 0
a. 0
b. 2
c. 4
d. 8