# Question Bank - Algebra - Hemant Malhotra

• Q 20) F(x)=x^2 + ax - 24 = 0
find how many integer values of x can take if -10 < a < =10
a) 14
b) 8
c) 12
d) 13

• Q21) If p and q are roots of 2x^2 + 6x + b = 0 where b < 0 find maximum value of p/q +q/p
a) 2
b) -2
c) 18
d) none of the above

• Q22) if ax^2 + bx + 6 = 0 has real roots and a ,b are real then find max value of 3a + b

• Q23) For how many integral values of k would the quadratic equation 3kx^2 - (k - 1)x + (k - 1) = 0 have real roots?

• Q24) For how many real numbers a does the quadratic equation x^2 + ax + 6a = 0 have only integer roots for x?

• 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(4a-1)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

• @hemant_malhotra

Q4. # of Positive Roots = 0 ( no variations)
# of Negative roots = 1 (Substitute f(-x) then check variations)

Thus only 1 real root :)

• @hemant_malhotra

Q16: We get the x and y - intercepts .
3x = +/-12, x = +/- 4
4y = +/- 12, y = +/-3

This is a rhombus with diagonals 6 and 8 thus A = 1/2 (8)(6) = 24 square units :)

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