# Question Bank - Algebra - Hemant Malhotra

• Q12) If the real numbers x and y satisfy x^3 - 3x^2 + 5x - 17 = 0 and y^3 - 3y^2 + 5y + 11 = 0, then the numerical value of x+y is

• Q 13) Find all positive integers x and y such that x^2 - 3xy - y = 0

• Q 14) How many ordered triples (a, b, c) satisfy the equation (a^2 − 1)^2 + (b^2 −4)^2 + (c^2 − 9)^2 = 0

• Q 15) In the expression m× (n)^p - q , the values of m,n,p, q are 0, 1, 2, and 3 although not necessarily in that order. What is the maximum possible value of the result?

• Q 16) what is the area enclosed by |3x| + |4y| = 12

• Q 17) a,b,c be nonnegative integers such that a+b+c=12. what is the maximum value of a × b × c + a × b + b × c + a × c

• 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

45

107

89

119

61

149