Question Bank  Algebra  Hemant Malhotra

Q 9) If a, b are integers then how many ordered pairs (a,b) satisfies the equation a^2 + ab + b^2 = 1
a) 6
b) 4
c) 2
d) 8

Q 10) Number of roots of x^12  x^6 + 1 = 0
a) 2
b) 4
c) 6
d) None of the above

Q 11) How many Integral Solution for the equation x7 + y < =10 , y >= 0

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