Question Bank  Algebra  Hemant Malhotra

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

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?