Question Bank  Algebra  Hemant Malhotra

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

Q37) Let a < b < c be the three real roots of the equation √2014 x^3  4029 x^2 + 2 = 0. Find b(a + c).

Q38) 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?

Q39) If a,b,c are non negative real numbers such that a + b + c = 1 then find maximum value of ab^2 + bc^2 + ca^2

Q40) From a stationery shop, Ram bought 2 pens, 3 pencils and 4 erasers for Rs.16. Shyam bought 3 pens, 6 pencils and 9 erasers for Rs.30. If Dilip bought 4 pens, 7 pencils and 10 erasers from the same shop, how much did he pay for them?
a) Rs.24
b) Rs.38
c) Rs.54
d) Rs.36

Q41) Hamsa has 50 coloured leaves, all either maple or oak. She found that the number of red oak leaves with spots is even and positive and the number of red oak leaves without any spot equals the number of red maple leaves without spots. All nonred oak leaves have spots, and there are five times as many of them as there are red spotted oak leaves. There are no spotted maple leaves that are not red. There are exactly 6 red spotted maple leaves and exactly 22 maple leaves that are neither spotted nor red. How many oak leaves did she collect?

Q42) If 3x + 2y = 18 and x, y ≥ 0, then the greatest value of x^4 y^5 is

@hemant_malhotra Is the ans 2^12 * 5^5 / 3^4 ?

Q43) If f(A) = A^2 + 6 and g(A) =A^3 – 11, then K(A) = f(A) – g(A) is
 Even function
 Odd function
 Neither even nor odd function
 Data Insufficient

Q44) Given that k < 15, how many integer values can k take if the equation x^2 – 6x + k = 0 has exactly 2 real roots?
(a) 15
(b) 14
(c) 16
(d) 13

Q45) The ticket office at a train station sells tickets to 200 destinations. One day, 3800 passengers buy tickets. Then minimum no of destinations receive the same number of passengers is ?

Q46) Two friends tried solving a quadratic equation x^2+ bx + c = 0. One started with the wrong value of b and got the roots as 4 and 14; the other started with the wrong value of c and got the roots as 17 and –2. Find the actual roots.

Q47) Let f be a function such that
f(a,b) = f(ab , b) if a=>b
= a if a< b.
Now, if f(n,5)= 3 and f(n,6)= 5, then find minimum value of n

Q48) 5 < x^2 + y^2 < 28
x  y < 3
How many integer solutions exist for the given set of inequalities?

5 < x^2 +y^2 < 28
xy=+3
so x=+3 +y
so x=3+y or 3+y
case1 when x=3+y
then 5< (3+y)^2 +y^2 < 28
so 5 < 9+y^2+6y+y^2 < 28
4 < 2y^2+6y < 19
so 2y^2+6y+4 > 0
so y^2+3y+2 > 0
(y+1)(y+2) > 0 so y > 1 or y < 2
now 2y^2+6y19 < 0
proceed like this

Q49) A quadratic expression g(x) = ax^2 + mx + n is such that g(2) < 4, g(2) > 4 and g(3) < 11. Which of the following best determines a
a. a < 1
b. a < 2
c. a > 1
d. a > 2

g(2) < 4
so 4a2m+n < 4 (1)
g(2) > 4
so 4a+2m+n > 4 so 4a2mn < 4  (2)
subtract these two so 8a+2n < 0 so 4a+n < 0
now g(3) < 11
so 9a+3m+n < 11
and 4a2m+n < 4
solve these two equations and eliminate m
so 18a+6m+2n < 22
12a6m+3n < 12
add both
30a+5n < 10
and 4a+n < 0
so 20a+5n < 0
subtract
10a < 10
so a < 1or try from graphs