Question Bank  Algebra  Hemant Malhotra

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

Q4. # of Positive Roots = 0 ( no variations)
# of Negative roots = 1 (Substitute f(x) then check variations)Thus only 1 real root :)

Q16: We get the x and y  intercepts .
3x = +/12, x = +/ 4
4y = +/ 12, y = +/3This is a rhombus with diagonals 6 and 8 thus A = 1/2 (8)(6) = 24 square units :)

Q21: p/q + q/p = (p^2 + q^2) / pq = ((p+q)^2  2pq)/pq
By Vieta's theorem, p + q = 6/2 = 3, pq = b/2
9/(b/2)  2 = 18/b  2
Max value of the above equation if b =  infinityThus it will be 18/(inf)  2 = 0 = 2 = 2

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 ?