Quant Boosters  Hemant Malhotra  Set 12

Q3) Find range of the function (x^2 + x + 3)/(x^2 + x + 1)

Method 1 :
y=x^2+x+3/(x^2+x+1))
yx^2+yx+y=x^2+x+3
x^2(y1)+x(y1)+y3=0
x real so D > =0
(y1)^24*(y1)(y2) > =0
y1)(y14y+12) > =0
(y1)(3y+11) > =0
(y1)(3y11)
so (1,11/3]Method 2 :
y=1+((2/(x^2+x+1))
now x^2+x+1=x^2+x+1/41/4+1
(x+1/2)^2+3/4
so min value is 3/4 and maax infinity
so y=1+2/inf
y=1
and y=1+2/(3/4))
y=1+8/3=11/3

Q4) How many integral values from 1 to 15 can the expression (x^2 + 34x 71)/(x^2 + 2x  7) not take?
a) 2
b) 3
c) 4
d) none

(x^2 +34x 71)/(x^2+2x7)=y
(1y)x^2+ (342y)x 71+7y=0
b^24ac > =0 for real values therefore
(342y)^24*(1y)(71+7y) > =0
now on simplifying
y^2 14y +45 > =0
that is y < = 5 and y > = 9 so y can't take values
6,7,8 so three integral values

Q5) f(x) * f(1/x) = f(x) + f(1/x) for all real x. If f(3) = 26 find f(4)

If f(x) * f(1/x) = f(x) + f(1/x)
then f(x) = 1+x^n
now f(3) = 26
so 1+(x)^n = 26
so +(x)^n = 27
so +(3)^n = 27 (negative case will be considered bcz in positive case 3^n=27 not possible)
so 3^n = 27
so n = 3
and f(x) = 1x^n
so f(4) = 14^3
= 63

Q6) If f(x − 1) + f(x + 1) = f(x) and f(2) = 6, f(0) = 1, then what is the value of f(50) ?
a) −7
b) 6
c) 1
d) 7

Since we know both f(0) and f(2), we can find f(1).
f(1) = f(0) + f(2) = 7
f(2) = f(1) + f(3)
f(3) = f(2) – f(1)
= 6 – 7
= −1
Also, f(3) = f(2) + f(4)
f(4) = f(3) – f(2) = −7
Continuing in a similar way, we can find out
f(0) = 1
f(1) = 7
f(2) = 6
f(3) = −1
f(4) = −7
f(5) = −6
f(6) = 1
f(7) = 7 and so on
After every 6 integral values of x, f(x) repeats itself.
f(6) = f(12) = f(18 ) = f(24) = f(30) = f(36) = f(42) = f(48 ) = 1
f(49) = 7
f(50) = 6

Q7) What is the maximum value for the equation 2x^2 + 12x + 15

Q8) For the equation x^2 + 6x + n to have real roots, how many values can n take ?

Q9) Find the solution set of the equation x^3  4x^2 + x + 6 > 0
a) (1, 2)
b) (1, 2) U (3, ∞)
c) (∞, 1) U (2, 3)
d) (1, 2) U (2, 3)

Q10) Find the largest value of t such that x^t + 1 divides 1 + x + x^2 + x^3 + ... + x^143

Q11) If x + 1/x = 3 then find the value of x^4 + 1/x^4

Q12) A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son's head are incident at the same point on the ground. If the heights of the lamp post, the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post then how far (in metres) is son standing form his father?

Q13) For how many values of y, y(y + 4)(y + 6)(y + 8) < 300 is satisfied

Q14) What is the minimum value of (p + q)(q + r)(r + p) if p, q and r are positive numbers

Q15) Find the sum of the first 10 terms of the series 1, 2, 5, 12, 27 ...

The given term can be written as (21), (42), (8  3), (16  4) and so on
so sum of first 10 terms will be = (2 + 4 + 8 + 16 +      )  (1 + 2 +3 +4 +5 +6 +7   )
2046  55 = 1991

Q16) If 2x + 9y ≤ 90, x ≥ 0, y ≥ 0 and x & y are integers, find the total number of solutions

Q17) x, y and z are real numbers such that 6 ≤ x ≤ 2, 4 ≤ y ≤ 4 and 3 ≤ z ≤ 7. If w = y/xz then which of the below options is necessarily true ?
a) 0 ≤ w ≤ 2
b) 1/2 ≤ w ≤ 1/2
c) 2/3 ≤ w ≤ 1/3
d) 2/3 ≤ w ≤ 2/3