Quant Boosters  Hemant Malhotra  Set 12

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Number of Questions  30
Solved ?  Not Yet!
Topic  Quant Mixed Bag
Source  Elite's Grid CAT Preparation Forum

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q1) A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?
(a) 364
(b) 231
(c) 455
(d) 472
(e) none

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Here we know value of V(a, a + b)
so 1st term should be less than 2nd term, but in V(66, 14) 66 > 14
so change this we know that V(a, b) = V(b, a)
so V(66, 14) = V(14, 66)
now proceed
V(66,14 ) = V(14,66= (33/26) * V(14,52)
= (33/26) * (26/19) * V(14, 38 )
= (33/19) * (19/12) * V(14, 24)
= (33/12) * (12/5) * V(14,10)
= (33/5) * (7/2) * V(10, 4)
=(231/10) * (5/3) * V(4, 6)
= (77/2) * 3 * V(4, 2)
= (231/2) * 2 * V(2, 2)
= 231 * 2
= 462

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q2) f (x + y) = f(x) * f(y) then find f(3) if f(1) = 4

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
If f(x + y) = f(x) * f(y) then f(x)= (f(1))^x always
Here f(1) = 4 = > f(x) = 4^x so f(3) = 4^3 = 64

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q3) Find range of the function (x^2 + x + 3)/(x^2 + x + 1)

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
(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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q5) f(x) * f(1/x) = f(x) + f(1/x) for all real x. If f(3) = 26 find f(4)

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q7) What is the maximum value for the equation 2x^2 + 12x + 15

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q8) For the equation x^2 + 6x + n to have real roots, how many values can n take ?

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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)

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q10) Find the largest value of t such that x^t + 1 divides 1 + x + x^2 + x^3 + ... + x^143

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q11) If x + 1/x = 3 then find the value of x^4 + 1/x^4

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
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?

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q13) For how many values of y, y(y + 4)(y + 6)(y + 8) < 300 is satisfied