Quant Boosters  Hemant Malhotra  Set 1

ABCD is cyclic quadrilateral
so BAD + BCD = 180
so OAD + OCD = 180  BAO = 180  54 = 126

Q11) How many integral solutions does the equation x^y = xy has
a) 1
b) 2
c) 3
c) 4
d) More than 4

x^y = xy
let y = 1
then x = x
we can put any integral value of x here so more than 4

Q12) Find the minimum number of coins required to pay the amounts of 67 paise, Rs 1.03 and 83 paise to three persons A, B and C, respectively, using only coins of the denominations of 2 paise, 5 paise, 10 paise, 25 paise and 50 paise.
a) 17
b) 20
c) 18
d) 19
e) 16

(Attack on highest coefficient )
2a + 5b + 10c + 25d + 50e = 67
now we need min so e = 1
2a + 5b + 10c + 25d = 17
so c = 1, b = 1, a = 1
so we need min 4 coins
now 2a + 5b + 10c + 25d + 50e = 103
so e = 1 and d = 1, c = 2, b = 0, a = 4 so 8 coins
now 2a + 5b + 10c + 25d + 50e = 83
so e = 1, d = 1
so 2a + 5b + 10c = 8 so a = 4
so 6 coins
so min coins needed = 4 + 8 + 6 = 18

Q13) For how many positive integer values of ‘x’ is x – 1 – 2 – 3 – 4 < 5

x – 1 – 2 – 3 – 4 < 5
so x  1 < 5 + 4 + 3 + 2
so x1 < 14
so 14 < x  1 < 14
so 13 < x < 15
so max 14 and min = 12 so 1 to 14
14 positive values

Q14) Because of an error , a printing machine cannot print numbers 6 and 8 . this machine is used to print page numbers on a volume of an encyclopedia. What number is printed on 503rd page of encyclopedia

0 to 9 means we are working on base 10. Now 2 values are missing so we are working on base 8
numbers used 0, 1, 2, 3, 4, 5, 7 and 9
so 6 will be printed as 7 and 7 as 9 here
now convert 503 in base 8 so 767
but 7 will be printed as 9 and 6 as 7
so OA = 979

Q15) A G.P with the first term as 'a' and common ratio 'r' > 1, is such that the average of the first n terms is 13 and the average of the first 'n + 1' terms is 30. If a = r = n, then what is the second term of the G.P.?

Average of first n terms is 13 so sum will be 13n
and (n + 1)th term will be a * r^(n + 1  1) = ar^n
now average of n + 1 terms is 30
so sum will be 30(n+1)
13n + a * r^n = 30(n+1) ;
now a = r = n
n^(n+1) = 17n + 30
n = 3
so a = 3 and r = 3
so 2nd term of GP = 9

Q16) The roots of 9y^2 + 6my + 2m = 0, where m is a whole number, are rational. How many values can m take ?
a) 1
b) 4
c) 2
d) More than 4

Roots are rational & Coefficient are rational then D will be perfect square
36m^2  72m = k^2
36(m^2  2m) = k^2
so m^2  2m should be perfect square
which is possible for m = 0 and m = 2
so 2 values possible

Q17) The lines x + 4y = 81 and y = mx + 9 intersect at points whose coordinates are integers. Find the number of positive integer values of m
a) 2
b) 3
c) 4
d) 5
e) More than 5

Put y = mx + 9 in first equation
so x + 4(mx + 9) = 81
so x(4m + 1) = 45
so x = 45/(4m + 1)
so factors of 45 =1, 3, 5, 9, 15, 45
so 4m + 1 form =5, 9 and 45
so m=1, 2, 11
x =9, 1, 5
and y = 18, 20, 19
so 3 values

Q18) If the equation px^2 + qx + p = 0, where p > 0, has positive roots, then which of the following must be true ?
a) q  2p > 0
b) q  2p < 0
c) 2p + q ≥ 0
d) p + q ≥ 0

Method 1 :
sum of roots = q/p.
sum of two positive numbers is positive so positive = q/p
p is positive so q must be negative
now in option b, q  2p ( negative  negative will always be < 0 )
so b is the answerMethod 2 :
positive roots
so D ≥ 0
q^2  4p^2 ≥ 0
(q  2p)(q + 2p) ≥ 0
(This is not the only condition)
now sum of roots and product of roots will be positive
so q/p so p > 0 , q < 0
so q  2p < 0Method 3 :
Pick value of p such that roots are positive
x^2 + qx + 1 = 0
take q = 2 so x^2  2x + 1 = 0 so roots 1, 1 so positive so q = 2 and p = 1
Put these values in the options and check out if it satisfies. Only b is always true

Q19) If 2 is one of roots of the quadratic equation x^2  ( 4  β)x + (β + 1/β) = 0, where β is some positive number. Which of the following could be the other root ?
a) 5/3
b) 1
c) 1/2
d) 2

Product of roots = β + 1/β, minimum value of which is 2
now one root is 2 so other root will be greater than 1
so only possible option is 5/3

Q20) A farmer mixes 25ml of water for every 75ml of pure milk and sells it to a teaseller. The tea seller sells tea containing milk which has 50% water and the rest pure milk. What percentage of the mixture the tea seller receives from the farmer is replaced with water?