Quant Boosters  Kamal Lohia  Set 2

2100 = 2² * 3 * 5² *7
35 = 5 * 7
Now we need to distribute 2² * 3 * 5 among two numbers in such a way that they are coprime.
So required number of pairs are = {(2 * 2 + 1)(2 * 1 + 1)(2 * 1 + 1)  1}/2 = 22.

Q21) How many sets of three distinct factors of the number N = 2^6 × 3^4 × 5^2 can be made such that the factors in each set has a highest common factor of 1 with respect to every other factor in that set?

N = (2^6) × (3^4) × (5^2)
Second is it is better here to make cases.
(p₁, p₂, p₃)  6 × 4 × 2 = 48
(p₁p₂, p₃, 1)  3(6 × 4 × 2) = 144
(p₁, p₂, 1)  (6 × 4) + (6 × 2) + (4 × 2) = 44
i.e. total 48 + 144 + 44 = 236

Q22) 1/ (1 x 2 x 3 x 4) + 1/(2 x 3 x 4 x 5) + … + 1/(15 x 16 x 17 x 18)

1/ (1 * 2 * 3 * 4) +1/(2 * 3 * 4 * 5) +... + 1/(15 * 16 * 17 * 18)
= (1/3)[1/1 * 2 * 3  1/2 * 3 * 4] + (1/3)[1/2 * 3 * 4  1/3 * 4 * 5] + .... + (1/3)[1/15 * 16 * 17  1/16 * 17 * 18]
= (1/3)[1/1 * 2 * 3 1/16 * 17 * 18]
= (1/3 * 6)[1  1/16 * 17 * 3]
= 815/16 * 17 * 54
= 815/14688

Q23) A square sheet measuring 10 cms * 10 cms is taken. Square corners are cut at all the corners. The resulting sheet is bent to form a box .Find the maximum possible value that the volume of the box formed can achieve?

If cut square's edge length is x, then you are to maximize x(10  2x)² = 2(2x)(5  x)(5  x) which will be maximize when 2x = 5  x = 5  x = 10/3 as sum of the three terms is constant i.e. 10 and their product will maximize when each of the term is equal.
So required maximum possible volume = 2(10/3)³ = 2000/27

Q24) The average age of a group of boys is 18 years, while that of girls is 16 years. The average age of the class is 16.4 years. Find the maximum number of boys in the class given that the strength of class is a two digit number.
a. 99
b. 59
c. 31
d. 19
e. 16

Regular Sol: Let B and G be the number of boys and girls in the class, then
18B + 16G = 16.4(B + G)
i.e. 1.6B = 0.4G
i.e. B/G = 1/4
As B + G is a two digit number, largest value is 95 and largest number of boys = B = 19. (d)Alternate Sol: B18(1.6) >16.4(0.4) >G16
That means B : G = 1 : 4
As strength of class is a two digit number and it should be of the form (1 + 4)k i.e. 5k. So clearly it is 5*19 = 95 and the number of boys in the class is 19. (d)

Q25) Out of 12 clerks in an office, one getting a salary of Rs.350 was sacked and a new one was appointed. The average salary was now reduced by 5. Determine the salary paid to new clerk.
a. 370
b. 340
c. 310
d. 290
e. 285

Regular Sol: Let salary of new one is x, so we have
12n  350 + x = 12(n  5) where n is the initial average salary.
i.e. x = 350  60 = 290. (d)Alt Sol: As average salary dropped by 5 i.e. salary of new clerk is less by 5*12 = 60 than the leaving clerk's 350 i.e. 350  60 = 290. (d)

Q26) Ram prepares solutions of alcohol in water according to customers' needs. This morning Ram has prepared 27 liters of a 12% alcohol solution and kept it ready in a 27 liter delivery container to be shipped to the customer. Just before delivery, he finds that the customer had asked for 27 liters of 21% alcohol solution. To prepare what the customer wants, Ram replaces a portion of 12% solution by 39% solution. How many liters of 12% solution is replaced? [XAT2012]
a. 5
b. 9
c. 10
d. 12
e. 15

Regular Sol: Let x liters of solution has been replaced, so
39x + 12(27  x) = 21(27)
i.e. 27x = 9(27)
i.e. x = 9 liters (b)Alt Sol: 12%(9) > 21% <  (18) 39%
So the quantities of 12% and 39% solutions are mixed in the INVERSE ratio of gaps i.e. 2 : 1.
As total solution is 27 liters, so the replaced mixture's quantity is (1/3) * 27 = 9 liters (b)

Q27) The owner of an art shop conducts his business in following manner, Every once a while he raises his prices by X% and then a while later he reduces all the new prices by X%. After one such updown cycle, the price of a painting decreased by Rs.441. After a second updown cycle the painting was sold for Rs. 1944.81. What was the original price of the painting? [CAT2001]
a. 2756.25
b. 2256.25
c. 2500
d. 2000

Regular Sol(with least calculation): Px²/100² = 441 = 21²
P(1  x²/100²)² = 1944.81 = (44.1)²
i.e. (100²  x²)/100x = 44.1/21 = 2.1
i.e. x² + 210x  10000 = 0
i.e. (x + 250)(x  40) = 0
i.e. x = 40
And P = 441*10000/1600 = 2756.25 (a)
Alt Sol: As in first updown cycle price reduced by 441, so in second such cycle the reduction in price will be lesser than 441 but it'll not be much lesser than 441. So taking it to be near 400, you can easily eliminate options and zero down to 2756.25 (a)

Q28) Find the remainder when 10³ + 11³ + 12³ + 13³ + 14³ + 15³ is divided by 75.
a. 0
b. 15
c. 25
d. 50

Sol: Just remember the property of numbers that (aⁿ + bⁿ + cⁿ + ...) is always divisible by (a + b + c + ...)
where
(i) all alphabets represent positive integers,
(ii) a, b, c, ... are in arithmetic progression and
(iii) n is odd.
So here (10³ + 11³ + 12³ + 13³ + 14³ + 15³) is divisible by (10 + 11 + 12 + 13 + 14 + 15) i.e. 75 as all the above mentioned three conditions are being satisfied.

Q29) N⁴ is a positive integer. Which of the following statement/s is/are certainly correct?
I. N is a positive integer.
II. N² is a positive integer.
III. N³ is a positive integer.
A. I only
B. I and III only
C. II only
D. none of these

Mind that information provided is "N⁴ is a positive integer", but nothing has been said about N. So it can be an imaginary number also. One possible set of values, which refutes all the three statements, is:
N = i
N² = 1
N³ = i
N⁴ = 1
Answer is D

Q30) TG² = GOG where T, G, O are distinct single digit positive integers. Find O.
A. 0
B. 2
C. 7
D. 8