# Quant Boosters - Gaurav Sharma - Set 6

• Q8) The points (0, 8/3), (1, 3) and (82, 30)
(a) form an obtuse angled triangle.
(b) form an acute angled triangle.
(c) form a right angled triangle.
(d) lie on a straight line.

• Slope of AB = slope of BC
So striaght line

• Q9) Sum of digits of a 5-digit number is 41.The probability that such a number is divisible by 11 is k/ 35 where k =

• The 5-digit combination whose sum = 41 are
● 99995 : No. of arrangements = 5
● 99986 : No. of arrangements = 20
● 99977 : No. of arrangements = 10
● 99887 : No. of arrangements = 30
● 98888 : No. of arrangements = 5
Total = 70

Now, for a number (abcde) to be divisible by 11, we should have (a + c + e) – (b + d) = 11
and (a + c + e) + (b + d) = 41
So, a + c + e = 26, b + d = 15
Thus (a, c, e) = (9, 9, 8 ) and (b, d) = (7, 8 ) or (6, 9)
Hence total 12 cases

Probability = 12/70 = 6/35
k = 6

• Q10) There is a frog who could climb either 1 stair or 3 stairs in one shot. In how many ways he could reach at 10th stair ?

• Fibonacci with a gap of 1 : 1, 1, 2, 3, 4, 6, 9, 13, 19, 28

Method 2 :
1 step in 1 way
2 steps in 1 way
3 steps in 2 ways
4 steps in 3 ways
5 steps in 4 ways
6 steps in 6 ways
7 steps in 9 ways
8 steps in 13 ways
9 steps in 19 ways
10 steps in 28 ways
So 28 should be the answer

Method 3 :

x + 3y = 10
(1 , 3 ) -> 4
(4 , 2 ) -> 6!/4!2! = 15
(7 , 1) -> 8!/7! = 8
(10 , 0) -> 1
TOTAL = 15 + 4 + 8 + 1 = 28

• Q11) If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n + 1)(n + 2) will be divisible by 8?

[OA : 5/8]

• Q12) If March 1, 2006 was a Wednesday, which day was it on March 1, 2002?

• Q13) N is a natural number of at least 5 digits and its leftmost digit is 6. When this 6 is removed from N, the number thus obtained is found to be 1/25 times of N. What is the sum of the digits of N?

• Q14) Number of terms in the expansion of (a + b + c)^10
a) 11
b) 21
c) 55
d) 66

• Q15) The product of all the factors of a number is equal to the tenth power of the number. N cannot be exactly divisible by which of the following?
a. 90
b.200
c. 210
d. 126

• Q16) 2n has 28 factors 3n has 30 factors How many factors will 6n have ?

• N = 2^a × 3^b
Factors of 2N = ( a + 2) ( b + 1)=28
Factors of 3N = ( a + 1 )( b + 2 )=30
So a = 5 and b = 3
So N = 2^5 × 3^3
Now Factors of 6N ( 2^6 × 3^4 )
will be 7 × 5 = 35

• Q17) If a three digit number ‘abc’ has 3 factors, how many factors does the 6-digit number ‘abcabc’ have?

• Q18) Let N = 2^15 * 3^12. How many factors of N^2 are less than N but do not divide N?

• Q19) If 2n has 28 factors, 3n has 30 factors, how many factors does 5n have ?

• Q20) The sum of the factors of a number is 124. What is the number?
a) Number lies between 40 and 50
b) Number lies between 50 and 60
c) Number lies between 60 and 80
d) More than one such number exists

• Q21) If coordinates of the vertices of a triangle are (2, 0), (6, 0) and (1, 5) then distance between its orthocentre and circumcentre is
(a) 4
(b) 6
(c) 5
(d) none of these

• Q22) Let (a1, a2, a3, ….) be a sequence such that a1 = 2 and an – an-1 = 2n for all n ≥ 2. Then a1 + a2 + …. + a20 is
(a) 420
(b) 1750
(c) 3080
(d) 3500

• Q23) In a class comprising 60 boys and some girls, the average age of boys is 14.8 years and that of girls is 14.1 years. If the average age of the class is 14.7 years, then how many girls are there in the class?
a) 10
b) 15
c) 20
d) 25

61

61

65

61

61

1

107

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