Quant Boosters - Swetabh Kumar - Set 2
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Number of Questions - 30
Topic - Quant Mixed Bag
Solved ? - Yes
Source - Compilation of my posts from various CAT prep forums.
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Q1) If log_2 4 * log_4 8 * log_8 16 * ... nth term = 49, what is the value of n?
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48
log 4 /log 2 * log 8/log 4 * log 16/log 8....
log 2^(n+1)/log 2 = 49
log_base 2_ 2^(n+1) = 49
n+1=49 so n=48
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Q2) a + 2b + 3c + 4d + 5e + ... + 26z = ?
Assume a = 26; b = 25, c = 24, d = 23 and so on
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3276
1 * 26 + 2 * 25 + 3 * 24 ... 26 * 1
= 27n-n^2 sigma from n = 1 to 26
= 27 * 26 * 27/2 - 26 * 27 * 53/6 = 3276
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Q3) If x and y are real numbers, then the minimum value of x^2 + 4xy + 6y^2 - 4y + 4 ?
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By differentiation,
dF/dx = 2x+4y=0 so x=-2y
dF/dy = 4x+12y-4=0 x+3y=1
solving both, y=1 x=-2
put it in F, so 4-8+6-4+4 = 2
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Q4) No of 5s in 1 to 240 in base 6.
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till 215, it is 3 * 6^3/6 = 108
216 is 1000 and 240 is 1040
so 00 se 40 mein 05, 15, 25, 35 so 4. so 108+4=112
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Q5) Remainder when 195! Is divided by 394?
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394 = 2 * 197
195! = 1 mod 197 and 0 mod 2 so 2k and 197k + 1 so 198
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Q6) No. of ways in which 3^16 can be written as a product of 3 factors?
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ordered = 18C2 = 153
(aab) : (1,1,x), (3,3,x), (3^2,3^2,x)...(3^8,3^8,1) 9 cases
so (153-3*9)/3! = 21
plus 9 cases when (a,a,b) form
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Q7) Sum of all even factors of 2160?
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2160 = 2^4 * 3^3 * 5
so (2 + 4 + 8 + 16)(1 + 3 + 9 + 27)(1 + 5) = 7200
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Q8) Remainder of 37!/41?
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39! = 1 mod 41
39 * 38 * 37! = 42 mod 41
13 * 19 * 37! = 7 mod 41 =130 mod 41
19 * 37! = 10 mod 41 = 133 mod 41
37! = 7 mod 41
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Q9) How many of the first 1200 natural no are either prime to 6 or to 15
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coprime to 6: 1200(1/2)(2/3) = 400
coprime to 15: 1200(2/3)(4/5) = 640
coprime to 30: 1200(1/2)(2/3)(4/5) = 320
so 400+640-2*320 = 400
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Q10) If the sum of a GP is given by 20767, the last term by 13851 and the first term by 19 then find the common ratio of the GP.
