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 |x-1| < 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 co-ordinates 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 answer

    Method 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 < 0

    Method 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 tea-seller. 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?


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