Quant Boosters - Gaurav Sharma - Set 4


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Number of Questions - 30
    Solved ? - Not yet
    Topic - Quant Mixed Bag
    Source - Genius Tutorial Preparation Forum


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q1) Compute 1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^4 + 8/10^5 + 13/10^6 + …
    a) 100/89
    b) 100/81
    c) 100/99
    d) None of these


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    S = 1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^4 + 8/10^5 + 13/10^6 + …
    S/10 = 1/10^1 + 1/10^2 + 2/10^3 + 3/10^4 + 5/10^5 + 8/10^6 + 13/10^7 + …
    S – S/10 = 1/10^0 + 0 + 1/10^2 + 1/10^3 + 2/10^4 + 3/10^5 + 5/10^6
    S – S/10 = 1/10^0 + 1/10^2 (1/10^0 + 1/10^1 + 2/10^2 + 3/10^3 + 5/10^6)
    S – S/10 = 1 + S/10^2
    S (1 – 1/10 – 1/10^2) = 1
    S = 100/89


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q2) In the given figure what is the radius of the inscribed circle

    0_1510560464840_dae2c9f5-7ea3-400f-81fd-7cd79c458216-image.png

    a) 3/2
    b) 5/2
    c) 7/5
    d) None of these


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    0_1510560431683_2005c507-1981-46d0-bdde-194dcd826b0a-image.png

    Ar(ACD) + Ar(CDB ) = Ar(ABC)
    5r/2 + 3r/2 = 12/2
    8r = 12
    r = 3/2


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q3) How many numbers from 1 – 2000 are such that at least two of their digits are same?
    a) 800
    b) 757
    c) 758
    d) 750


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Single digit number with no digit repeated = 9
    Two digit numbers with no digit repeated = 9C1 x 9C1 = 81
    Three digit numbers with no digit repeated = 9C1 x 9C1 x 8C1 = 9 x 9 x 8 = 648
    Four digit numbers less than 2000, with no digit repeated = 9C1 x 8C1 x 7C1 x 1 = 9 x 8 x 7 = 504
    Total numbers with no digit repeated = 9 + 81 + 684 + 504 = 1242
    Numbers with at least two digits same = 2000 – Numbers with no repetition
    2000 – 1242 = 758


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q4) A triangle has sides 6, 7 and 8. The line through it’s incenter parallel to the shortest side is drawn to meet other two sides at P and Q. Then find the length of the line segment PQ
    a) 4
    b) 30/7
    c) 15/7
    d) Cannot be determined


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    0_1510560102280_425bcf34-9bcc-486a-adc8-7fb9de55f427-image.png

    Area of triangle = r x s
    21r/2 = 6h/2 = 3h
    r/h = 2/7
    APQ and ABC are similar thus,
    ( h – r)/h = PQ/6
    Or 1 – r/h = PQ/6 = > 1 – 2/7 = PQ/6
    PQ = 30/7


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q5) N = 99^3 – 36^3 – 63^3. How many factors does N have?
    a) 48
    b) 84
    c) 96
    d) 134


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    N = 99^3 – 36^3 – 63^3
    N = 99^3 + (– 36) ^3 + (– 63) ^3
    N = 3 x 99 x 36 x 63 [if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc]
    N = 2^2 x 3^7 x 7 x 11
    Number of factors of N = (2 + 1) (7 + 1) (1 + 1) (1 + 1) = 96


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q6) A person is travelling on the given grid and has to go from A to B through C – D. in each move he can take a step in either the north direction of the east direction. Following the given instructions, in how many different ways can he travel from A to B

    0_1510560194577_ef81306e-9b3e-4973-937d-ccdd23c09a06-image.png


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Number of ways to travel from A to C = 6! / (3! X 3!) = 20
    Number of ways to travel from C to D = 1
    Number of ways to travel from D to B = 6!/(4! X 2!) = 15
    Total ways = 20 x 15 = 300


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q7) AC is the diameter of a circle with center O. OE and OF are perpendicular to AD and AB respectively such that A – F – B and A – E – D. If perimeter of ABCD is x, then what will be the perimeter of AEOF?
    a) x/2
    b) 2x/3
    c) x/3
    d) root(2)x/3


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    0_1510560255159_83d8d08b-1b7e-449e-bd15-fdf8d1a1e815-img_5_q10.png

    Triangle AEO ~ Triangle ADC ( AA Similarity)
    EO/DC = AE/AD = AO/AC = 1/2
    Similarly, Triangle AFO ~ Triangle ABC (by AA Similarity)
    FO/BC = AF/AB = AO/AC = 1/2
    AD + DC + CB + AB = x
    1/2 (AD + DC + CB + AB ) = x/2
    AE + OE + OF + AF = x/2


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q8) How many positive divisors does 7^12 + 7^13 + 7^14 + 7^15 have
    a) 200
    b) 199
    c) 195
    d) 197


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    7^12 + 7^13 + 7^14 + 7^15 = 7^12 (1 + 7 + 49 + 343)
    = 7^12 x 400
    = 7^12 x 2^4 x 5^2

    Number of factors:
    (12 + 1) (4 + 1) (2 + 1) = 195


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q9) There are 10 numbers and none of them are divisible by 3. What will be the remainder when sum of their squares is divided by 3?
    a) 0
    b) 1
    c) 2
    d) 1 or 2


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    The Square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1. In other words, a perfect square leaves remainder 0 or 1 on division by 3.
    Here it is said that none of them is divisible by 3. Hence each square term will give a reminder of 1
    So for 10 terms, 1 + 1 + 1 … + 1 (10 times) = 10
    10 mod 3 = 1


  • Director, Genius Tutorials, Karnal ( Haryana ) & Delhi | MSc (Mathematics)


    Q10) Find the value of [root (1)] x [root (2)] x [root (3)] x … x [root (100)]
    a) 2^87 x 3^59 x 5^13 x 7^14
    b) 2^88 x 3^58 x 5^12 x 7^15
    c) 2^81 x 3^59 x 5^12 x 7^17
    d) 2^79 x 3^61 x 5^21 x 7^12


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