Quant Boosters - Hemant Malhotra - Set 11


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    When you solve, you will get x=5/4 but that value will not satisfy initial equation so zero roots


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q11) Find integral value of a for which x^2 - 2(4a - 1)x + 15a^2 - 2a - 7 > 0 is valid for any x
    a) 2
    b) 3
    c) 4
    d) none of the above


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    if f(x) > 0 for all values of x
    then D < 0
    4(4a-1)^2-4(15a^2-2k-7) < 0
    a^2-6a+8 < 0
    2 < a < 4
    so integral value is 3


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q12) If roots p,q,r in HP of equation x^3 - 3ax^2 + 3bx - c = 0 then
    a) q = 1/p
    b) q = b
    c) q = c/b
    d) q = b/c


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    approach 1
    p+q+r = 3a
    pq + qr + rp = 3b
    pqr = c
    q = 2pr/p+r
    1 = 2c/q / [3b-c/q]
    3b = 3c/q
    q=c/b

    approach 2
    roots are in HP
    so 1/p,1/q,1/r are the roots of
    (1/x)^3-3z(1/x)^2+3b(1/x)-c=0
    so -cx^3+3bx^2-3ax+1=0 are in AP
    so 1/p+1/q+1/r=3b./c
    so 3/q=3b/c
    so q=c/b


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q13) For all real values of find min and max value of (x^2-3x+4)/(x^2+3x+4)


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    x^2-3x+4/(x^2+3x+4) = y
    now make a quadratic in x
    x^2(y-1)+3x(y+1)+4(y-1) = 0
    now real x so D > = 0
    9(y+1)^2-16(y-1)^2 > = 0
    so (7y-1)(y-7) < = 0
    so 1/7 < = y < = 7


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q14) Number of possible value of integer p for which x^2 + px + 16 = 0 has integral roots
    a) 4
    b) 6
    c) 2
    d) not


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    For roots to be integral Discriminant should be perfect square
    D=k^2
    a^2-64=k^2
    a^2-k^2=64
    Now easy ? Answer is 6


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q15) In how many ways we arrange 6 boys and 4 girls such that
    a) all girls are together
    b) all boys are together
    c) all boys are together , all girls together
    d) no girls are together


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    a) all girls are together so make a group of them and take it as a one unit
    so they could be arranged in 4! ways so now we have BBBBBB(G) so total 7 persons , number of ways to arrange those 7!
    so total ways =7! * 4

    b) all boys are together so make it as one entity so they could be arranged in 6! ways
    so total number of ways = 6! * 5!

    c) all boys are together and all girls are together
    so BBBBBBGGGG
    6!*4!
    and GGGGBBBBBB
    so 4 * 6!
    so total ways = 2 * 4! * 6!

    d) no two girls are together
    so _ B _ B _ B _ B _ B _ B _
    first arrange 6 boys so ways 6!
    now 6 boys will leave 7 spaces
    now select 4 spaces and arrange 4 girls there
    so 7c4*4!
    so total ways = 6! * 7c4 * 4!


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q16) Five boys and five girls form a line with the boys and girls alternating . The number of ways of making the line


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    GBGBGBGBGB
    so 5!*5!
    now BGBGBGBGBG
    so 5!*5!
    so total ways = 5! * 5! + 5! * 5! = 2 * (5!)^2


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q17) Number of arrangements that can be made with the letters of the word " MATHEMATICS " in which all vowels come together


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    there are 4 vowels A,E,A,I , number of ways to arrange them = 4!/2!
    now considering four vowels as one letter ( bcz we want them together ) we have 8 letters
    M,T,H,M,T,C,S and one letter combining the vowels
    so number of ways to arrange them = 8!/(2!)*2! ( bcz of 2 M and 2 T )
    so total ways =8!/2!*2! * 4!/2!


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q18) Find the rank of the word " SACHIN" in dictionary


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    In dictionary, the words at each stage are arranged in alphabetic order
    We must consider words beginning wth A,C,H,I,N,S in order

    case1- Words starting with A
    now we will arrange CHINS
    in 5! ways =120

    case2 - Words starting with C
    same we will arrange AHINS
    so 5! ways

    case3- Words starting with I
    5!

    case4- Words starting with N
    5!

    case5 - words starting with S and SACHIN will appear in this list
    SACHIN is first word in list of words beginning with S

    so 5!+5!+5!+5!+5!+1=601


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q19) Find the rank of the word DASMESH


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    First consider DSMESH, 6!/2! ways=360
    DAE (arrange remaining in 4!/2!) = 12
    DAH (4!/2!) = 12
    DAM (4!/2!) = 12
    DASE (3!) = 6
    DASH (3!) = 6
    DASMEHS
    DASMESH
    so 360+50=410


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q20) Distribute 10 similar apple to A,B and C such that A,B and C get at-least 2 apples.


 

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