Maths-
General
Easy

Question

There are 10 points in a plane and A is one of them If no three of the points are collinear then the number of triangles formed with A as vertex is

  1. 45    
  2. 36    
  3. 84    
  4. 120    

The correct answer is: 36


    10 minus 1 C subscript 3 minus 1 end subscript equals 36

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    2 cross times 8 subscript C subscript 2 end subscript equals 2 cross times 28 equals 56

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    A guard of 12 men is formed form  soldiers If A and B are three times as often together on guard as C,D, E then =


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    An examination paper, which is divided into two groups consisting of 3 and 4 questions respectively carries the note it is not necessary to answer all the questions One question atleast should be answered form each group The number of ways can an examinee select the questions is

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    blank to the power of straight n straight C subscript straight r plus 1 end subscript plus 2 to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript equals

    nC subscript straight r plus 1 end subscript plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript

    blank to the power of straight n straight C subscript straight r plus 1 end subscript plus 2 to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript equals

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    nC subscript straight r plus 1 end subscript plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript
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    A person wishes to make up as many different parties of 10 as he can out of 20 friends, each party consisting of the same number In how many of the parties the same man is found ?

    20 minus 1 C subscript 10 minus 1 end subscript

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    20 minus 1 C subscript 10 minus 1 end subscript
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    A father with 6 children takes 3 at a time to a park without taking the same children How often each child goes to the park?

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    chemistry-

    Action of H N O subscript 2 space o n space C H subscript 3 N H subscript 2 gives:

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    chemistry-General
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    chemistry-

    Primary alcohols can be obtained from the reaction of the R M g X with:

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    The quadratic equation whose roots are I and m where l=lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

    Given, the quadratic equation whose roots are I and m where
    l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses
    l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses
space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction
space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2
l equals 3 space a n d space m equals 2
t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n
equals x squared minus left parenthesis l plus m right parenthesis x plus l m
equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis
equals x squared minus 5 x plus 6 space

    The quadratic equation whose roots are I and m where l=lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

    Maths-General
    Given, the quadratic equation whose roots are I and m where
    l =lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses
    l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses
space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses
space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction
space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2
l equals 3 space a n d space m equals 2
t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n
equals x squared minus left parenthesis l plus m right parenthesis x plus l m
equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis
equals x squared minus 5 x plus 6 space
    General
    Maths-

    L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction

    L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x minus begin display style fraction numerator sin squared x. space cos squared x over denominator cos squared x end fraction end style over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x left parenthesis 1 minus begin display style space cos squared x end style right parenthesis over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
u sin g space L space H o s p i t a l space t h e o r e m comma
limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style left parenthesis 1 minus cos cubed x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x squared end fraction close parentheses
equals limit as x rightwards arrow 0 of fraction numerator 3 cos squared x. sin x over denominator 2 x end fraction
equals limit as x rightwards arrow 0 of 3 over 2 cos squared x. open parentheses fraction numerator sin x over denominator x end fraction close parentheses
equals 3 over 2 limit as x rightwards arrow 0 of cos squared x
equals 3 over 2 cos squared 0
equals 3 over 2
N o w comma limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
equals 1 cross times 3 over 2 equals 3 over 2

    L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction

    Maths-General
    L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x minus begin display style fraction numerator sin squared x. space cos squared x over denominator cos squared x end fraction end style over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of fraction numerator tan cubed x left parenthesis 1 minus begin display style space cos squared x end style right parenthesis over denominator x to the power of 5 end fraction
equals limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
u sin g space L space H o s p i t a l space t h e o r e m comma
limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style left parenthesis 1 minus cos cubed x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x squared end fraction close parentheses
equals limit as x rightwards arrow 0 of fraction numerator 3 cos squared x. sin x over denominator 2 x end fraction
equals limit as x rightwards arrow 0 of 3 over 2 cos squared x. open parentheses fraction numerator sin x over denominator x end fraction close parentheses
equals 3 over 2 limit as x rightwards arrow 0 of cos squared x
equals 3 over 2 cos squared 0
equals 3 over 2
N o w comma limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses
equals 1 cross times 3 over 2 equals 3 over 2
    General
    Maths-

    L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction Is equal to

    L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction
equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction
equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets
equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space
equals straight pi cross times 1 cross times 1
equals straight pi

    L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction Is equal to

    Maths-General
    L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction
equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction
equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets
equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space
equals straight pi cross times 1 cross times 1
equals straight pi
    parallel

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