Maths-
General
Easy

Question

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equals

  1. log space 10
  2. 3 log space 10
  3. 2 log space 10
  4. 4 log space 10

Hint:

In the given question if we put x=0 in the function the value is indeterminate so first we will simplify the given function, separate the trigonometric and logarithmic function then use some trigonometric identities and then use the theorems to find the value of the limit.

The correct answer is: 2 log space 10


    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equalslimit as x rightwards arrow 0 of fraction numerator x.10 to the power of x minus x over denominator 1 minus cos x end fraction
    equals limit as x rightwards arrow 0 of fraction numerator x to the power of 2 space end exponent open parentheses 10 to the power of x minus 1 close parentheses over denominator left parenthesis 1 minus cos space x right parenthesis space x end fraction
equals limit as x rightwards arrow 0 of fraction numerator x squared over denominator 1 minus cos space x end fraction space limit as x rightwards arrow 0 of fraction numerator 10 to the power of x minus 1 over denominator x end fraction space space space space space space open square brackets limit as x rightwards arrow 0 of fraction numerator a to the power of x minus 1 over denominator x end fraction equals log space a close square brackets
equals log space 10 space limit as x rightwards arrow 0 of fraction numerator x squared over denominator 2 sin squared begin display style x over 2 end style end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets space
equals log space 10 space 1 half limit as x rightwards arrow 0 of fraction numerator begin display style x over 2 end style over denominator sin space begin display style x over 2 end style end fraction fraction numerator begin display style x over 2 end style.4 over denominator sin begin display style x over 2 end style end fraction
equals log space 10 fraction numerator space 1 over denominator 2 end fraction space 4 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 open square brackets limit as x rightwards arrow 0 of fraction numerator x over denominator sin space x end fraction equals 1 close square brackets
equals 2 log space 10

    Related Questions to study

    General
    Maths-

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equalslimit as x rightwards arrow infinity of fraction numerator open square brackets n open parentheses open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared close parentheses squared close square brackets over denominator open parentheses begin display style fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction end style close parentheses cubed end fraction
    =equals limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction end style over denominator begin display style fraction numerator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed over denominator 216 end fraction end style end fraction
equals limit as x rightwards arrow infinity of fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction cross times fraction numerator 216 over denominator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis 2 n plus 1 right parenthesis cubed end fraction
u sin g space L apostrophe s space H o s p i t a l space r u l e comma
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d n squared over denominator d n end fraction end style left parenthesis n plus 1 right parenthesis plus n squared begin display style fraction numerator d left parenthesis n plus 1 right parenthesis over denominator d n end fraction end style over denominator begin display style fraction numerator d open parentheses 2 n plus 1 close parentheses cubed over denominator d n end fraction end style end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n left parenthesis n plus 1 right parenthesis plus n squared left parenthesis 1 plus 0 right parenthesis over denominator 3. left parenthesis 2 n plus 1 right parenthesis squared. left parenthesis 2 plus 0 right parenthesis end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n squared plus 2 n plus n squared over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 3 n squared plus 2 n over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction
a g a i n space u sin g space L apostrophe s space H o s p i t a l space r u l e comma
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d 3 n squared over denominator d n end fraction end style plus begin display style fraction numerator d 2 n over denominator d n end fraction end style over denominator 6 begin display style fraction numerator d left parenthesis 2 n plus 1 right parenthesis squared over denominator d n end fraction end style end fraction
equals 27 over 2 space limit as x rightwards arrow infinity of fraction numerator 6 n plus 2 over denominator 24 left parenthesis 2 n plus 1 right parenthesis.2 end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 6 plus begin display style 2 over x end style over denominator 48 plus begin display style 1 over x end style end fraction
equals 27 over 2 cross times 6 over 48
equals 27 over 16

    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals

    Maths-General
    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equalslimit as x rightwards arrow infinity of fraction numerator open square brackets n open parentheses open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared close parentheses squared close square brackets over denominator open parentheses begin display style fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction end style close parentheses cubed end fraction
    =equals limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction end style over denominator begin display style fraction numerator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed over denominator 216 end fraction end style end fraction
equals limit as x rightwards arrow infinity of fraction numerator n to the power of 5 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction cross times fraction numerator 216 over denominator n cubed open parentheses n plus 1 close parentheses cubed open parentheses 2 n plus 1 close parentheses cubed end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis 2 n plus 1 right parenthesis cubed end fraction
u sin g space L apostrophe s space H o s p i t a l space r u l e comma
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d n squared over denominator d n end fraction end style left parenthesis n plus 1 right parenthesis plus n squared begin display style fraction numerator d left parenthesis n plus 1 right parenthesis over denominator d n end fraction end style over denominator begin display style fraction numerator d open parentheses 2 n plus 1 close parentheses cubed over denominator d n end fraction end style end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n left parenthesis n plus 1 right parenthesis plus n squared left parenthesis 1 plus 0 right parenthesis over denominator 3. left parenthesis 2 n plus 1 right parenthesis squared. left parenthesis 2 plus 0 right parenthesis end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 2 n squared plus 2 n plus n squared over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 3 n squared plus 2 n over denominator 6 left parenthesis 2 n plus 1 right parenthesis squared end fraction
a g a i n space u sin g space L apostrophe s space H o s p i t a l space r u l e comma
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator d 3 n squared over denominator d n end fraction end style plus begin display style fraction numerator d 2 n over denominator d n end fraction end style over denominator 6 begin display style fraction numerator d left parenthesis 2 n plus 1 right parenthesis squared over denominator d n end fraction end style end fraction
equals 27 over 2 space limit as x rightwards arrow infinity of fraction numerator 6 n plus 2 over denominator 24 left parenthesis 2 n plus 1 right parenthesis.2 end fraction
equals 27 over 2 limit as x rightwards arrow infinity of fraction numerator 6 plus begin display style 2 over x end style over denominator 48 plus begin display style 1 over x end style end fraction
equals 27 over 2 cross times 6 over 48
equals 27 over 16
    General
    Maths-

    If fraction numerator b over denominator c plus a end fraction plus fraction numerator c over denominator a plus b end fraction equals 1then

    In triangleABC,
    fraction numerator c over denominator a plus b end fraction plus fraction numerator b over denominator c plus a end fraction equals 1
rightwards double arrow fraction numerator c left parenthesis c plus a right parenthesis plus b left parenthesis a plus b right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis c plus a right parenthesis end fraction equals 1
rightwards double arrow c squared plus a c plus a b plus b squared equals left parenthesis a plus b right parenthesis left parenthesis a plus c right parenthesis
rightwards double arrow c squared plus a c plus a b plus b squared equals a squared plus a c plus a b plus b c
rightwards double arrow c squared plus b squared minus a squared equals b c
rightwards double arrow 2. b c. cos A equals b c space space space open square brackets b y space cos i n e space f o r m u l a close square brackets
rightwards double arrow 2 cos A equals 1
rightwards double arrow cos A equals 1 half
rightwards double arrow A equals 60 degree

    If fraction numerator b over denominator c plus a end fraction plus fraction numerator c over denominator a plus b end fraction equals 1then

    Maths-General
    In triangleABC,
    fraction numerator c over denominator a plus b end fraction plus fraction numerator b over denominator c plus a end fraction equals 1
rightwards double arrow fraction numerator c left parenthesis c plus a right parenthesis plus b left parenthesis a plus b right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis c plus a right parenthesis end fraction equals 1
rightwards double arrow c squared plus a c plus a b plus b squared equals left parenthesis a plus b right parenthesis left parenthesis a plus c right parenthesis
rightwards double arrow c squared plus a c plus a b plus b squared equals a squared plus a c plus a b plus b c
rightwards double arrow c squared plus b squared minus a squared equals b c
rightwards double arrow 2. b c. cos A equals b c space space space open square brackets b y space cos i n e space f o r m u l a close square brackets
rightwards double arrow 2 cos A equals 1
rightwards double arrow cos A equals 1 half
rightwards double arrow A equals 60 degree
    General
    maths-

    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator sin squared begin display style space end style x end fraction minus fraction numerator 1 over denominator sinh squared space x end fraction equals

    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator sin squared begin display style space end style x end fraction minus fraction numerator 1 over denominator sinh squared space x end fraction equals

    maths-General
    parallel
    General
    Maths-

    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-

    In straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan invisible function application A over 2 plus tan invisible function application B over 2 close parentheses equals

    As, a+ b+ c = 2s, tanA over 2=square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root , tanB over 2=square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root
    we have, (a +b +c) (tanA over 2+tanB over 2)
    equals 2 s open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus c right parenthesis left parenthesis s minus a right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator square root of s minus b end root square root of s minus b end root plus square root of s minus a end root square root of s minus a end root over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator s minus b plus s minus a over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of fraction numerator s squared left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator 2 s minus left parenthesis a plus b right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of s left parenthesis s minus c right parenthesis end root open curly brackets fraction numerator c over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 c. square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root
equals 2 c. c o t C over 2 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 open square brackets square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root equals c o t C over 2 close square brackets

    In straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan invisible function application A over 2 plus tan invisible function application B over 2 close parentheses equals

    Maths-General
    As, a+ b+ c = 2s, tanA over 2=square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root , tanB over 2=square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root
    we have, (a +b +c) (tanA over 2+tanB over 2)
    equals 2 s open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus c right parenthesis left parenthesis s minus a right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator square root of s minus b end root square root of s minus b end root plus square root of s minus a end root square root of s minus a end root over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator s minus b plus s minus a over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of fraction numerator s squared left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator 2 s minus left parenthesis a plus b right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of s left parenthesis s minus c right parenthesis end root open curly brackets fraction numerator c over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 c. square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root
equals 2 c. c o t C over 2 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 open square brackets square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root equals c o t C over 2 close square brackets
    General
    Maths-

    Let f : [-10, 10] →R, where f(x) = sin x + open square brackets fraction numerator x to the power of 2 end exponent over denominator a end fraction close square brackets be an odd function. Then set of values of parameter a is/are

    Let f : [-10, 10] →R, where f(x) = sin x + open square brackets fraction numerator x to the power of 2 end exponent over denominator a end fraction close square brackets be an odd function. Then set of values of parameter a is/are

    Maths-General
    parallel
    General
    physics-

    Light wave enters from medium 1 to medium 2. Its velocity in 2nd medium is double from 1st. For total internal reflection the angle of incidence must be greater than

    Light wave enters from medium 1 to medium 2. Its velocity in 2nd medium is double from 1st. For total internal reflection the angle of incidence must be greater than

    physics-General
    General
    maths-

    If the function not stretchy integral subscript 0 end subscript superscript x end superscript blankf(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x + not stretchy integral subscript 0 end subscript superscript x end superscript blankf(t) dt = a has at least two roots of opposite signs in (-1, 1) is

    If the function not stretchy integral subscript 0 end subscript superscript x end superscript blankf(x) dt → 5 as |x| → I, then the value of ‘a’ so that the equation 2x + not stretchy integral subscript 0 end subscript superscript x end superscript blankf(t) dt = a has at least two roots of opposite signs in (-1, 1) is

    maths-General
    General
    Maths-

    In straight triangle A B C comma fraction numerator a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction equals

    fraction numerator a left parenthesis cos B plus cos C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator c minus b cos A plus b minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction space space space space space space space open square brackets c equals a cos B plus b cos A rightwards double arrow b cos A equals c minus a cos B comma space b equals a cos C plus c space cos A rightwards double arrow c space cos space A equals b minus a cos C close square brackets
equals fraction numerator b plus c minus b cos A space minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator b plus c minus cos A left parenthesis b plus c right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator left parenthesis b plus c right parenthesis left parenthesis 1 minus cos A right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator 1 minus cos A over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space
equals sin squared A over 2 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 open square brackets cos space 2 x equals 1 minus 2 sin squared x rightwards double arrow 2 sin squared x equals 1 minus 2 cos 2 x rightwards double arrow sin squared x equals fraction numerator 1 minus cos 2 x over denominator 2 end fraction close square brackets

    In straight triangle A B C comma fraction numerator a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction equals

    Maths-General
    fraction numerator a left parenthesis cos B plus cos C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator c minus b cos A plus b minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction space space space space space space space open square brackets c equals a cos B plus b cos A rightwards double arrow b cos A equals c minus a cos B comma space b equals a cos C plus c space cos A rightwards double arrow c space cos space A equals b minus a cos C close square brackets
equals fraction numerator b plus c minus b cos A space minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator b plus c minus cos A left parenthesis b plus c right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator left parenthesis b plus c right parenthesis left parenthesis 1 minus cos A right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator 1 minus cos A over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space
equals sin squared A over 2 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 open square brackets cos space 2 x equals 1 minus 2 sin squared x rightwards double arrow 2 sin squared x equals 1 minus 2 cos 2 x rightwards double arrow sin squared x equals fraction numerator 1 minus cos 2 x over denominator 2 end fraction close square brackets
    parallel
    General
    chemistry-

    On Bohr’s stationary orbits -

    On Bohr’s stationary orbits -

    chemistry-General
    General
    chemistry-

    For a valid Bohr orbit, its cicumfrence should be -

    For a valid Bohr orbit, its cicumfrence should be -

    chemistry-General
    General
    chemistry-

    Arrange the following particles in increasing order of values of e/m ratio: Electron (e), proton (p), neutron (n) and α-particle (α) -

    Arrange the following particles in increasing order of values of e/m ratio: Electron (e), proton (p), neutron (n) and α-particle (α) -

    chemistry-General
    parallel
    General
    Maths-

    In straight triangle A B C comma open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin invisible function application open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals

    By the sine rule, we have
    fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction
    let fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction =k
    rightwards double arrowa = k sin A, b =k sin B and c = k sin C
    Now, open parentheses fraction numerator b minus c over denominator a end fraction close parentheses space sin space fraction numerator B plus C over denominator 2 end fraction
    equals open parentheses fraction numerator k space sin space B space minus space k space sin space C space over denominator k sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals open parentheses fraction numerator space sin space B space minus space space sin space C space over denominator sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals fraction numerator 2 cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses over denominator 2 sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction sin open parentheses fraction numerator B plus C over denominator 2 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 open square brackets sin space A minus space sin space B equals 2 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator cos open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses sin space open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses sin space open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses over denominator sin A over 2 cos A over 2 end fraction space space space space space space space space space space space space space space space space space space space space space open square brackets A plus B plus C equals straight pi comma space fraction numerator straight A plus straight B plus straight C over denominator 2 end fraction equals straight pi over 2 comma space fraction numerator straight B plus straight C over denominator 2 end fraction equals straight pi over 2 minus straight A over 2 close square brackets
equals fraction numerator sin begin display style A over 2 end style sin begin display style fraction numerator B minus C over denominator 2 end fraction end style cos begin display style A over 2 end style over denominator sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction 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 open square brackets cos open parentheses straight pi over 2 minus A over 2 close parentheses equals sin A over 2 comma sin space open parentheses straight pi over 2 minus A over 2 close parentheses equals cos A over 2 close square brackets
equals sin open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses

    In straight triangle A B C comma open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin invisible function application open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals

    Maths-General
    By the sine rule, we have
    fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction
    let fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction =k
    rightwards double arrowa = k sin A, b =k sin B and c = k sin C
    Now, open parentheses fraction numerator b minus c over denominator a end fraction close parentheses space sin space fraction numerator B plus C over denominator 2 end fraction
    equals open parentheses fraction numerator k space sin space B space minus space k space sin space C space over denominator k sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals open parentheses fraction numerator space sin space B space minus space space sin space C space over denominator sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals fraction numerator 2 cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses over denominator 2 sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction sin open parentheses fraction numerator B plus C over denominator 2 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 open square brackets sin space A minus space sin space B equals 2 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator cos open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses sin space open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses sin space open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses over denominator sin A over 2 cos A over 2 end fraction space space space space space space space space space space space space space space space space space space space space space open square brackets A plus B plus C equals straight pi comma space fraction numerator straight A plus straight B plus straight C over denominator 2 end fraction equals straight pi over 2 comma space fraction numerator straight B plus straight C over denominator 2 end fraction equals straight pi over 2 minus straight A over 2 close square brackets
equals fraction numerator sin begin display style A over 2 end style sin begin display style fraction numerator B minus C over denominator 2 end fraction end style cos begin display style A over 2 end style over denominator sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction 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 open square brackets cos open parentheses straight pi over 2 minus A over 2 close parentheses equals sin A over 2 comma sin space open parentheses straight pi over 2 minus A over 2 close parentheses equals cos A over 2 close square brackets
equals sin open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses
    General
    physics-

    An antenna behaves as resonant circuit only when its length is

    An antenna behaves as resonant circuit only when its length is

    physics-General
    General
    Maths-

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals

    Maths-General
    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3
    parallel

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