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Let f colon R not stretchy rightwards arrow R comma f left parenthesis x right parenthesis equals open curly brackets table attributes columnspacing 1em end attributes row cell vertical line x minus left square bracket x right square bracket vertical line comma left square bracket x right square bracket end cell row cell vertical line x minus left square bracket x plus 1 right square bracket vertical line comma left square bracket x right square bracket end cell end table closetable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell text  is odd  end text end cell row cell 1 text  is even where [.]  end text end cell end table denotes greatest integer function, then integral subscript negative 2 end subscript superscript 4   f left parenthesis x right parenthesis d x is equal to

Maths-General

  1. 5    
  2. 5 over 2  
  3. 3 over 2    
  4. 3  

    Answer:The correct answer is: 5Since these two lines are intersecting so shortest distance between the lines will be 0.
    Hence (c) is the correct answer.

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    The value of 01|sin 2πx|dx is equal to 

    Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.    

    The value of 01|sin 2πx|dx is equal to 

    maths-General
    Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.    
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    π/4π/4ex(xsinx)e2x1dx is equal to

    Let direction cosines of straight line be l, m, n \ 4l + m + n = 0 l – 2m + n = 0 Þ l3=m-3=n-9 Þ l-1=m+1=n3 \ Equation of straight line is x-2-1=y+11=z+13. Hence (c) is the correct choice.    

    π/4π/4ex(xsinx)e2x1dx is equal to

    maths-General
    Let direction cosines of straight line be l, m, n \ 4l + m + n = 0 l – 2m + n = 0 Þ l3=m-3=n-9 Þ l-1=m+1=n3 \ Equation of straight line is x-2-1=y+11=z+13. Hence (c) is the correct choice.    
    General
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    The value of 0100{x}dx (where {x} is the fractional part of x) is 

       

    The value of 0100{x}dx (where {x} is the fractional part of x) is 

    maths-General
       
    General
    maths-

    0 with hat on top subscript 1 superscript 1 fraction numerator sin invisible function application x minus x squared over denominator 3 minus vertical line x vertical line end fraction d x equals

    0 with hat on top subscript 1 superscript 1 fraction numerator sin invisible function application x minus x squared over denominator 3 minus vertical line x vertical line end fraction d x equals

    maths-General
    General
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     Q with dot on top subscript p superscript p left parenthesis cos invisible function application a x minus sin invisible function application b x right parenthesis squared(where a, b are integers) =

    The centre of the sphere is (1, 2, –3) so if other extremity of diameter is (x1, y1, z1), then
    fraction numerator x subscript 1 end subscript plus 2 over denominator 2 end fraction = 1, fraction numerator y subscript 1 end subscript minus 1 over denominator 2 end fraction = 2, fraction numerator z subscript 1 end subscript plus 1 over denominator 2 end fraction = –3
    \ Required point is (0, 5, 7).
    Hence (c) is the correct answer.

     Q with dot on top subscript p superscript p left parenthesis cos invisible function application a x minus sin invisible function application b x right parenthesis squared(where a, b are integers) =

    maths-General
    The centre of the sphere is (1, 2, –3) so if other extremity of diameter is (x1, y1, z1), then
    fraction numerator x subscript 1 end subscript plus 2 over denominator 2 end fraction = 1, fraction numerator y subscript 1 end subscript minus 1 over denominator 2 end fraction = 2, fraction numerator z subscript 1 end subscript plus 1 over denominator 2 end fraction = –3
    \ Required point is (0, 5, 7).
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    General
    maths-

    The foot of the perpendicular on the line 3 x plus y equals lambda drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

    Let direction ratios of the required line be <a, b, c>
    Therefore a - 2 b - 2 c = 0
    And 2 b + c = 0
    Þ c = - 2 b
    a - 2 b + 4b = 0 Þ a = - 2 b
    Therefore direction ratios of the required line are <- 2b, b, - 2b> = <2, - 1, 2>
    direction cosines of the required line
    open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses = open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses

    The foot of the perpendicular on the line 3 x plus y equals lambda drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

    maths-General
    Let direction ratios of the required line be <a, b, c>
    Therefore a - 2 b - 2 c = 0
    And 2 b + c = 0
    Þ c = - 2 b
    a - 2 b + 4b = 0 Þ a = - 2 b
    Therefore direction ratios of the required line are <- 2b, b, - 2b> = <2, - 1, 2>
    direction cosines of the required line
    open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses = open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses
    General
    maths-

    The shortest distance between the two straight linex4/32=y+6/53=z3/24 and 5y+68=2z39=3x45 is  

    The shortest distance between the two straight linex4/32=y+6/53=z3/24 and 5y+68=2z39=3x45 is  

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    General
    physics-

    A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

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    A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

    physics-General
    Velocity of the bob at the point A v=5gL(i) v22=v2-2ghii h=L(1-cosθ)(iii) Solving Eqs.i, iiand iii, we get cosθ=-78 or   θ=cos-1-78=151°    
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    physics-

    A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle theta at which the speed of the bob is half of that at A, satisfies

    V to the power of 2 end exponent equals U to the power of 2 end exponent minus 2 g left parenthesis L minus L cos invisible function application theta right parenthesis
    fraction numerator 5 g L over denominator 4 end fraction equals 5 g L minus 2 g L left parenthesis 1 minus cos invisible function application theta right parenthesis

    5 equals 20 minus 8 plus 8 cos invisible function application theta
    cos invisible function application theta equals negative fraction numerator 7 over denominator 8 end fraction
    fraction numerator 3 pi over denominator 4 end fraction less than theta less than pi

    A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle theta at which the speed of the bob is half of that at A, satisfies

    physics-General
    V to the power of 2 end exponent equals U to the power of 2 end exponent minus 2 g left parenthesis L minus L cos invisible function application theta right parenthesis
    fraction numerator 5 g L over denominator 4 end fraction equals 5 g L minus 2 g L left parenthesis 1 minus cos invisible function application theta right parenthesis

    5 equals 20 minus 8 plus 8 cos invisible function application theta
    cos invisible function application theta equals negative fraction numerator 7 over denominator 8 end fraction
    fraction numerator 3 pi over denominator 4 end fraction less than theta less than pi
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    A piece of wire is bent in the shape of a parabola y=kx2 (y-axis vertical) with a bead of mass m on it. The bead can side on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the y-axis is

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    physics-General
    macosθ=mgcos(90-θ) ag=tanθag=dydx ddxkx2=agx=a2gk    
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    G i v e n comma blank s equals t to the power of 3 end exponent plus 5
    S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent
    a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t
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    a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent
    a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent
    therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent
    therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent

    A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length s equals t to the power of 3 end exponent plus 5 comma where s is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

    physics-General
    G i v e n comma blank s equals t to the power of 3 end exponent plus 5
    S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent
    a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t
    therefore T a n g e n t i a l blank a c c e l e r a t i o n blank a t blank t equals 2 blank s comma
    a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent
    a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent
    therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent
    therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent
    General
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    A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

    As the prisms Q and R are of the same material and have identical shape they combine to form a slab with parallel faces. Such a slab does not cause any deviation.

    A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

    physics-General
    As the prisms Q and R are of the same material and have identical shape they combine to form a slab with parallel faces. Such a slab does not cause any deviation.
    General
    physics-

    In the given figure, what is the angle of prism

    Angle of prism is the angle between incident and emergent surfaces.

    In the given figure, what is the angle of prism

    physics-General
    Angle of prism is the angle between incident and emergent surfaces.
    General
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    A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

    In minimum deviation position refracted ray inside the prism is parallel to the base of the prism

    A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true

    physics-General
    In minimum deviation position refracted ray inside the prism is parallel to the base of the prism
    General
    maths-

    The equation of the plane containing the line fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text  is  end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0 where al + bm + cn is equal to

     

    Since these two lines are intersecting so shortest distance between the lines will be 0.
    Hence (c) is the correct answer.

    The equation of the plane containing the line fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text  is  end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0 where al + bm + cn is equal to

     

    maths-General
    Since these two lines are intersecting so shortest distance between the lines will be 0.
    Hence (c) is the correct answer.