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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

  1. 0
  2. 2 straight Q with hat on top to the power of 1 fraction numerator sin invisible function application straight x over denominator 3 minus vertical line straight x vertical line end fraction dx    
  3. straight Q with hat on top to the power of 1 fraction numerator negative 2 straight x squared over denominator 3 minus vertical line straight x vertical line end fraction dx    
  4. 2 straight Q with hat on top to the power of 1 fraction numerator sin invisible function application straight x minus straight x squared over denominator 3 minus vertical line straight x vertical line end fraction dx    

The correct answer is: straight Q with hat on top to the power of 1 fraction numerator negative 2 straight x squared over denominator 3 minus vertical line straight x vertical line end fraction dx

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General
maths-

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
maths-

Let f:RR,f(x)=|x[x]|,[x]|x[x+1]|,[x] is odd 1 is even where [.]  denotes greatest integer function, then 24f(x)dx is equal to

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

Let f:RR,f(x)=|x[x]|,[x]|x[x+1]|,[x] is odd 1 is even where [.]  denotes greatest integer function, then 24f(x)dx 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.    
General
maths-

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.    
General
maths-

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-

The shortest distance between the two straight linefraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction is

The given equations are
fraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and
fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction
rightwards double arrow fraction numerator x minus begin display style 4 over 3 end style over denominator begin display style 5 over 3 end style end fraction equals fraction numerator y plus begin display style 6 over 5 end style over denominator begin display style 8 over 5 end style end fraction equals fraction numerator z minus begin display style 3 over 2 end style over denominator begin display style 9 over 2 end style end fraction
The shortest distance between the two cartesian lines
=fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell x subscript 2 minus x subscript 1 end cell cell y subscript 2 minus y subscript 1 end cell cell z subscript 2 minus z subscript 1 end cell row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell end table close vertical bar space comma w h e r e space D equals left parenthesis a subscript 1 b subscript 2 minus a subscript 2 b subscript 1 right parenthesis squared plus left parenthesis b subscript 1 c subscript 2 minus b subscript 2 c subscript 1 right parenthesis squared plus left parenthesis c subscript 1 a subscript 2 minus c subscript 2 a subscript 1 right parenthesis squared
equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell 4 over 3 minus 4 over 3 end cell cell fraction numerator negative 6 over denominator 5 end fraction plus 6 over 5 end cell cell 3 over 2 minus 3 over 2 end cell row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar
equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row 0 0 0 row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar
equals fraction numerator 1 over denominator square root of D end fraction cross times 0
equals 0

The shortest distance between the two straight linefraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction is

Maths-General
The given equations are
fraction numerator x minus 4 divided by 3 over denominator 2 end fraction equals fraction numerator y plus 6 divided by 5 over denominator 3 end fraction equals fraction numerator z minus 3 divided by 2 over denominator 4 end fraction and
fraction numerator 5 y plus 6 over denominator 8 end fraction equals fraction numerator 2 z minus 3 over denominator 9 end fraction equals fraction numerator 3 x minus 4 over denominator 5 end fraction
rightwards double arrow fraction numerator x minus begin display style 4 over 3 end style over denominator begin display style 5 over 3 end style end fraction equals fraction numerator y plus begin display style 6 over 5 end style over denominator begin display style 8 over 5 end style end fraction equals fraction numerator z minus begin display style 3 over 2 end style over denominator begin display style 9 over 2 end style end fraction
The shortest distance between the two cartesian lines
=fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell x subscript 2 minus x subscript 1 end cell cell y subscript 2 minus y subscript 1 end cell cell z subscript 2 minus z subscript 1 end cell row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell end table close vertical bar space comma w h e r e space D equals left parenthesis a subscript 1 b subscript 2 minus a subscript 2 b subscript 1 right parenthesis squared plus left parenthesis b subscript 1 c subscript 2 minus b subscript 2 c subscript 1 right parenthesis squared plus left parenthesis c subscript 1 a subscript 2 minus c subscript 2 a subscript 1 right parenthesis squared
equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row cell 4 over 3 minus 4 over 3 end cell cell fraction numerator negative 6 over denominator 5 end fraction plus 6 over 5 end cell cell 3 over 2 minus 3 over 2 end cell row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar
equals fraction numerator 1 over denominator square root of D end fraction open vertical bar table row 0 0 0 row 2 3 4 row cell 5 over 3 end cell cell 8 over 5 end cell cell 9 over 2 end cell end table close vertical bar
equals fraction numerator 1 over denominator square root of D end fraction cross times 0
equals 0
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

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°    

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°    
General
physics-

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

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

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
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.
General
physics-

A small body of mass m slides down from the top of a hemisphere of radius r. The surface of block and hemisphere are frictionless. The height at which the body lose contact with the surface of the sphere is

A small body of mass m slides down from the top of a hemisphere of radius r. The surface of block and hemisphere are frictionless. The height at which the body lose contact with the surface of the sphere is

physics-General
General
physics-

Average torque on a projectile of mass m, initial speed u and angles of projection theta, between initial and final position P and Q as shown in figure about the point of projection is

Time of flight. T equals fraction numerator 2 u sin invisible function application theta over denominator g end fraction
Horizontal range, R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
Change in angular momentum,
open vertical bar d stack L with rightwards arrow on top close vertical bar equals open vertical bar stack L with rightwards arrow on top subscript f end subscript minus stack L with rightwards arrow on top subscript i end subscript close vertical bar about point of projection
equals left parenthesis m u sin invisible function application theta right parenthesis cross times fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
equals fraction numerator m u to the power of 3 end exponent sin invisible function application theta sin invisible function application 2 theta over denominator g end fraction
T o r q u e blank open vertical bar stack tau with rightwards arrow on top close vertical bar equals fraction numerator c h a n g e blank i n blank a n g u l a r blank m o m e n t u m over denominator t i m e blank o f blank f l i g h t end fraction
equals open vertical bar fraction numerator d stack L with rightwards arrow on top over denominator T end fraction close vertical bar

Average torque on a projectile of mass m, initial speed u and angles of projection theta, between initial and final position P and Q as shown in figure about the point of projection is

physics-General
Time of flight. T equals fraction numerator 2 u sin invisible function application theta over denominator g end fraction
Horizontal range, R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
Change in angular momentum,
open vertical bar d stack L with rightwards arrow on top close vertical bar equals open vertical bar stack L with rightwards arrow on top subscript f end subscript minus stack L with rightwards arrow on top subscript i end subscript close vertical bar about point of projection
equals left parenthesis m u sin invisible function application theta right parenthesis cross times fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
equals fraction numerator m u to the power of 3 end exponent sin invisible function application theta sin invisible function application 2 theta over denominator g end fraction
T o r q u e blank open vertical bar stack tau with rightwards arrow on top close vertical bar equals fraction numerator c h a n g e blank i n blank a n g u l a r blank m o m e n t u m over denominator t i m e blank o f blank f l i g h t end fraction
equals open vertical bar fraction numerator d stack L with rightwards arrow on top over denominator T end fraction close vertical bar
General
physics-

A string of length L is fixed at one end and the string makes fraction numerator 2 over denominator pi end fraction rev/s around the vertical axis through, the fixed and as shown in the figure, then tension in the string is

T sin invisible function application theta equals M omega to the power of 2 end exponent R (i)
T sin invisible function application theta equals M omega to the power of 2 end exponent L blank s i n blank theta (ii)
T equals M omega to the power of 2 end exponent L
equals M bullet 4 pi to the power of 2 end exponent n to the power of 2 end exponent L
equals M bullet 4 pi to the power of 2 end exponent open parentheses fraction numerator 2 over denominator pi end fraction close parentheses to the power of 2 end exponent L
equals 16 blank M L

A string of length L is fixed at one end and the string makes fraction numerator 2 over denominator pi end fraction rev/s around the vertical axis through, the fixed and as shown in the figure, then tension in the string is

physics-General
T sin invisible function application theta equals M omega to the power of 2 end exponent R (i)
T sin invisible function application theta equals M omega to the power of 2 end exponent L blank s i n blank theta (ii)
T equals M omega to the power of 2 end exponent L
equals M bullet 4 pi to the power of 2 end exponent n to the power of 2 end exponent L
equals M bullet 4 pi to the power of 2 end exponent open parentheses fraction numerator 2 over denominator pi end fraction close parentheses to the power of 2 end exponent L
equals 16 blank M L
General
physics-

A thin prism P subscript 1 end subscript with angle 6 to the power of ring operator end exponent and made from glass of refractive index 1.54 is combined with another thin prism P subscript 2 end subscript of refractive index 1.72 to produce dispersion without deviation. The angle of prism P subscript 2 end subscript will be

fraction numerator A to the power of ´ end exponent over denominator A end fraction equals fraction numerator left parenthesis mu subscript y end subscript minus 1 right parenthesis over denominator left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis end fraction rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator 6 end fraction equals negative fraction numerator left parenthesis 1.54 minus 1 right parenthesis over denominator left parenthesis 1.72 minus 1 right parenthesis end fraction
Þ A' equals negative 4.5 to the power of o end exponent equals 4 to the power of o end exponent 3 0 to the power of ´ end exponent

A thin prism P subscript 1 end subscript with angle 6 to the power of ring operator end exponent and made from glass of refractive index 1.54 is combined with another thin prism P subscript 2 end subscript of refractive index 1.72 to produce dispersion without deviation. The angle of prism P subscript 2 end subscript will be

physics-General
fraction numerator A to the power of ´ end exponent over denominator A end fraction equals fraction numerator left parenthesis mu subscript y end subscript minus 1 right parenthesis over denominator left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis end fraction rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator 6 end fraction equals negative fraction numerator left parenthesis 1.54 minus 1 right parenthesis over denominator left parenthesis 1.72 minus 1 right parenthesis end fraction
Þ A' equals negative 4.5 to the power of o end exponent equals 4 to the power of o end exponent 3 0 to the power of ´ end exponent
General
physics-

A triangular prism of glass is shown in the figure. A ray incident normally to one face is totally reflected, if theta equals 4 5 to the power of o end exponent. The index of refraction of glass is

For total internal reflection theta greater than C
rightwards double arrow sin invisible function application theta greater than sin invisible function application C rightwards double arrow sin invisible function application theta greater than fraction numerator 1 over denominator mu end fraction
or mu greater than fraction numerator 1 over denominator sin invisible function application theta end fraction rightwards double arrow mu greater than fraction numerator 1 over denominator sin invisible function application 4 5 to the power of o end exponent end fraction rightwards double arrow mu greater than square root of 2 rightwards double arrow mu greater than 1.41

A triangular prism of glass is shown in the figure. A ray incident normally to one face is totally reflected, if theta equals 4 5 to the power of o end exponent. The index of refraction of glass is

physics-General
For total internal reflection theta greater than C
rightwards double arrow sin invisible function application theta greater than sin invisible function application C rightwards double arrow sin invisible function application theta greater than fraction numerator 1 over denominator mu end fraction
or mu greater than fraction numerator 1 over denominator sin invisible function application theta end fraction rightwards double arrow mu greater than fraction numerator 1 over denominator sin invisible function application 4 5 to the power of o end exponent end fraction rightwards double arrow mu greater than square root of 2 rightwards double arrow mu greater than 1.41
General
physics-

Which of the following diagrams, shows correctly the dispersion of white light by a prism

Because in dispersion of white light, the rays of different colours are not parallel to each other. Also deviation takes place in same direction.

Which of the following diagrams, shows correctly the dispersion of white light by a prism

physics-General
Because in dispersion of white light, the rays of different colours are not parallel to each other. Also deviation takes place in same direction.
General
physics-

A ray of light incident normally on an isosceles right angled prism travels as shown in the figure. The least value of the refractive index of the prism must be

A ray of light incident normally on an isosceles right angled prism travels as shown in the figure. The least value of the refractive index of the prism must be

physics-General