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If \int \frac{dx}{(x^{2}-2x+10)^{2}}=A(\tan^{-1}(\frac{x-1}{3})+\frac{f(x)}{x^{2}-2x+10})+C

Where C is a constant of integration , then :

  • Option 1)

    A=\frac{1}{54}\: \: and\: \: f(x)=3(x-1)

  • Option 2)

    A=\frac{1}{81}\: \: and\: \: f(x)=3(x-1)

  • Option 3)

    A=\frac{1}{27}\: \: and\: \: f(x)=9(x-1)

  • Option 4)

    A=\frac{1}{54}\: \: and\: \: f(x)=9(x-1)^{2}

 
put              correct option is (1)    Option 1) Option 2) Option 3) Option 4)

If the area ( in sq. units) bounded by the parabola y^{2}=4\lambda x and 

the line y=\lambda x , \lambda>0 , is \frac{1}{9} , then \lambda is equal to :  

  • Option 1)

    2\sqrt6

  • Option 2)

    48

  • Option 3)

    24

  • Option 4)

    4\sqrt3

 
the parabola  and  the line  If , then Area =          =>         =>        =>         =>          =>       Option 1) Option 2) 48 Option 3) 24 Option 4)

Let \alpha \epsilon (0,\pi /2) be fixed. If the integral  \int \frac{\tan x+\tan \alpha }{\tan x-\tan \alpha}dx= A(x) \cos 2\alpha +B(x) \sin 2\alpha +C, where C is a constant of integration, then the functions A(x) and B(x) are respectively :

 

  • Option 1)

    x+\alpha and \log_{e}|\sin (x+\alpha )|

     

     

     

     

  • Option 2)

    x-\alpha and  \log_{e}|\sin (x-\alpha )|

  • Option 3)

    x-\alpha and \log_{e}|\cos (x-\alpha )|

  • Option 4)

    x+\alpha and \log_{e}|\sin (x-\alpha )|

 
I =   comparing with LHS    Option 1)  and          Option 2)  and   Option 3)  and  Option 4)  and 

A value of \alpha such that 

\int_{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}=\log_{e}\left ( \frac{9}{8} \right ) is :

 

  • Option 1)

    -2 

     

     

  • Option 2)

    \frac{1}{2}

  • Option 3)

    -\frac{1}{2}

  • Option 4)

    2

 
using partial fractions, & Option 1) -2      Option 2) Option 3) Option 4) 2

The integral \int \frac{2x^{3}-1}{x^{4}+x}dx is equal to : (Here C is a constant of integration) 


 

  • Option 1)

    \log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

  • Option 2)

    \log_{e}\frac{\left | x^{3}+1 \right |}{x}+C

  • Option 3)

    \frac{1}{2}\log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

     

  • Option 4)

    \frac{1}{2}\log_{e}\frac{\left ( x^{3}+1 \right )^{2}}{\left | x^{3} \right |}+C

 
                                     Option 1) Option 2) Option 3)   Option 4)

If the area (in sq. units) of the region \left \{ \left ( x,y \right ):y^{2}\leq 4x,x+y\leq 1,x\geq 0,y\geq 0 \right \} is a\sqrt{2}+b, then a-b is equal to : 

  • Option 1)

    -\frac{2}{3}

  • Option 2)

    \frac{10}{3}

  • Option 3)

    6

  • Option 4)

    \frac{8}{3}

          Option 1)Option 2)Option 3)Option 4)

If \int_{0}^{\frac{\pi }{2}}\frac{\cot x}{\cot x+cosec x}dx=m\left ( \pi +n \right ), then  m\cdot n is equal to :

 

 

 

  • Option 1)

    -1

  • Option 2)

    1

  • Option 3)

    -\frac{1}{2}

  • Option 4)

    \frac{1}{2}

 
                                                                                                                                                                                                         So,          and      Option 1) Option 2) Option 3) Option 4)

The integral \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}sec^{\frac{2}{3}}x\: cosec^{\frac{4}{3}}x\: dx is equal to :

  • Option 1)

    3^{\frac{5}{6}}-3^{\frac{2}{3}}

  • Option 2)

    3^{\frac{4}{3}}-3^{\frac{1}{3}}

  • Option 3)

    3^{\frac{7}{6}}-3^{\frac{5}{6}}

  • Option 4)

    3^{\frac{5}{3}}-3^{\frac{1}{3}}

 
Let  =>                                            So, option (3) is correct. Option 1) Option 2) Option 3) Option 4)

If \int x^{5}e^{-x^{2}}dx=g(x)e^{-x^{2}}+C, where C is a constant 

of integration, then g(-1) is equal to :

  • Option 1)

    -1

  • Option 2)

    1

  • Option 3)

    -\frac{5}{2}

  • Option 4)

    -\frac{1}{2}

 
Let          Integrating by parts    =>  =>                      So, option (3) is correct.     Option 1) -1 Option 2) 1 Option 3) Option 4)

The area ( in sq. units ) of the region bounded by the curves y=2^{x} and

y=|x+1|, in the first quadrant is :

  • Option 1)

    \log_{e}2+\frac{3}{2}

  • Option 2)

    \frac{3}{2}

  • Option 3)

    \frac{1}{2}

  • Option 4)

    \frac{3}{2}-\frac{1}{\log_{e}2}

 
 and    The required area is  So, option (4) is correct.   Option 1) Option 2) Option 3) Option 4)

\lim_{n\rightarrow \infty }(\frac{(n+1)^{\frac{1}{3}}}{(n)^{\frac{4}{3}}})+\frac{(n+2)^{\frac{1}{3}}}{(n)^{\frac{4}{3}}}+............+\frac{(2n)^{\frac{1}{3}}}{(n)^{\frac{4}{3}}}) 

is equal to :

  • Option 1)

    \frac{3}{4}(2)^{\frac{4}{3}}-\frac{3}{4}

  • Option 2)

    \frac{4}{3}(2)^{\frac{4}{3}}

  • Option 3)

    \frac{3}{4}(2)^{\frac{4}{3}}-\frac{4}{3}

  • Option 4)

    \frac{4}{3}(2)^{\frac{3}{4}}

 
=>  =>  =>  option (1) is correct. Option 1) Option 2) Option 3) Option 4)

The region represented by \left | x-y \right |\leq 2 and \left | x+y \right |\leq 2 is 

bounded by a :

  • Option 1)

    square of side length 2\sqrt2 units

  • Option 2)

    rhombus of side length 2 units

  • Option 3)

    square of area 16 sq. units

  • Option 4)

    rhombus of area 8\sqrt2 sq. units

 
region bounded by  and  Graph of this is    Area = 8 side =     option (1) is correct. Option 1) square of side length  units Option 2) rhombus of side length 2 units Option 3) square of area 16 sq. units Option 4) rhombus of area  sq. units

The value of \int_{0}^{2\pi}[sin2x(1+cos3x)]dx, where [t] denotes

greatest integer function, is :

  • Option 1)

    \pi

  • Option 2)

    -\pi

  • Option 3)

    -2\pi

  • Option 4)

    2\pi

 
 .....................(1)    ..............................(2) Add (1) and (2)           So, option (2) is correct. Option 1) Option 2) Option 3) Option 4)

The area ( in sq. units ) of the region A=\left \{ (x,y):\frac{y^{2}}{2}\leq x\leq y+4 \right \}  is :

  • Option 1)

    \frac{53}{3}

  • Option 2)

    30

  • Option 3)

    16

  • Option 4)

    18

 
required are     Option 1) Option 2) Option 3) Option 4)

If     \\ \int e^{\sec x}\left ( \sec x \tan x f(x) +\sec x \tan x +\sec^{2}x \right )dx=e^{\sec x}f(x)+C   , then a possible choice of f(x)  is  :

  • Option 1)

    \sec x + \tan x +\frac{1}{2}

  • Option 2)

    \sec x - \tan x -\frac{1}{2}

  • Option 3)

    \sec x + x \:\tan x -\frac{1}{2}

  • Option 4)

    x \sec x + \tan x +\frac{1}{2}

     

 
differentiating both sides  Option 1) Option 2) Option 3) Option 4)  

If  f:R\rightarrow R  is a differentiable function and F(2)=6   , then \lim_{x\rightarrow 2}\int_{6}^{f(x)}\frac{2t\:dt}{(x-2)}   is :

 

 

 

  • Option 1)

    24f^{'}(2)

  • Option 2)

    2f^{'}(2)

  • Option 3)

    0

  • Option 4)

    12f^{'}(2)

 
Option 1) Option 2) Option 3) Option 4)

The value of the integral 

\int_{0}^{1}x\cot^{-1}(1-x^{2}+x^{4})dx is :

  • Option 1)

    \frac{\pi}{2}-\frac{1}{2}\:log_{e}\:2

  • Option 2)

    \frac{\pi}{4}-\:log_{e}\:2

  • Option 3)

    \frac{\pi}{2}-\:log_{e}\:2

  • Option 4)

    \frac{\pi}{4}-\frac{1}{2}\:log_{e}\:2

 
Option 1) Option 2) Option 3) Option 4)

The area (in sq. units) of the region  A=\left \{ \left ( x,y \right ):x^{2}\leq y\leq x+2 \right \} is :

  • Option 1)

    \frac{10}{3}

  • Option 2)

    \frac{9}{2}

  • Option 3)

     \frac{31}{6}

  • Option 4)

     \frac{13}{6}

 
Point of intersection of  and                                         and                                                      Required Area,                                                                                                                  Option 1) Option 2) Option 3)   Option 4)  

The value of  \int_{0}^{\pi /2}\frac{sin^{3}x}{sinx+cos x}dx    is :

  • Option 1)

     \frac{\pi -2}{8}       

  • Option 2)

    \frac{\pi -1}{4}

  • Option 3)

     \frac{\pi -2}{4}

  • Option 4)

    \frac{\pi -1}{2}

 
                                             Option 1)          Option 2) Option 3)   Option 4)

The integral \int sec^{\frac{2}{3}}x\; cosec^{\frac{4}{3}}x\; dx is equal to :

(Here C is a constant of integration)

  • Option 1)

     -3\; tan^{\frac{-1}{3}}x+C            

  • Option 2)

    -\; \frac{3}{4}\; tan^{\frac{-4}{3}}x+C

  • Option 3)

    -3\; cot^{\frac{-1}{3}}x+C

  • Option 4)

     3\; tan^{\frac{-1}{3}}x+C

                                              put,                                                                                                          or                  Option 1)             Option 2)Option 3)Option 4) 
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