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Engineering
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\lim_{x\rightarrow 2}\left ( \frac{\sqrt{1-\cos \left \{ 2(x-2) \right \}}}{x-2} \right )

  • Option 1)

    equals-\sqrt{2}\;

  • Option 2)

    \; \; equals\frac{1}{\sqrt{2}}\;

  • Option 3)

    does not exist

  • Option 4)

    \; equals\sqrt{2}\;

 
As we learnt in Evalution of Trigonometric limit - -     For x-2    it is For x-2    it is So LHL PHL So limit does not exist. Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) does not exist This option is correct. Option 4) This option is incorrect.
Engineering
118 Views   |  

For x\in \left ( 0,\frac{5\pi }{2} \right ),   define  f(x)=\int_{0}^{x}\sqrt{t}\sin t\, dt\; \; Then\; f\; has

  • Option 1)

    local minimum at \pi and local maximum at 2\pi

  • Option 2)

    local maximum at \pi and local minimum at 2\pi

  • Option 3)

    local maximum at \pi and 2\pi

  • Option 4)

    local minimum at \pi and 2\pi

 
As we learnt in Rate Measurement - Rate of any of variable with respect to time is rate of measurement. Means according to small change in time how much other factors change is rate measurement: - wherein Where dR / dt  means Rate of change of radius.     & x=0 at   at      It is   Option 1) local minimum at and local maximum at This option is incorrect. Option 2) local...
Engineering
118 Views   |  

\frac{d^{2}x}{dy^{2}}  equals  to

  • Option 1)

    \left ( \frac{d^{2}y}{dx^{2}} \right )\left ( \frac{dy}{dx} \right )^{-2}\;

  • Option 2)

    \; \; -\left ( \frac{d^{2}y}{dx^{2}} \right )\left ( \frac{dy}{dx} \right )^{-3}\;

  • Option 3)

    \; \; \left ( \frac{d^{2}y}{dx^{2}} \right )^{-1}\;

  • Option 4)

    \; -\left ( \frac{d^{2}y}{dx^{2}} \right )^{-1}\left ( \frac{dy}{dx} \right )^{-3}

 
As we learnt in Second order derivative for parametric function - When we find  -     Option 1) This option is incorrect. Option 2) This option is correct. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
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The values of p\; and\; q for which the function

 f(x)=\left\{\begin{matrix} \frac{\sin (p+1)x+\sin x}{x} &,x< 0 \\ q&,x=0 \\ \frac{\frac{q}{\sqrt{x+x^{2}-\sqrt{x}}}}{x^{3/2}}&,x> 0 \end{matrix}\right.

is continuous for all x in R,are

 

  • Option 1)

    p=-\frac{3}{2},q=\frac{1}{2}\;

  • Option 2)

    \; \; p=\frac{1}{2},q=\frac{3}{2}\;

  • Option 3)

    \; p=\frac{1}{2},q=-\frac{3}{2}\;

  • Option 4)

    \; \; p=\frac{5}{2},q=\frac{1}{2}

 
As we learnt in Continuity - If the function is continuous, Its graph does not break but for discontinuous functions there is a break in the graph. - wherein     Since f(x) is continuous so that = P+2 = q = P = q = Option 1) This solution is correct. Option 2) This solution is incorrect  Option 3) This solution is incorrect  Option 4) This solution is incorrect 
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