If \(y=\log \left(x+\sqrt{x^2+a^2}\right)\) then \(\frac{d y}{d x}\) =

If \(y=500 e^{7 x}+600 e^{-7 x}\), then \(\frac{d^2 y}{d x^2}\) is equal to :

For the function \(f(x)=2 x^3-3 x^2-12 x+5\), the difference of maximum and minimum value of \(f(x)\) is

If \(\cos ^{-1} \sqrt{3} x+\cos ^{-1} x=\frac{\pi}{2}\), then the value of \(x\) is :

The area of the region (in square units) bounded by \(x=1, x=2\) and the curve \(y^2=4 x\) in the first quadrant is:

The corner points of the bounded feasible region determined by the system of linear constraints are \((0,8),(4,4)\), \((12,12),(0,20)\). Let \(z=p x+q y\), where \(p, q>0\). Condition on \(p\) and \(q\) so that the maximum of \(z\) occurs at both the points \((12,12)\), \((0,20)\) is

At what rate of interest will the present value of a perpetuity of Rs. 1000 payable at the end of every six months be Rs. 20000?

Let \(f: R -\left\{\frac{3}{5}\right\} \rightarrow R\) be defined by \(f(x)=\frac{3 x+2}{5 x-3}\), then

An LED has a voltage drop of 2 V across it and a current of 10 mA flows when it operates with a 6 V battery through a limiting resistance R. The value of R is :

Match List I with List II

LIST I	LIST II
A. Tubectomy	I. Barrier method
B. Copper ions	II. Surgical method of sterilisation in human male.
C. Cervical Cap	III. Surgical method of sterilisation in human female.
D. Vasectomy	IV. Suppress motility of sperms

Choose the correct answer from the options given below:

In a diffraction pattern due to a narrow slit of width a, alternate dark and bright fringe pattern is obtained on a screen kept at distance D from the slit. Choose the correct option.
(A) The angular width of central maximum is given by 2D/a, linear width is given by 2D\lambda/a
(B) When the size of aperture, a is quite large, the diffraction effect is negligible,
(C) For light of smaller wavelength, angular width of central maximum increases, while linear width decreases,
(D) Angular width of central maximum does not decrease while linear width decreases on decreasing distance of screen from slit.
Choose the correct answer from the options given below:

Ferric oxide in blast furnace's upper half is mainly reduced by :

Which one of the followings is not the male sex accessory duct?