If \(f(x)=\sin x-\cos x,\ x\in[0,2\pi]\) then (A) \(f(x)\) is increasing in \(\left(0,\frac{3\pi}{4}\right)\) (B) \(f(x)\) is decreasing in \(\left(0,\frac{3\pi}{4}\right)\) (C) \(f(x)\) is decreasing in \(\left(\frac{3\pi}{4},\frac{7\pi}{4}\right)\) (D) \(f(x)\) is decreasing in \(\left(\frac{7\pi}{4},2\pi\right)\) Choose the correct answer from the options given below:

For \(x\in\mathbb{R}\), if \(f(x)=-(x-1)^2+2\), then (A) \(f\) is an increasing function on \((-\infty,1]\) (B) \(f\) has no critical points (C) \(f\) has a maximum value at \(x=1\) (D) \(f\) has a minimum value at \(x=1\) Choose the correct answer from the options given below:

If \(x=a\sec^3\theta,\ y=a\tan^3\theta\), then \(\frac{d^2y}{dx^2}\) equals.

The domain of the function \(y=\sin^{-1}(x-1)+\cos^{-1}\sqrt{x-1}\) is:

A balloon which always remains spherical, has a variable diameter \(\frac{3}{2}(5x+7)\). Then the rate of change of its volume with respect to \(x\) is

Match List-I with List-II List-I | List-II (A) The number of possible matrices of order \(3\times3\) with each entry 1 or 0 | (I) \(2^4\) (B) The number of possible matrices of order \(2\times3\) with each entry 1 or 0 | (II) \(2^9\) (C) The number of possible matrices of order \…

Match List-I with List-II List-I | List-II (A) \(\tan^{-1}\frac{2}{11}+\tan^{-1}\frac{7}{24}\) | (I) \(\frac{3\pi}{4}\) (B) \(\tan^{-1}2+\tan^{-1}3\) | (II) \(\pi\) (C) \(\tan^{-1}1+\tan^{-1}2+\tan^{-1}3\) | (III) \(\tan^{-1}\frac{1}{2}\) (D)…

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

The projection of the vector \(5\hat i+\hat j-3\hat k\) on the vector \(\hat i+2\hat j-3\hat k\) is