Which of the following is not correct?

Match List-I with List-II

List I (Function)	List II (Derivative with respect to ' x ')
(A) \(f(x)=x^x\)	(I) \(a x^{a-1}\)
(B) \(f(x)=a^x\)	(II) 0
(C) \(f(x)=a^a\)	(III) \(a^x \log _e a\)
(D) \(f(x)=x^a\)	(IV) \(x^x\left(1+\log _e x\right)\)

Choose the correct answer from the options given below:

The interval in which \(f(x)=\sin x-\cos x, 0 \leq x \leq 2 \pi\) is strictly decreasing is:

By using the determinants the area of the triangle with vertices \((0,0),(0,1)\) and \((1,0)\) is :

For the system of linear equations
x + y + z = 5000
6x + 7y + 8z = 35800
6x + 7y - 8z = 7000
the values of x, y and z are:

If matrix \(A=\left[\begin{array}{ccc}x & 2 & 3 \\ a & y & -5 \\ b & c & 0\end{array}\right]\) isa skew-symmetric matrix, then
(A) x + y + c = 5
(B) c = 5
(C) a + b + c = 0
(D) a + b - c = 10
Choose the correct answer from the options given below:

If \(\vec{a}\) and \(\vec{b}\) are two unit vectors and \(\vec{a}+\vec{b}\) is also a unit vector, the magnitude of \(\vec{a}-\vec{b}\) is

If A is any \(m \times n\) matrix and B is a matrix such that AB and BA are both defined, then B is a matrix of order :

Arrange the following in increasing order of basic strength in aqueous solution :
(A) \(C _6 H _5 NH _2\)
(B) \(C _2 H _5 NH _2\)
(C) \(\left( C _2 H _5\right)_2 NH\)
(D) \(NH _3\)
Choose the correct answer from the options given below:

The hormone responsible for milk ejection during lactation is:

Read the given passage and answer the following questions:
Secondary treatment or biological treatment: The primary effluent is passed into large tanks where air is pumped into it. This allows vigorous growth of aerobic microbes. While growing, these microbes consume the major part of the organic matter in the effluent. This significantly change the BOD (biochemical oxygen demand) of the effluent. The effluent in then passed into a tank a small part of the sludge is pumped back into the aeration tank to serve as the inoculum. The remaining major part of the sludge is pumped into large tanks called sludge digesters. Here, other kinds of bacteria grow anaerobically in the sludge. The biogas produced can be used source of energy as it is inflammable. The effluent from the secondary treatment plant is generally released into natural water bodies like rivers and streams.
Which of the following gas is not produced in anaerobic digestion?

Identify " \(Y\) " in the following reaction:
\(R-NH_2+HNO_2 \xrightarrow{NaNO_2+HCl} X\xrightarrow{H_2 O} Y+N_2+HCl\)

The particular solution of the differential equation \(x\left(1+y^2\right) d x-y\left(1+x^2\right) d y=0\), given that \(y(0)=1\) is :