The average cost function for a commodity is given by \(A C=0.05 x^2-5 x+1000+\frac{3000}{x}\) in terms of output \(x\). The fixed cost is :

If the product of matrices \(\left[\begin{array}{cc}\cos ^2 \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta\end{array}\right]\) and \(\left[\begin{array}{cc}\cos ^2 \beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & \sin ^2 \beta\end{array}\right]\) is a null matrix, then \(\theta\) and \(\beta\) differ by:

If \(\mathbb{Z}\) and \(\mathbb{R}\) set of integers and set of real numbers respectively, then match List I with List II.

List - I	List - II
(A) \(5 x-3<3 x+1, x \in \mathbb{Z}\)	(I) \(x \in(-\infty,-3)\)
(B) \(3 x+17 \leq 2(1-x), x \in \mathbb{R}\)	(II) \(x \in(-\infty,-1)\)
(C) \(13 x+17<2(1-x), x \in \mathbb{R}\)	(III) \(\{\ldots,-4,-3, \ldots, 0,1\}\)
(D) \(\frac{2 x+3}{5}-2>\frac{3(x-2)}{5}, x \in \mathbb{Z}\)	(IV) \(\{\ldots,-4,-3,-2\}\)

Choose the correct answer from the options given below:

The function \(f(x)=2 x^3-3 x^2-12 x+4\) has :

Let \(f(x)=x^3\) be a function with domain \(\{0,1,2,3\}\) then domain of \(f^{-1}\) is:

The order and degree of the differential equation
\(x^2 \frac{d^2y}{dx^2} - 3 \left(\frac{dy}{dx}\right)^2 + y = 0\) are:

The solution of the differential equation \(\frac{d y}{d x}=\sqrt{\frac{y}{x}}\) is

Consider the figure as shown below. If work done in bringing a point charge from infinity to points A, B and C are \( W_A, W_B \) and \( W_C \) respectively, then, which of the following relation is correct?

Which of the following frequency would be suitable for the beyond the horizon communication using sky waves?

Answer the question on the basis of passage given below:
Second period elements of p-Block differs from the rest of their group members due to their small atomic size, high electronegativity, high ionization enthalpy and absence of d-orbitals. The elements from the second period like C, N, O have unique property of forming \(p \pi-p \pi\) multiple bonds whereas other heavier elements of p-block form \(d \pi-p \pi\) multiple bonds. This changes the properties of second period elements to great extent. Also the single bond strength of the p-block elements with their own atoms gets affected by their atomic size and presence of lone pairs.
Inert pair effect also affects the properties of the p-Block elements especially for those elements which are present from 4th period onward and this happens due to poor screening effect of inner (n-1)d subshell electrons.
Among the elements mentioned below, some are not stable in their +3 oxidation state and undergoes disproportionation reactions easily:
A. N
B. \(P\)
C. As
D. Sb
E. Bi
Choose the correct answer from the options given below:

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) be a \(3 \times 3\) matrix,
where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\).
If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is ___________ .

If 200 MeV energy is released in the fission of a single nucleus of uranium-235, how many fissions per second will produce a power of 32 kW?

Which transition metal group is known as coinage metals?