The general solution of the differential equation \(\frac{d y}{d x}-\frac{y}{x}=x^2\) is :

The area (in sq. units) of the region bounded by the curve \(y=2 x^3, x\)-axis and ordinates \(x=-1\) and \(x=1\) is.

Which of the following are types of statistical inferences?
(A) Point estimation
(B) Interval estimation
(C) Marginal estimation
(D) Hypothesis testing
Choose the correct answer from the options given below :

At what rate of interest will the present value of a perpetuity of ₹ 500 payable at the end of every 6 months be ₹ 20,000 ?

The area of the triangle whose vertices are \((3,8),(-4,2)\) and \((5,1)\) is

If the cost function of a product is given by \(C(x)=\frac{3}{4} x^2-5 x+21\), then the marginal cost when \(x=10\) is

Match the Matrix in List - I with its Adjoint's Determinant in List - II.
List - I (Matrix A) :

LIST 1	LIST 2
(A) \(\left[\begin{array}{ll}3 & 1 \\ 4 & 2\end{array}\right]\)	(I) 9
(B) \(\left[\begin{array}{cc}5 & -1 \\ 4 & 2\end{array}\right]\)	(II) 8
(C) \(\left[\begin{array}{cc}6 & -1 \\ 2 & 1\end{array}\right]\)	(III) 14
(D) \(\left[\begin{array}{ll}4 & 1 \\ 3 & 3\end{array}\right]\)	(IV) 2

Choose the correct answer from the options given below:

Read the passage and answer the following
Microbes are organisms that cannot be seen with the naked eye. They include bacteria, archaea, protists, viruses, prions, and fungi. They account for most of the diversity of life on earth. Far more microbes inhabit a single person's body than there are people on the entire planet. Microbes also dominate in terms of their ability to live and thrive in extreme environments, including clouds, deep-sea volcanoes, springs, ice caps, and animal intestines. Most animals, including humans, exhibit numerous associations with microbes. Our "human microbiome" includes our skin, hair, mouth, and gut.
Which of the following microbe is harmful?

Select the correct statements for [Fe(CN)₆]^3- complex :
(A) Paramagnetic
(B) sp³d² hybridization
(C) Magnetic moment = 5.92 BM
(D) d²sp³ hybridization
Choose the correct answer from the options given below :

\(\int \tan ^{-1} \sqrt{x} d x\) equals to : (Here \(C\) is an arbitrary constant)

Match List I with List II

LISTI	LIST II
A. \(\int \sin ^2(2 x+1) d x\)	I. \(\frac{x}{2}+\frac{\sin (2 x+1) \cos (2 x+1)}{4}+C\)
B. \(\int \cos ^2(2 x+1) d x\)	II. \(\frac{\cos 3(2 x+1)}{24}-\frac{3 \cos (2 x+1)}{8}+C\)
C. \(\int \sin ^3(2 x+1) d x\)	III. \(\frac{x}{2}-\frac{\sin (2 x+1) \cos (2 x+1)}{4}+C\)
D. \(\int \cos ^3(2 x+1) d x\)	IV. \(\frac{\sin 3(2 x+1)}{24}+\frac{3 \sin (2 x+1)}{8}+C\)

Choose the correct answer from the options given below:

In an a. c. series LCR circuit, the applied voltage is \(v=(100 \sqrt{2}) \sin (\omega t+\pi / 4)\) volt. Given that \(R=30 \Omega, X_L=50 \Omega\) and \(X_C=10 \Omega\). The value of power dissipated in the circuit is

One cannot see through fog clearly because :