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

The solution of the differential equation \(\frac{d y}{d x}=\left(1+x^2\right)\left(1+y^2\right)\) is (Here \(C\) is an arbitrary constant)

Let \(f: R \rightarrow R\) be defined as \(f(x)=5 x^2\). Choose the correct answer :

Consider \(z=x+3 y\) subject to the constraints \(2 x+3 y \geq 6,2 x+y \leq 8, x \geq 0, y \geq 0\). The minimum value of \(z\) is :

Match List - I with List - II

List - I	List - II
(A) \(\int \frac{d x}{a^2-x^2}\)	(I) \(\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C\)
(B) \(\int \frac{d x}{x^2-a^2}\)	(II) \(\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C\)
(C) \(\int \frac{d x}{x^2+a^2}\)	(III) \(\sin ^{-1}\left(\frac{x}{a}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{a^2-x^2}}\)	(IV) \(\frac{1}{a} \tan ^{-1} \frac{x}{a}+C\)

Choose the correct answer from the options given below:

Match List-I with List-II
Consider two vectors \(\vec{a}=\hat{i}+2 \hat{j}-\hat{k}\) and \(\vec{b}=-3 \hat{i}-6 \hat{j}+3 \hat{k}\).

List-I	List-II
(A) Angle between \(\vec{a}\) and \(\vec{b}\)	(I) \(\cos ^{-1}\left(\frac{1}{\sqrt{6}}\right)\)
(B) Angle between \(\vec{a}\) and \(x\)-axis	(II) \(\cos ^{-1}\left(\frac{2}{\sqrt{6}}\right)\)
(C) Angle between \(\vec{b}\) and \(x\)-axis	(III) \(\pi\)
(D) Angle between \(\vec{a}\) and \(y\)-axis	(IV) \(\cos ^{-1}\left(-\frac{1}{\sqrt{6}}\right)\)

Choose the correct answer from the options given below:

If \(P\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]\), then matrix \(P\) is equal to :

A battery of emf 4 V and internal resistance \(1 \, \Omega\) is connected across a wire of resistance \(9.0 \, \Omega\). The number of electrons passing through the cross-section of the wire in 4 s is:

A beam of electrons move horizontally from South to North in the region of the magnetic field acting vertically downward. The force acting on the electron due to the magnetic field is :

Match List-I with List-II:

List-I (Application)	List-II (Electromagnetic Waves)
(A) Used in radiography	(I) UV rays
(B) Detection by stimulating nerve endings of the human retina	(II) Microwaves
(C) Water purification	(III) Visible light
(D) Radar communication	(IV) X rays

Choose the correct answer from the options given below:

Read the following paragraph and answer the given questions.
Cancer detection is based on biopsy and histopathological studies. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Do you know why mother's milk is considered very essential for the new-born infant? The yellowish fluid colostrum secreted by mother during the initial days of lactation has abundant antibodies (IgA) to protect the infant. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal. The exaggerated response of the immune system to certain antigens present in the environment is called allergy. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Drugs like barbiturates, amphetamines, benzodiazepines, and other similar drugs, that are normally used as medicines to help patients cope with mental illnesses like depression and insomnia, are often abused. Morphine is a very effective sedative and painkiller, and is very useful in patients who have undergone surgery.
Which one of the following is not used for detection of cancer?

The first clinical gene therapy was given in 1990 to a 4-year-old girl with __________.

Electric field at a point due to a uniformly charged infinite plane sheet is