The value of the integral \(\int \frac{2 x}{\left(x^2+1\right)\left(x^2+2\right)} d x\) is:

For the system of equations \(A X=B\), which of the following is correct?

The number of all possible matrices of order 2 × 2 with each entry 0 or 1 is:

The sum of order and degree of the differential equation \(\frac{\left\{1+\left(\frac{d y}{d x}\right)^2\right\}^{\frac{5}{2}}}{\frac{d^2 y}{d x^2}}= p\) is :

Rakshita plans to buy a house for Rs. 1,00,00,000 with down payment of 20% of the value of the house paid by her mother. Rest of the amount she wishes to pay in 25 years by equal monthly instalments at an interest of 9% per annum compounded monthly Then the EMI paid by her is \(\left(\right.\) Given \(\left.(1.0075)^{300}=9\right)\) :

Consider the function \(f: N \rightarrow N\) given by \(f(x)=\left\{\begin{array}{ll}x+1, & \text { if } x \text { is odd } \\ x-1, & \text { if } x \text { is even }\end{array}\right.\)

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{j}\).
Then a vector \(\vec{c}\) satisfying \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}=8\) is equal to:

Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. 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. 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.
Malignant malaria is caused by -

If \(A\) and \(B\) are matrices of same order, then ( \(A B^{ T }-B A^{ T }\) ) is always:

If A is a square matrix of order 3 such that A(adj A) = \(\left[\begin{array}{ccc}-2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2\end{array}\right]\), then \(|A|\) is equal to

What is the shape of a wavefront when light emerging out of a convex lens when a parallel beam of light incident over it ?

The general solution of the differential equation \(\frac{x d y}{d x}+4 y=x^3,(x \neq 0)\) is :

What is the first step in acid catalyzed dehydration of alcohol?