If \(\int e^x\left(\frac{x-1}{(x+1)^3}\right) d x=\frac{A e^x}{(x+1)^B}+C\), where \(C\) is constant of integration, then which of the following are correct?
(A) \(A=-1\)
(B) \(A=1\)
(C) \(B=3\)
(D) \(B=2\)
Choose the correct answer from the options given below :

If \(\vec{a}\) and \(\vec{b}\) are two non-zero orthogonal vectors, then \(|\vec{a}+\vec{b}|\) is equal to:

There are two values of a for which the determinant, \(\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 0 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86\), then the sum of these values of a is:

The volume of spherical balloon is increasing at the rate of 4 \(cm ^3\) / sec. The rate of increase of its surface area, when the radius is 3 cm will be :-

If \(\left(t_1, y_1\right),\left(t_2, y_2\right)\),...,\(\left(t_n, y_n\right)\) denote the time series and \(y_t\) are the trend values of the variables \(y\), then

The area of region bounded by two curves \(y^2=x\) and \(x^2=y\) is:

The maximum value of \(f(x)=\left(\frac{1}{x}\right)^x\) is

Read the following paragraph and answer the given questions.
RNAi takes place in all eukaryotic organisms as a method of cellular defense. The source of this complementary RNA could be from an infection by viruses having RNA genomes or mobile genetic elements (transposons) that replicate via an RNA intermediate. Insulin from an animal source, though caused some patients to develop allergy or other types of reactions to the foreign protein. Insulin consists of two short polypeptide chains: chain A and chain B, that are linked together by disulphide bridges. Transgenic mice are being developed for use in testing the safety of vaccines before they are used on humans. Transgenic mice are being used to test the safety of the polio vaccine. If successful and found to be reliable, they could replace the use of monkeys to test the safety of batches of the vaccine. Specific Bt toxin genes were isolated from Bacillusthuringiensis} and incorporated into the several crop plants such as cotton. The choice of genes depends upon the crop and the targeted pest, as most Bt toxins are insect-group specific. The toxin is coded by a gene cryIAc named cry. There are a number of them, for example, the proteins encoded by the genes cryIAc and cryIIAb control the cotton bollworms, that of cryIAb controls corn borer.

Which of these is not present in mature insulin?

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

The angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes \(\sqrt{3}\) and 4 respectively and \(\vec{a} \cdot \vec{b}=2 \sqrt{3}\) is :

Vitamin B12 is a coordination compound of _______

The capacitance of a capacitor becomes \( \frac{9}{7} \) times of its original value when a dielectric slab of thickness \( t = \frac{d}{4} \) is inserted between the plates, where d is the separation between the plates. The dielectric constant of the slab is

The value of \(\int_0^1 \log _e\left(\frac{1}{x}-1\right) d x\) is :