If \(\vec{a}\) and \(\vec{b}\) represent the diagonals of a square, then

A and B are two students. Their chances of solving a problem correctly are \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively.
If the probability of their making a common error is \(\frac{1}{20}\) and they obtain the same answer, then the probability that their answer is correct is:

Slope of the tangent to the parabola \(y^2=x+2\) at a point in the 1st quadrant and lying on the line \(y=x\) is:

Consider the LPP: Minimize \(Z=x+2 y\) subject to \(2 x+y \geq 3, x+2 y \geq 6, x, y \geq 0\). The optimal feasible solution occurs at:

If \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\). The value of \(\frac{d^2 y}{d x^2}\) at \(\theta=\frac{\pi}{6}\) is :

The sides of an equilateral triangle are increasing at the rate of \(2 cm / sec\). The rate at which the area increases when the side is 10 cm , is

If \(y=\log _e\left(\frac{e^2}{x^2}\right)\) for \(x \neq 0\), then \(\frac{d^2 y}{d x^2}\) equals

Answer the question on the basis of passage given below :
Living systems are made up of various complex bio-molecules like carbohydrates, proteins, nucleic acids, lipids etc. Proteins and carbohydrates are essential constituents of our food. Carbohydrates are primarily produced by plants and form a very large group of naturally occurring organic compounds. Some common examples are cane sugar, glucose, starch etc.
Insulin, generally given to diabetic patients is a :

The equivalent resistance between P and Q is:

Choose the correct statements from the following :
(A) A dielectric develops a net dipole moment in the presence of an electric field.
(B) Electrostatic potential is constant throughout the volume of the conductor.
(C) Electric field \( \vec{E} \) depends upon the distance between two thin infinite long uniformly charged plane sheets held parallel to each other.
(D) Electric field due to a uniformly charged spherical shell is zero at all points inside the shell.
(E) Electric field is in the direction in which the potential decreases steepest.

The higher stability of metal carbonyls is very evident on the basis of spectrochemical series. Identify the incorrect statement(s) about metal carbonyls.
A. The metal-carbon bond has both \(\sigma\) and \(\pi\) characters.
B. Metal-carbon \(\sigma\) bond is formed by donation of lone pair of electrons on carbonyl carbon into vacant \(d\)-orbital of metal.
C. Metal-ligand bond has synergic effect.
D. Transition metals never have zero oxidation state in carbonyls.
E. Metal-carbon \(\pi\) bond is formed by donation of electrons from antibonding molecular orbital of CO to metal.
Choose the correct answer from the options given below :

If \(A\) is any square matrix of order 3 and \(B=\left[\begin{array}{ccc}\sin \theta & \cos \theta & 0 \\ -\cos \theta & \sin \theta & 0 \\ 0 & 0 & a\end{array}\right]\), \(a\) is any constant, then \(|A B|\) is equal to:

HIV primarily destroys which cells of the immune system?