The relation \(R=\left\{(a, b): a \leq b^2\right\}\) on the set of real numbers is:

\(\int\left(e^{x \log a}+e^{a \log x}\right) d x\) is equal to (where \(\left.a>1\right)\)

The rate of change of the area of a circle with respect to its radius r, when r = 3 cm is :

If the binomial distribution \(X \sim B(n, p)\) has mean 3 and variance \(\frac{3}{2},(p+q)=1\), then which of the following is/are TRUE?
(A) \(q=\frac{1}{2}, n=6\)
(B) \(P(X \leq 5)=\frac{63}{64}, p=\frac{1}{2}\)
(C) \(q=\frac{1}{3}, p=\frac{2}{3}\)
(D) \(P(X=4)=\frac{15}{64}, n=6\)
Choose the correct answer from the options given below:

The solution of the differential equation \(x \frac{d y}{d x}+2 y=x^2\) is: ( \(C\) is constant of integration)

\(\int \frac{d x}{9 x^2-16}\) is equal to:

The interval on which the function \(f(x) = 2x^3 + 12x^2 + 18x - 7\) is decreasing, is:

For the given five values 16, 25, 19, 34, 46, the three years moving averages are:

A. Transformers can not be used to step up DC.
B. The frequency of output across the secondary is always equal to the frequency of input across the primary.
C. The output power of a transformer may be increased to a value which is more than the input power.
D. Transformation ratio is equal to turns ratio.
E. In a step up transformer, secondary current is less than primary current.
Choose the correct answer from the options given below:

A 600 pF capacitor is charged by a 150 V battery. The charge on the capacitor is:

Copying genetic information from one strand of DNA into mRNA is:

A moving coil galvanometer has a resistance of 40 \( \Omega \). It gives a full scale deflection with a current of 2 mA. How is to be modified to convert it to a voltmeter of 5 V range?

Read the passage carefully and answer
The crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. Ligands are treated as point charges in case of anions or dipoles in case of neutral molecules. The five d orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands (either anions or the negative ends of dipolar molecules like NH3 and H2O) in a complex, it becomes asymmetrical and the degeneracy of the d orbitals is lifted. It results in splitting of the d orbitals. The pattern of splitting depends upon the nature of the crystal field.
If the fourth electron in d subshell enters one of the eg orbitals in octahedral complexes, it indicates that