The inverse of the matrix \(A=\left[\begin{array}{ccc}\frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{8}\end{array}\right]\) is

If the matrix \(A = \begin{bmatrix} 0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0 \end{bmatrix}\) is skew-symmetric, then a + b + c is equal to:

The shortest distance between the lines \(l_1\) and \(l_2\) given by
\(l_1: \vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})\) and \(l_2: \vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+6 \hat{k})\) is :

If the matrix \(\left[\begin{array}{ll}1 & 2 \\ 3 & x\end{array}\right]\) is a singular matrix, then the value of \(x\) is :

The function \(f(x)=x^2 e^{-2 x}\) increases on

The rise and fall of share market is an example of

\(\text { If } I=\int \frac{x}{x-\sqrt{x^2-4}} d x=\alpha x^3+\beta\left(x^2-4\right)^{\frac{3}{2}}+\gamma\) where \(\gamma\) is a constant of integration, then

Read the passage carefully and answer the questions
The chemical reactions proceed at different rates which depend on various factors like concentration, temperature, etc. Rate law is an expression which relates the rate of reaction with concentration of various reacting species in the reaction. For a first order reaction, the concentration of the reactant after time \(t\) is related with its initial concentration by the relation, \(C=C_0 e^{-k t}\). The rate constants for different orders have different expressions. For a first order reaction, \(k=\frac{2.303}{t} \log \frac{a}{a-x}\), where \(a\) is the initial concentration and \(x\) is the extent of reaction.
Which plot is linear with negative slope for a first order reaction

Arrange the following steps of retrovirus replication sequentially.
A. Production of viral DNA by reverse transcriptase
B. Introduction of viral RNA in host cell
C. Production of new viral RNA by host cell
D. Incorporation of viral DNA into host genome
Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by the curve \(y=x^5\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is equal to:

Read the passage given below and answer the question.
Phenols and Alcohol have the same function group (-OH) but phenols are stronger acids than alcohol due the resonance stabilization of phenoxide ion than alkoxide ion (-RO\(^{-}\)). Electron withdrawing groups tend to increase the acidic strength while electron releasing groups tend to decrease it.
Compound 'A' on reaction with Conc. HCl and anhydrous \(ZnCl_2\), gives turbidity immediately.
Compound 'A' is :

In agarose gel electrophoresis, the separated DNA fragments can be visualised by staining the DNA with a compound, known as

The degree of the differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3=\left(\frac{d^3 y}{d x^3}\right)^2\) is :