Derivative of \(x^x\) with respect to \(x\) \(log\) \(x\) is

Match List I With List II

List I	List II
A. \(\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^{\frac{3}{2}}\)	I. order + degree \(=2\)
B. \(2\left(\frac{d^3 y}{d x^3}\right)^2+3\left(\frac{d^2 y}{d x^2}\right)+y\left(\frac{d y}{d x}\right)^2=e^x\)	II. order + degree \(=3\)
C. \(\frac{d y}{d x}+\frac{1}{d y / d x}=3\)	III. order + degree \(=4\)
D. \(\frac{d y}{d x}+x^2=5\)	IV. order + degree \(=5\)

The solution of \(y^{\prime}-y^{\prime \prime}=2 x\) is:
A. \(y=x^2+2 x+2\)
B. \(y=x^2+2 x+1\)
C. \(y=x+2\)
D. \(y=x^2-2 x+1\)
Choose the correct answer from the options given below :

If \(A\) is a non-singular square matrix of order 3 such that \(A^3=4 A^2\), then the value of \(|A|\) is:

If the elementary column transformation \(C_1 \rightarrow 2 C_1\) is carried out on the following matrix equation.
\(\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}-1 & 2 \\ 5 & 0\end{array}\right]=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) Then the transformed equation will be:

If \(x=t^{1 / 2}, y=t^{3 / 2}\), then \(\frac{d y}{d x}=\)

If the optimal value of the objective function \(z=p x+y\) of an L.P.P occurs at two corner points \((2,11)\) and \((4,5)\) of its bounded feasible region, then its optimal value is

When a dielectric slab is introduced in between the plates of a charged capacitor with battery connected across the plates, following are the observations. Choose which observation is incorrect?

The intensity ratio of two interfering sources in Young's double slit experiment is \(4 : 1\). The ratio of \( \frac{I_{min}}{I_{max}} \) is:

The formation of micelle takes place only :
(A) quad above a particular temperature called kraft temperature
(B) quad above a particular concentration called critical micelle concentration
(C) quad when catalyst is added
Choose the correct answer from the options given below :

Read the following passage carefully and answer the given questions.
Theory of Evolution
The evolution of life is a gradual process through which simple organisms are transformed into complex life forms over billions of years. It involves genetic changes, adaptation to the environment, and the emergence of diversity in species. Fossil records provide evidence of extinct life forms and transitional stages. Evolution is influenced by factors such as mutation, genetic recombination, and environmental pressure, shaping the biodiversity we see today. Charles Darwin and Alfred Russel Wallace were pioneering naturalists who developed the theory of evolution by natural selection.
Which one of the following statements is not a connotation of the theory of special creation?

In a Young's double slit experiment, a light of wavelength 550 nm is used to get fringes on screen 1.2 m away from the slits. The distance between the two slits is 1.5 mm. The fringe width is

If maximum value of \(f(x)=2 x^3+3 x^2-6 a x+10\) occurs at \(x=-3\), then the value of \(a\) is ___________