The matrix A \(=\left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 5\end{array}\right]\) is :

Let \(\vec{a}=3 \hat{i}+\hat{j}-4 \hat{k}\) and \(\vec{b}=6 \hat{i}+5 \hat{j}-2 \hat{k}\) be two vectors.
Then a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) with magnitude 3 units is :

If \(A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\), and \(A^2=8 A+K I\) the value of \(K\) is :

Match List I with List II

LIST-I	LIST-II
A. \(\left[1 + \left(\frac{dy}{dx}\right)^2\right] = \frac{d^2y}{dx^2}\)	I. order 2, degree 3
B. \(\left(\frac{d^3y}{dx^3}\right)^2 - 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^4\)	II. order 2, degree 1
C. \(\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^3 = 0\)	III. order 1, degree 2
D. \(\left(\frac{dy}{dx}\right)^2 + 6y^3 = 0\)	IV. order 3, degree 2

Choose the correct answer from the options given below:

For \(x \neq-1\), if \(\int \frac{x e^x d x}{(1+x)^2}=\frac{a e^x}{(1+x)^b}+c\), where \(a, b\) are fixed numbers and \(c\) is the integration constant, then \(a+b\) is equal to

Let f : R → R be defined as f(x) = [x], where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct?
(A) f is one-one but not onto
(B) f is not onto
(C) f is not one-one
(D) f is one-one and onto
Choose the correct answer from the options given below:

Consider the following hypothesis test.
\(H _0: \mu \leq 12\)
\(H _a: \mu>12\)
If a sample of 25 is taken with sample mean 15 and a sample standard deviation of 6, then the value of t-test statistic is:

The conductivity of 0.001 mol \(\text{L}^{-1}\) acetic acid is \(5.01 \times 10^{-5} \text{ S cm}^{-1}\).
Calculate its dissociation constant if \(\Lambda_m^{\circ}\) for acetic acid is
390.5 \(\text{S cm}^2 \text{mol}^{-1}\).

The following experiment suggests that light propagates as waves

If the lines \(\frac{1-x}{3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{6-z}{5}\) are perpendicular, then the value of \(\lambda\) is:

In a Young's double-slit experiment, the slits are separated by 0.56 mm and the screen is placed 2.8 m away from the slits. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.

The electric field in a region is \( \vec{E} = (6\hat{i} + 3\hat{j} + 5\hat{k}) \text{ N/C} \).
The electric flux due to this electric field through an area of \( 100 \text{ cm}^2 \) lying in the X-Y plane is :

Read the passage carefully and answer the questions based on the passage:
Species diversity on earth is not uniformly distributed. It is generally highest in the tropics and decreases towards the poles. Earth's fossil history reveals the incidences of mass extinctions in the past. Earth's rich biodiversity is vital for the very survival of mankind. It is believed that communities with high diversity tend to be less variable and more productive. The reasons of conserving biodiversity are narrowly utilitarian, broadly utilitarian and ethical. Biodiversity conservation may be in-situ as well as ex-situ.
Which one of the following does not exhibit narrowly utilitarian argument for conserving biodiversity?