If \(c_{i j}\) denotes the cofactor of element \(c_{i j}\) of the matrix \(A=\left[\begin{array}{ccc}1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3\end{array}\right]\), then the value of \(c_{21} \cdot c_{33}\) is

If \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k},|\vec{b}|=5\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\), then the area of the triangle formed by these two vectors as two sides is :

If \(\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|\), then x is equal to :

Let \(\vec{a}=4 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+\hat{j}-2 \hat{k}\). Then
(A) \(\vec{a}\) is a unit vector
(B) \(\vec{a} \times \vec{b}=-\hat{i}+2 \hat{j}+2 \hat{k}\)
(C) \(\vec{a}\) and \(\vec{b}\) are parallel vectors
(D) \(\vec{a}\) and \(\vec{b}\) are neither parallel nor perpendicular vectors
Choose the \(\backslash\) textbf\{correct\} answer from the options given below :

Let A be a square matrix of order 3, then \(|x A|\) is equal to:

Let \(\vec{a}, \vec{b}\) be two vectors such that \(|\vec{a}|=2,|\vec{b}|=3, \vec{a} \cdot \vec{b}=4\). Then \(|\vec{a}-\vec{b}|\) is equal to :

The value of \(\int_{-3}^2 x^2|2 x| d x\) 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 of the following hot spots does not cover India's biodiversity regions?

Identify the basic features of the photon picture of electromagnetic radiation from the following:
(A) Each photon has energy \(E = h\nu\) in vacuum.
(B) Photons are electrically neutral and are not deflected by electric and magnetic fields.
(C) All photons of light of a particular frequency \(\nu\) or wavelength \(\lambda\) have the same energy and momentum.
(D) If the intensity of light of a given wavelength is increased, there is an increase in the number of photons incident on a given area in a given time.
Choose the correct answer from the options given below:

The set of values of \(x\) satisfying \(26 \equiv 5(\bmod x)\) is __________ .

Two cells with internal resistances \( r_1 \) and \( r_2 \) are connected across the load resistance R as shown in the figure. The values of \( r_1, r_2 \) and R are (1/2) \( \Omega \), (1/3) \( \Omega \) and (9/5) \( \Omega \), respectively. The current through R is:

Match List - I with List - II.

List – I	List – II
(A) Ethanol and Diethyl ether	(I) Lucas reagent
(B) Methanol and Ethanol	(II) Sodium metal
(C) n-Butyl alcohol and tert. Butyl alcohol	(III) Iodoform test
(D) Phenol and Cyclohexanol	(IV) Ferric Chloride

Choose the correct answer from the options given below :

Which of the following material takes longer time for decomposition?