A die is thrown twice and the sum of the numbers appearing is observed to be 6 .
The probability that the number 4 has appeared at least once is:

If A and B are non-singular square matrices of the same order, then \(\left(A B^{-1}\right)^{-1}=\)

For the given five values 3.6, 4.3, 4.3, 3.4, 4.4, the three years moving averages are:

The direction ratios of the perpendicular from the point \((1,0,1)\) to the plane \(x-y+z=0\) are :

If the matrix \(M=\left[\begin{array}{ccc}0 & -1 & 3 \alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0\end{array}\right]\) is skew-symmetric, then

The angle between the vectors \(\hat{i}-2 \hat{j}+\hat{k}\) and \(3 \hat{i}+2 \hat{j}+\hat{k}\) is:

For the objective function \(Z = 4x + 6y\) subject to the constraints
\(3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0\),
the maximum value of Z occurs at (a, b)
and the minimum value of Z occurs at (p, q)
then the value of \(\frac{a}{p}+\frac{b}{q}\) is:

Match List-I with List-II

List-I	List-II
A. Reactance	(I) LCR Circuit
B. Inductance	(II) \(\frac{\rho l}{\pi r^2}\)
C. Capacitance	(III) \(\frac{\varepsilon_0 A}{d}\)
D. Impedance	(IV) \(\frac{\mu_0 N^2 A}{L}\)

Choose the correct answer from the options given below :

The point which provides the optimal solution of the linear programming problem maximize \(z=21 x+35 y\)
\(\begin{array}{l}3 x+2 y \leq 30 \\4 x+5 y \leq 60 \\x \geq 0, y \geq 0\end{array}\)
has the coordinates

In the context of the Photoelectric effect, identify the statements which support quantum nature of light.
(A) The electric charge of photoelectrons is quantized
(B) The maximum kinetic energy of photoelectrons depends on the frequency of the incident light, but is independent of its intensity
(C) The photoelectric emission is an instantaneous process without any apparent time lag, even when the incident radiation is made exceedingly dim
(D) There is a minimum frequency of light below which no photoelectrons are emitted
Choose the correct answer from the options given below:

Let A and B be square matrices of order 3, \(\left[\left(A-A^T\right)+\left(B-B^T\right)\right]\) then det is equal to

Read the passage carefully and answer the Questions
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1) d, n s, n p\) or \(n s, n p, n d\)orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. 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. 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 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.
The spin only magnetic moment of \(\left[ MnBr _4\right]^{2-}\) is 5.9 BM. Predict the geometry of the complex ion?

Reaction :
Benzene \(\xrightarrow[\text { Anhyd. } AlCl l _{ s } / CuCl ]{ CO _{ Cl } HCl } X\)
\(X\) in above reaction is