The value of \(\left|\begin{array}{lll}1 & b & c+a \\ 1 & c & a+b\end{array}\right|\) is

When a slab of insulating material 4 mm thick is introduced between the plates of a parallel plate capacitor of separation 4 mm , it is found that the distance between the plates has to be increased by 3.2 mm to restore the capacity to its original value. The dielectric constant of the material is

In a LPP, let \(R\) be the feasible region.
A. If R is unbounded then a max./min. value of objective function may not exist.
B. If R is bounded then a max. and min. value of objective function will always exist.
C. If a solution exists, it must occur at a corner point.
D. If R is bounded then max. will exist but min. may or may not exist for an objective function.
Choose the correct answer from the options given below:

Consider an LPP: Maximise \(Z=50 x+15 y\) subjected to constraints
\(\begin{array}{l}x+y \leq 60 \\5 x+y \leq 100 \\x, y \geq 0\end{array}\)
If the maximum value of \(Z\) occurs at \(x=a\) and \(y=\beta\), then the value of \(a+\beta\) is :

Which of the following is not correct?

For the linear programming problem,
Minimize (Z) =60x + 30y
subject to : \(2 x - y \geq-5 ; 3 x + y \geq 3 ; 2 x -3 y \leq 12; x , y \geq 0\)
the optimal value of \(z\) is

The area of the triangle with vertices \((1,4),(2,7)\) and \((4,13)\) is:

Read the passage carefully and answer the questions:
The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complex to yield three orbitals of lower energy, \(t_{2 g}\) set and two orbitals of higher energy, \(e_g\) set. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting and the energy separation is denoted by \(\Delta_0\). Thus energy of the two \(e_g\) orbitals will increase by \((3 / 5) \Delta_0\) and that of three \(t_{2 g}\) will decrease by \((2 / 5) \Delta_0\). The crystal field splitting \(\Delta_0\) depends upon the field produced by the ligand and charge on the metal ion. Some ligands are able to produce strong fields, in which case the splitting will be large, whereas others produce weak fields and consequently result in small splitting of \(d\) orbitals. Relative magnitude of crystal field splitting energy \(\Delta_{\circ}\) and pairing energy, \(P\) (energy required for electron pairing in a single orbital) determine the formation of low spin ( \(\Delta_{\circ}>P\) ) or high spin ( \(\Delta_{\circ}<P\) ) complex. In tetrahedral coordination entity formation, the \(d\)-orbital splitting is inverted and is smaller as compared to the octahedral field splitting. The crystal field theory attributes the colour of the complex to \(d-d\) transition of the electron. The colour of the coordination compounds depends on the crystal field splitting. In the absence of ligand, crystal field splitting does not occur and hence the substance is colourless
What will be the crystal field splitting energy for tetrahedral \(\left[ CoCl _4\right]^{2-}\) if its value for octahedral \(\left[ CoCl _6\right]^{4-}\) is \(18,000 cm^{-1}\) ?

If \(A\) and \(B\) are symmetric matrices of order \(3 \times 3\) then the matrix \(2 A B-B A\) is :

Correct increasing order of wavelength of absorption in visible region for complex is
A. \(\left[ Ni ( CN )_6\right]^{4-}\)
B. \(\left[ Ni \left( H _2 O \right)_8\right]^{2+}\)
C. \(\left[ Ni ( CO )_0\right]^{2+}\)
D. \(\left[ Ni \left( NH _3\right)_6\right]^{2+}\)
Choose the correct answer from the options given below:

Arrange these in ascending order of wavelength while propagating through different medium.
(A) Light propagating in air
(B) Light propagating in water
(C) Light propagating in diamond
(D) Light propagating in glass
Choose the correct answer from the options given below :

If A is a square matrix such that \(A^2=A\) and I is the identity matrix of same order as A, then the value of
\((A-2 I)^2-(2 A+I)^2+11 A\) is :

The non-negative remainder when \(7^{30}\) is divided by 5 is :