If \(a, b\) and \(c\) are all different from zero and \(\left|\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right|=0\), then the value of \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\) is :

The effective rate that is equivalent to a nominal rate of 16 % compounded semi-annually is :

If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right], a \neq 0\) then \(|\operatorname{adj} A|\) is equal to

If \(\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 7 & 3\end{array}\right|\) then the value of x is:

For what value of k, the following system have a unique solution? (where R is set of real numbers)
x + y + z = 1
2x + 3y + 4z = 3
x - y + kz = 5

The value of the integral \(I=\int_0^1 \frac{1}{\sqrt{x^2+2 x+3}} d x\) is :

The probability distribution of a random variable \(X\) is given below :

x	0	1	2	3
\(P ( x )\)	\(\frac{k}{3}\)	\(\frac{k}{2}\)	\(\frac{k}{4}\)	\(\frac{k}{7}\)

Then the value of \(k\) is

The electric field and magnetic field of a plane electromagnetic wave propagating in vacuum are expressed as
\(E_x=E_0 \sin (k z-\omega t)\)
\(B_y=B_0 \sin (k z-\omega t)\)
Which of the following statements are correct?
(A). The wave is propagating along the z- direction.
(B). The electric field Ex and magnetic field By are perpendicular to each other and perpendicular to the z direction.
(C) The electric and magnetic fields Ex and By are perpendicular to each other, and parallel to the direction of propagation.
(D) k is related to the wave length À of the wave by the usual equation \(k = A / 2 Tt\)
Choose the correct answer from the options given below:

Read the passage carefully and answer the Questions
\(KMnO _4\) is prepared by the fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidizing agent like \(KNO _3\) to give a dark-green manganate ion which disproportionate to give permanganate as follows.
\(\begin{array}{l}2 MnO _2+4 KOH + O _2 \rightarrow 2 K_2 MnO _4+2 H _2 O \\ 3 K_2 MnO _4+4 H ^{+} \rightarrow 2 KMnO _4+ MnO _2+2 H _2 O \end{array}\)
On heating \(K _2 MnO _4\) decomposes at 513 K to give \(K _2 MnO _4\). Permanganate ion is tetrahedral and diamagnetic
Acidified \(KMnO _4\) acts a strong oxidizing agent which oxidizes oxalic acid, ferrous ions, nitrite ion and iodides.
Reaction of iodide ion \(\left( I ^{-}\right)\)with \(KMnO _4\) will.....

Which of the following is/are responsible for the formation of Rainbow ?
(A) Dispersion of light
(B) Interference of light
(C) Reflection of light
(D) Refraction of light
Choose the correct answer from the options given below :

An electric dipole consists of charges \( \pm 2 \times 10^{-6} \, \text{C} \), separated by a distance of 2 mm. It is placed on the right side near a long line charge of density \( 4 \times 10^{-4} \, \text{C m}^{-1} \). The negative charge of dipole is at a distance of 2 cm from the line charge as shown in figure. A positive charge (+2q) is placed on the left at 2 cm at point C. Given \( q = 2 \times 10^{-8} \, \text{C} \).

(A) Force \( |F_A| \) exerted on negative charge at point A due to line charge
(B) Force \( |F_B| \) exerted on positive charge at point B due to line charge
(C) Net force on the dipole \( |F_{net}| \) due to line charge
(D) Force \( |F_C| \) exerted on charge +2q at point C due to line charge
If the forces \( |F_A|, |F_B|, |F_C|, |F_{net}| \) are arranged in decreasing order, then:
Choose the correct answer from the options given below:

A radioactive isotope phosphorous-32 has an initial activity \( 10 \mu \text{Ci} \) and has half life of \( 14 \text{ days} \). What is the activity of the source after \( 42 \text{ days} \) ?

The order of the differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^2=\left(\frac{d^3 y}{d x^3}\right)\) is: