Let \(x\) denote the number of doublets in three throws of a pair of dice with the following probability distribution.

\(x\)	\(0\)	1	2	3
\(P(x)\)	\(\frac{25}{72} k\)	\(\frac{15}{72} k\)	\(\frac{3}{72} k\)	\(\frac{1}{360} k\)

If value of \(k\) is equal to \(\frac{m}{n}, \operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to

Match List - I with List - II.

LIST-I ((Function)y)	LIST-II (\((\) Derivative \() \frac{d y}{d x}\))
(A) \(y=\cos \sqrt{x}\)	(I) \(\frac{\sec (\tan \sqrt{x}) \tan (\tan \sqrt{x}) \sec ^2 \sqrt{x}}{2 \sqrt{x}}\)
(B) \(y=\sqrt{\sin x}\)	(II) \(-3 x^2 \cos \left(x^3\right) \sin \left(\sin \left(x^3\right)\right)\)
(C) \(y=\sec (\tan \sqrt{x})\)	(III) \(\frac{\cos x}{2 \sqrt{\sin x}}\)
(D) \(y=\cos \left(\sin \left(x^3\right)\right)\)	(IV) \(-\frac{1}{2 \sqrt{x}} \sin \sqrt{x}\)

Choose the correct answer from the options given below:

The number of arbitrary constants in general solution of a differential equation of third order is :

The water and milk in two vessels are in the ratio 1:1 and 3: 8 respectively. In what ratio, the mixtures in the vessels be mixed to obtain a new mixture containing water and milk in the ratio 4 : 7?

The slope of the normal to the curve \(x^2+3 y+y^2=5\) at \((1,1)\) is :

If \(x=a \sin 2 t(1+\cos 2 t)\) and \(y=b \cos 2 t(1-\cos 2 t)\), then \(\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{4}}\) is equal to

If it is 7:00 pm currently in the clock, what will the clock show (in am or pm) after 674 hours?

The shortest wavelength in the Balmer series of a hydrogen atom is (Given: \(R=1.1 \times 10^7 m^{-1}\) )

A 1.0 m long metallic rod is rotated with an angular frequency of \( 500 \, \text{rad s}^{-1} \) with one end hinged at the center and the other end at the circumference of a circular metallic ring of radius 1 m about an axis passing through the center and perpendicular to the plane of the ring. A constant and uniform magnetic field of 0.6 T parallel to the axis exists everywhere. The emf developed between the center and the ring is

The magnetic energy stored in a current carrying solenoid is (if \(B=\) magnetic field, \(1=\) length of solenoid, \(A=\) area of cross section of solenoid) :

Answer the question on the basis of passage given below :
Nitrogen differs from rest of the members of group 15 due to its small size, high electronegativity, high ionization enthalpy and non-availability of d orbitals. Nitrogen has unique ability to form pa - pa multiple bonds with itself and with other elements having small size and high electronegativity. Heavier elements of this group do not form pa - pr bonds as their atomic orbitals are so large and diffuse that they cannot have effective overlapping.
Group 15 elements forms oxides of type \(E _2 O _3\). Acidic oxides are formed by
A. Bi
B. Sb
C. P
D. As
E. N
Choose the correct answer from the options given below:

If \(A =\left(\begin{array}{cc}-a & b \\ c & a\end{array}\right)\) and \(A ^2= I\), where \(I\) is an Identity matrix of order 2, then which of the following is correct?

If \(\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a\end{array}\right|=86\), then product of all values of \(a\) is :