Match (List I) with (List II)

(List I)	(List II)
A. For any vector \(\vec{a}, \vec{a} \times \vec{a}\)	\(I\). \(|\vec{a}|^2\)
B. For any vector \(\vec{a}, \vec{a} \cdot \vec{a}\)	\(II.\) \(0\)
C. If either \(\vec{a}=\overrightarrow{0}\) or \(\vec{b}=\overrightarrow{0}\), then \(\vec{a} \cdot \vec{b}\) is	\(III.\) \(|\vec{a}||\vec{b}|\)
D. If the angle between \(\vec{a}\) and \(\vec{b}\) is \(\theta=0\), then \(\vec{a} \cdot \vec{b}\) is	\(IV.\) \(\overrightarrow{0}\)

Choose the correct answer from the options given below:

The integrating factor of the differential equation \(x \frac{d y}{d x}-y=2 x^2\) is :

A man plans to take a housing loan of \(\operatorname{Rs} 99,53,000\) from a bank costing \(18 \%\) per annum compounded monthly. The loan is to be paid back in 30 years in equal monthly installments (EMI). The EMI by reducing balance method is :
[Given \((1.015)^{-360}=0.0047\)]

Curd is at \(80^{\circ} F\),five minutes later it came down at \(60^{\circ} F\).
After another 5 minutes, its temperature became \(50^{\circ} F\).
Given that the rate of change of temperature is proportional to (T - S).
where S is temperature of surroundings and T is temperature of the curd at any time t.
Then the temperature of the surroundings is :

(A) f(x) = |x| is continuous function on R (the set of real numbers)
(B) f(x) = [x] is continuous function on R
(C) f(x) = sin x is continuous function on R
(D) f(x) = tanx is continuous function on R
(E) f(x) = sinx + cosx is continuous function on R

If the matrix \(A = \begin{bmatrix} 0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0 \end{bmatrix}\) is skew-symmetric, then a + b + c is equal to:

The rate of change of area of a circle with respect to its circumference when radius is \(4\) cm , is:

The solution of the differential equation \(\frac{d r}{d t}=-r t, r(0)=r_0\) is

If red light is replaced by blue light in single slit diffraction then

In reference to Bt cotton, select the correct statement(s) :
(A) The alkaline pH of an insect gut solublises the crystals of Bt toxin
(B) The acidic pH of an insect gut solublises the crystals of Bt toxin
(C) The Bt toxin does not kill the tobacco budworm
(D) Specific Bt toxin genes were isolated from Bacillus thuringiensis
Choose the correct answer from the options given below :

The current flowing through a wire as a function of time is given as : \( I = 3t^2 + 2t + 5 \). The charge flowing through any cross-section of the wire in first 2 s is :

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.
Planar structures with \(N - N\) single bond are shown by
A. \(N _2 O _3\)
B. \(N _2 O _4\)
C. \(N _2 O _5\)
D. \(N _2 O\)
E. \(NO\)
Choose the correct answer from the options given below:

Read the passage carefully and answer the questions
Replacement of a hydrogen atom in a hydrocarbon by an alkoxy or carboxyl group yields a class of compounds known as ethers. Ethers are classified as symmetrical or unsymmetrical on the basis of groups attached to the oxygen atoms. Diethyl ether, a symmetrical ether, has been widely used as an inhalation anesthetic. Ethers can be prepared by acid catalyzed intermolecular dehydration of alcohols and Williamson's synthesis. Acid catalyzed dehydration of alcohols is not generally preferred as it gives a mixture of elimination and substitution products. In Williamson's synthesis, an alkyl halide is allowed to react with sodium alkoxide. Ethers containing substituted Alkyl groups may also be prepared by this method. The C-O bond in ether is weakly polar and is cleaved under drastic conditions with excess of hydrogen halides. In electrophilic substitution, the alkoxy group deactivates the aromatic ring and directs the incoming group to ortho and para positions.
Which is most reactive hydrogen halide for cleavage of ethers?