If \(\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+\hat{j}-2 \hat{k}, \vec{c}=\hat{i}+3 \hat{j}-\hat{k}\), such that \(\vec{a}\) is perpendicular to \((\lambda \vec{b}+\vec{c})\), then the value of \(\lambda\) is :

The area (in sq.units) of the region bounded by the curves \(y = x\) and y = \(x^3\) is:

Match List-l with List-ll
Let \(A\) be any invertible square matrix. Then

List-l	List-ll
(A) \(A-A^T\)	(I) \(|A| A^{-1}\)
(B) \(A A^T\)	(II) Skew-symmetric
(C) \(\operatorname{det}\left(A^{-1}\right)\)	(III) Symmetric
(D) \(\operatorname{adj} A\)	(IV) \([\operatorname{det}(A)]^{-1}\)

Choose the correct answer from the options given below :

The interval in which the function given by \(f(x) = \sin^4 x + \cos^4 x\), x \(\in\) [0, \(\frac{\pi}{2}\)] is decreasing is:

A matrix which is symmetric as well as skew symmetric is:

In the adjoining figure, the inequality representing AB is :

Match List - I with List - II. Evaluate the integrals.

\(List - I\)	\(List - II\)
(A) \(\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\)	\((I)\) \(2 \pi\)
(B) \(\int_0^{1 / 2} \frac{d x}{\sqrt{x-x^2}}\)	\((II)\) \(\frac{\pi}{4}\)
(C) \(\int_{-\pi}^\pi x \sin x d x\)	\((III)\) \(0\)
(D) \(\int_0^1 \frac{1}{1+x^2} d x\)	\((IV)\) \(\frac{\pi}{2}\)

A line passes through two points A(2, -3, -1) and B(8, -1, 2). The co-ordinates of a point on this line at a distance of 14 units from A are:
Α. (14, 1, 5)
Β. (86, 25, 41)
C. (-10, -7, -7)
D. (-82, -31, -43)
E. (10, 7, 7)
Choose the correct answer from the options given below:

The first diffraction minima due to a single slit diffraction is at \(\theta=30^{\circ}\) for a light of wavelength \(5000 \AA\). The width of the slit is

Answer the question on the basis of passage given below :
Formaldehyde is the first member of the aldehydic series. It is present in green leaves of plants where its presence is supposed to be due to the reaction of \(CO_2\) with water in the presence of sunlight and chlorophyll. Commercially, formaldehyde is obtained by the controlled oxidation of methane in the presence of molybdenum oxide (\(MoO\)) acting as a catalyst. Formaldehyde is a constituent of a number of polymers also called resins. These are non volatile solids or semi solids which are obtained directly from certain plants and can be synthesized in the laboratory.
What is formalin ?

The unit of rate constant for first order reaction is:

How much energy will be carried out by electromagnetic radiation of wavelength \( 3 \times 10^{-6} \text{ m} \)?

Identify the examples of evolution caused due to anthropogenic actions:
(A) Bacillus thuringiensis and Lepidopterans
(B) Industrial melanism in moths
(C) Resistant microbes against antibiotics
(D) Pesticides and insects
Choose the correct answer for the options given below: