The area (in sq. units) of a triangle formed by vertices \(O, A\), and \(B\) where \(\overrightarrow{O A}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\overrightarrow{O B}=-3 \hat{i}-2 \hat{j}+\hat{k}\) is

\(\vec{n}\) is a vector of magnitude \(2 \sqrt{3}\) such that it makes equal angles with coordinate axes. The vector equation of the plane passing through (1, -1, 2) and perpendicular to \(\vec{n}\) is:

Which of the following is not correct about the Compound Annual Growth Rate (CAGR)?

The derivative of \(f(\cot x)\) with respect to \(g(cosec x)\) at \(x=\frac{\pi}{4}\) (where \(\left.f^{\prime}(1)=2, g^{\prime}(\sqrt{2})=4\right)\) is:

The general solution of the differential equation \(\left(x^2-y x^2\right) d y+\left(y^2+x^2 y^2\right) d x=0\) is :

The feasible region of the linear programming problem is represented below :

The constraints of this LPP are

Match List 1 with List 2.

List - 1 (Matrix)	List - 2 (Determinants)
(A) \(\operatorname{Matrix}(A)=\left[\begin{array}{cc}1 & 7 \\ -3 & 5\end{array}\right]\)	(I) 24
(B) \(\operatorname{Matrix}(B)=\left[\begin{array}{cc}-2 & 5 \\ -3 & -3\end{array}\right]\)	(II) 32
(C) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(III) 21
(D) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(IV) 26

Choose the correct answer from the options given below :

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.
Match List I with List II

LIST I	LIST II
A. Bakelite formaldehyde resin	I. Condensation polymers
B. Melamine formaldehyde resin	II. Phenol and formaldehyde
C. Urea formaldehyde resin	III. Melamine and formaldehyde
D. Bakelite \(\&\) Melamine formaldehyde resin	IV. Unbreakable cups and laminated sheets

Choose the correct answer from the options given below:

The value of \(\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(-\frac{\pi}{6}\right)\right]\) is :

In Rutherford's scattering experiment, if impact parameter \( b = 0 \), then the value of the scattering angle will be :

With reference to the microorganisms involved in production, identify the odd combination.

Match List I with List II

LIST I	LIST II
A. One species is benefitted and other harmed.	I. Amensalism
B. One species is benefitted and other neither harmed nor benefitted.	II. Parasitism
C. Both the interacting species are benefitted.	III. Commensalism
D. One species is harmed and other remains unaffected.	IV. Mutualism

Select the conditions(s) in which decomposition does not takes place in
A. In warm and moist environment
B. In presence of decomposers
C. In absence of decomposers
D. In absence of detritivores
E. In presence of complex compounds like lignin and chitin.
Choose the correct answer from the options given below: