If \(A=\left[\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right]\), then the value of \(\operatorname{det}(\operatorname{adj}(2 A))\) is :

Match List - I with List - II regarding the value of the following.

List - I	List - II
(A) \(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)\)	(I) \(\frac{\pi}{2}\)
(B) \(\cos ^{-1}\left(\cos \frac{5 \pi}{4}\right)\)	(II) \(\frac{\pi}{4}\)
(C) \(\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}\)	(III) \(\frac{5 \pi}{6}\)
(D) \(\tan ^{-1} \frac{x}{y}-\tan ^{-1} \frac{x-y}{x+y}\)	(IV) \(\frac{3 \pi}{4}\)

Choose the correct answer from the options given below :

If \(f: R \rightarrow R\) is defined by f(x) = sin x + x then f(f(x)) is:

Which of the following statements are correct ? (A) \(\sin ^{-1}(-x)=-\sin ^{-1} x, x \in[-1,1]\) (B) \(\tan ^{-1}(-x)=-\tan ^{-1} x,-1<x<1\) (C) \(\cos ^{-1}(-x)=\pi-\cos ^{-1} x,-1 \leq x \leq+1\) (D) \(\sec ^{-1}(-x)=\pi-\sec ^{-1} x,|x|>1\) (E) \(\cot ^{-1}(-x)=\pi-\cot ^{-1} x, x \in R\) Choose the correct answer from the options given below :

Out of students in a school, it is known that \(65 \%\) reside in hostel and \(35 \%\) are day scholars. Previous year results report that \(40 \%\) of the students who reside in hostel attain A grade and \(20 \%\) of day scholars attain A grade in their annual examination.
At the end of the year, one student is chosen at random from the class.
Probability of the selected student attaining grade \(A\) is:

\(\int_{0}^{1} (x^3 + 3x^2)e^x dx =\)

The corner points of the feasible region determined by the system of linear constraints are \((0, 0), (0, 40), (20, 40), (60, 20), (60, 0)\). The objective function is \(Z = 4x + 3y\).
Compare the quantity in Column A and Column B
Column A : Column B :
Maximum of Z 325

When an intrinsic semi conductor is doped with an acceptor impurity :
(A) The hole concentration increases
(B) The conductivity of semiconductor decreases
(C) The electron concentration increases
(D) The energy gap decreases
Choose the correct answer from the options given below :

Match List I with List - II.

List - I	List - II
(A) L.C. Oscillation	(I) Inductor supplied with AC
(B) \( Q = \frac{\omega_o L}{R} \)	(II) \( \frac{d^2q}{dt^2} + \frac{1}{LC}q = 0 \)
(C) Voltage leads current	(III) \( \frac{i_p}{i_s} = \frac{V_s}{V_p} \)
(D) Transformer	(IV) Sharpness of resonance curve

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.
Question: How is formalin useful?

Two identical short bar magnets each of magnetic moment \( 12.5 \text{ Am}^2 \) are placed with 10 cm separation between their centers as shown in the figure. The magnitude of magnetic field at O is

Match List - I with List - II.

List - I	List - II
(A) To study crystal structure	(I) IR ray
(B) To study structure of nuclei	(II) X-ray
(C) Take haze photography	(III) UV-ray
(D) Can cause fluorescence	(IV) \( \gamma \)-ray

Choose the correct answer from th options given below :

If \(\frac{d}{d x}\left[a x^3+a x^2+a x+1\right]=9 x^2+6 x+3\), then \(a\) is equal to :