If \(z=3 x+4 y\) be the objective function of a linear programming problem (LPP) and (3,1), (2,4), (0,4), (5,0) be corner points of the bounded feasible region. Then the maximum value of objective function is

The equation of the normal to the curve \(y=4 x^3+2 \sin x\) at \((0,3)\) is :

The point estimate of the population standard deviation as per the below mentioned data from a simple random sample 6, 10, 15, 12, 9, 8 will be :-

Match List-I with List-II

List-I (Inverse Trigonometric Function)	List-II (Principal Value)
(A) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)	(I) \(\pi / 6\)
(B) \(\cos ^{-1}\left(-\frac{1}{2}\right)\)	(II) \(-\pi / 6\)
(C) \(\tan ^{-1}(-\sqrt{3})\)	(III) \(2 \pi / 3\)
(D) \(\cot ^{-1}(\sqrt{3})\)	(IV) \(-\pi / 3\)

Choose the correct answer from the options given below :

If \(\left|\begin{array}{ccc}-1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1\end{array}\right|\) is a singular matrix, then the relation between a and b is:

\(\int_{\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\tan x}{\tan x+\cot x} d x\) is equal to:

If \(A=\left[a_{i j}\right]_{2 \times 2}\) where \(a_{i j}=\left\{\begin{array}{ll}1, & i \neq j \\ 0, & i=j\end{array}\right.\) and \(I\) is the identity matrix of order 2 , then \(\left(A^2-3 A+4 I\right)\) is:
(A) Symmetric Matrix
(B) Skew-symmetric Matrix
(C) Non-singular Matrix
(D) Square Matrix
Choose the correct answer from the options given below:

Which of the following is not the property of \( \beta \) ray?

For the given values 27, 35, 42, 45, 51, 34, 43; the five yearly moving averages are:

The volume of a cube is increasing at the rate of \(27 cm^3 / s\). How fast is the surface area increasing when the length of the cube is 12 cm ?

Read the passage carefully and answer the questions based on the passage:
Alcohols are versatile compounds as they can act as both nucleophile and electrophile. They undergo reactions based on cleavage of \(\mathrm{O}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{O}\) bonds. The cleavage of \(\mathrm{O}-\mathrm{H}\) bond reflects the acidic character of alcohols. However they are weaker acids than water, phenol and carboxylic acids. Alkoxide ion formed by loss of proton from the OH group of alcohols reacts with water to form the corresponding conjugate acid - base pair. They act as Bronsted base due to presence of unshared electron pairs on oxygen.Both alcohols and phenol react with carboxylic acids, acid chlorides and anhydrides to form esters. The introduction of acetyl group in alcohols or phenol is known as acetylation. Alcohols react with hydrogen halides to form alkyl halides which form the basis of Lucas test. They also undergo acid catalyzed dehydration to form alkenes. Alcohols can be easily oxidized with variety of oxidizing agents. The nature of the product depends on the type of oxidizing agent and the nature of alcohol.
Oxidation of alcohols involves formation of carbon oxygen double bond with

The reagent involved in Lucas test is

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 \(z=4 x+3 y\).
Compare the quantity in Column - A and Column - B.

Column - A	Column - B
Maximum value of z	350