The maximum value of \(z=2 x+3 y\) subject to the constraints \(3 x-3 y \geq 0,2 x+2 y \leq 12, x \geq 0, y \geq 0\) occurs at the point:

Let \(A, B, C, D\) and \(E\) be matrices of \(2 \times n, 3 \times k, 2 \times p\),
\(2 \times p\) and \(3 \times 3\) respectively.
Choose the correct statement(s) from the following :
(A) \(E B+D B\) will be defined if \(k=3, p=n\).
(B) \(E(3 \times 3) \cdot B(3 \times k) \rightarrow(3 \times k) \rightarrow n \times 3\).
(C) If \(n=p=2\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 2\).
(D) If \(n=p\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 3\).
Choose the correct answer from the options given below :

The radius of a spherical ball is increasing at the rate of 1 m/sec. At the radius equal to 3m, the volume of the ball is increasing at the rate given by :

Match List I with List II

List - I	List - II
(A) \(\int x^2 e^{x^3} d x\)	(I) \(\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C\)
(B) \(\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x\)	(II) \(\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C\)
(C) \(\int \frac{d x}{a^2-x^2}\)	(III) \(\frac{1}{3} e^{x^3}+C\)
(D) \(\int \frac{d x}{x^2+a^2}\)	(IV) \(e^x \tan ^{-1} x+C\)

For the given \(5\) values, \(15, 18, 21, 27, 39;\) the three year moving averages are:

Derivative of \(\sin \left(\cos \left(x^2\right)\right)\) with respect to \(x^2\) is :

The value of \(\tan ^2\left(\sec ^{-1} 2\right)\) is:

The number of years required for a sum of money to get tripple at the effective rate of \(4\%\) is : (Given \((1.04)^{28}=3\) )

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:

The magnetic flux through a coil perpendicular to the plane is varying according to the relation \( \phi = (7t^3 + 3t^2 - 2t - 2) \, \text{Wb} \). If the resistance of the coil is \( 10\Omega \), the induced current through the coil at t = 2 s, will be

Number of defective bulbs in a lot of 500 bulbs follows a binomial distribution with probability of a randomly selected bulb to be defective equal to 0.3.
A sample of 50 bulbs is drawn. Probability of 2 defective bulbs in the sample is:

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
In polar C-O bond in carbonyl compounds, the nature of carbon and oxygen atom is

If the money is worth \(8 \%\) per annum compounded semi-annually, then the present value of a sequence of payments of ₹ 1,000 made at the end of every 6 months and continuing forever, is: