Consider the Linear Programming Problem
Maximize \(z=x+y\)
Subject to the constraints \(x-y \leq-1, x \geq y, x \geq 0, y \geq 0\)
Then which one of the following is TRUE?

The side of an equilateral triangle increases at the rate of \(3 cm / sec\), then the rate at which the area of the triangle increases when the side is 4 cm , is :

The value of \(\int_0^1 \log _e\left(\frac{1}{x}-1\right) d x\) is :

If \(f(x) = sin\) \(x - cos\) \(x\), \(x \in[0,2 \pi]\) then
(A) f(x) is increasing in \(\left(0, \frac{3 \pi}{4}\right)\)
(B) f(x) is decreasing in \(\left(0, \frac{3 \pi}{4}\right)\)
(C) f(x) is decreasing in \(\left(\frac{3 \pi}{4}, \frac{7 \pi}{4}\right)\)
(D) f(x) is decreasing in \(\left(\frac{7 \pi}{4}, 2 \pi\right)\)
Choose the correct answer from the options given below:

The value of the determinant \(\left|\begin{array}{lll}a^2 & a & 1 \\ b^2 & b & 1 \\ c^2 & c & 1\end{array}\right|\) is:

If \(A=\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]\) and \(I\) is an identity matrix of order \(2\) , then \(A-3 I\) equals :

Match List I with List II

LIST I	LIST II
A. \(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x} + \sqrt{x}} dx\)	I. -\(\frac{1}{2}\log 3\)
B. \(\int_{1}^{2} x^2 dx\)	II. \(\frac{7}{3}\)
C. \(\int_{-4}^{-1} \frac{1}{x} dx\)	III. \(\frac{1}{2}\)
D. \(\int_{0}^{1} \frac{1}{2x-3} dx\)	IV. -\(\log 4\)

Choose the correct answer from the options given below:

Two pipes P and Q can fill a tank in 26 minutes and 52 minutes respectively.
Both the pipes are opened together for some time and then pipe P is closed.
If the tank is filled in 26 minutes, then after how many minutes pipe P is closed?

In corrosion :
(A) A metal is oxidised by loss of electrons to oxygen
(B) It is an Electrochemical phenomenon
(C) The object under corrosion behaves as anode
(D) Coating the metal with Zn can save iron object from corrosion
Choose the correct answer from the options given below :

Industrial melanism in peppered moths is an example of:

The maximum value of the objective function \(Z=2 x+y\) of an LPP, subject to the constraints \(x \leq 6, y \leq 2, x-y \leq 0\), \(x \geq 0, y \geq 0\), is:

The frequency of revolution of an electron revolving in a circular path in a magnetic field of \( 7.8 \times 10^{-4} \, \text{T} \) will be (Take \( e = -1.6 \times 10^{-19} \, \text{C} \), \( m_e = 9.1 \times 10^{-31} \, \text{kg} \), \( \pi = 22/7 \))

Question based on the following passage :
P-Block elements are placed in group 13 to 18 in the periodic table. Group 15 is nitrogen family.
Nitrogen can react with hydrogen, oxygen, halogens etc. It shows multiple oxidation states in its various oxides i.e. \(N_2O, N_2O_3, N_2O_5\) etc. Important allotropic forms of phosphorus are white, red and black phosporus. Various compounds of phosphorus are phosphine, \(PCl_3, PCl_5\) and oxoacids of phosphorus. Oxidation states \(+1\) to \(+5\) are seen in various oxoacids of phosphorus. Group 17 elements are collectively known as halogens. Bond dissociation enthalpy of \(F_2\) is smaller than \(Cl_2\).
The gases present in group 18 are chemically unreactive, they form only few compounds. Neil Bartlett (in 1962) observed the reaction of a noble gas. Since then, many noble gas compounds have been synthesised.
The bond angle in \(PH_3\) is less than \(PH_4^+\) due to :