The corner points of the bounded feasible region for an LPP are (0,4), (4, 4), (6, 6), (0, 12).
If the objective function is \(Z=p x+q y, p>0, q>0\) then the condition on p and q so that the maximum of Z occurs at (6, 6) and (0, 12) is

Consider the region bounded by the lines \(y-1=x, x=-2, x=3\) and \(x\)-axis. Then,
(A) The area of the bounded region is given by \(\int_{-2}^3(x+1) d x\)
(B) The numerical value of the area is \(\frac{15}{2}\) sq. units
(C) The numerical value of the area is 8 sq . units
(D) The numerical value of the area is \(\frac{17}{2}\) sq. units
Choose the correct answer from the options given below:

For the function \(f(x)=x^x, x>0\), which of the following are TRUE?
(A) \(f^{\prime}(x)=x^x(1+\log x)\)
(B) \(x=e\) is the critical point
(C) \(f\) is increasing in \(\left(\frac{1}{e}, \infty\right)\)
(D) \(f\) is increasing in \((0, \infty)\)
Choose the correct answer from the options given below :

The value of \(\lambda\), so that the lines \(\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}\) are perpendicular is:

The mean number of heads in two tosses of a coin is:

The value of the integral \(\int\left(x^6+x^4\right) d\left(x^2\right)\) is equal to:

If \(y =\sin ^{-1} x\), then

In the case of co-evolved plant pollinator mutualism where one of the species got extinct, what would be the impact on the other species?

If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\), then the value of \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) is:

Read the following passage carefully.
Restriction enzyme
Restriction enzymes cut the strand of DNA a little away from the center of the palindrome sites, but between the same two bases on the opposite strands. This leaves single stranded portions at the ends. There are overhanging stretches called sticky ends on each strand. These are named so because they form hydrogen bonds with their complementary cut counterparts. This stickiness of the ends facilitates the action of the enzyme DNA ligase. The word palindromes refer to group of letters that forms the same words when read both forward and backward. Restriction endonucleases are used in genetic engineering as molecular scissors.
Which one of the following option is correct about restriction enzyme?

For the function, \(f(x)=\frac{-3}{4} x^4-8 x^3-\frac{45}{2} x^2-350\), which of the following statements are correct?
(A) \(x=-3\) and \(x=-5\) are the only critical points of the given function.
\((B) x=-3\) is a point of local minimum.
(C) The local minimum value at \(x =-3\) is 231.
\((D) x=-5\) is a point of local maximum.
Choose the correct answer from the options given below:

The number of all possible matrices of order 2 × 2 with each entry 0, 1 or 2 are:

The value of \(k\) for which the following system of equations does not possess a unique solution is:
\(\begin{array}{l}k x+3 y-z=1 \\ x+2 y+z=2 \\ -k x+y+2 z=-1\end{array}\)