The value of \(x\) for which the matrix \(A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{bmatrix}\) is singular is:

The image of the point \((3,-2,1)\) in the plane \(3 x-y+4 z=2\) is:

A container contains 100 liters of milk.
From this container, 10 liters of milk is taken out and replaced by water.
This process is repeated two more times.
The amount of milk left in the container now is :

The integrating factor of the differential equation \(\left(x \log _e x\right) \frac{d y}{d x}+y=2 \log _e x\) is:

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})\) and \(\vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(3 \hat{i}+4 \hat{j}+5 \hat{k})\) is equal to :

The probability distribution of number of heads in two tosses of a coin is:

A can solve \(90 \%\) problems and B can solve \(70 \%\) problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to

Consider the following statements with respect to Probability distribution:
(A) In a given sample space not more than four random variables can be defined
(B) Mean expectation value is a parameter and not a statistic
(C) In a Gaussain distribution if only mean and mode are same it is said to be normally distributed
(D) Probability of 'r' success in 'n' Bernoulli is P(r) = \({ }^n C_r q^r p^{n-r}\)
(E) Variance enables us to study the variability of a random variable from the mean expectation

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When resources are unlimited in a habitat, what type of growth is seen in the population?

If the corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5), then which of the following are correct for the objective function \(Z=4 x+6 y \) ?
(A) The minimum value of the objective function occurs at (0, 2) and (3, 0) only.
(B) The minimum value of the objective function occurs at the mid-point of the line segment joining the points (0, 2) and (3, 0) only.
(C) The minimum value of the objective function occurs at every point of the line segment joining the points (0, 2) and (3,0).
(D) The difference between the maximum value and minimum value of the objective function is 60.
Choose the correct answer from the options given below :

Which enzyme modified by genetic engineering is used as a 'clot buster'?

Match List I with List II

LIST - I	LIST - II
A. Zone refining	I. Titanium
B. Mond's process	II. Zinc
C. Electrolytic refining	III. Nickel
D. van-Arkel method	IV. Germanium

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The area enclosed between \(y^2=4 x\) and \(x^2=4 y\) is: