If \(\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=0\), then value of x is

If the area of an equilateral triangle is increasing at the rate of \(4 \sqrt{3} cm^2 / sec\), then the rate of increase of its perimeter when the side is 4 cm , is

If three points \(A \left(a_1, b_1\right), B \left(a_2, b_2\right)\) and \(C \left(a_3, b_3\right)\) are collinear and D is the determinant of their coordinates with a column of 1s, then:

If \(z\)-coordinate of a point P on the line joining the points A (2, 2, 1) and B (5, 1, -2) is -1, than \(x\)-coordinate of point P is

In the formula for Z -test statistic \(Z=\frac{\bar{x}-\mu_0}{\sigma / \sqrt{n}}\), mark the incorrect statement.

Two positive numbers whose sum is 60 and the product of the square of one number and other number is maximum are:

The values of \(\lambda\) for which the system of equation \(x+2 y+z=14,-x+y+z=10, x+\lambda y+z=2\) has unique solution is :

The marginal cost of production of \(X\) units of a commodity is \(56+\frac{3}{2} x\).
It is known that fixed costs are Rs.115.
Then the total cost of producing 50 units is :-

The probability that A speaks the truth is \(\frac{4}{5}\). He throws a die and reports that it is a five. The probability that it is actually a five is:

Read the following passage carefully and answer the given questions.
Insulin consists of two short polypeptide chains- chain A and chain B, that are linked together by disulphide bridges. Some strains of Bacillus thuringiensis produce proteins that kill certain insects such as lepidopterans, coleopterans and dipterans. Bt toxin protein is converted into an active form of toxin due to the alkaline pH of the gut which solubilise the crystals. Transgenic animals that produce useful biological products can be created by the introduction of the portion of DNA (or genes) which codes for a particular product such as human protein (alpha-1-antitrypsin) used to treat emphysema. Plants, bacteria, fungi and animals whose genes have been altered by manipulation are called Genetically Modified Organisms (GMO). GM plants have been useful in many ways. Genetic modification has: (i) made crops more tolerant to abiotic stresses. (ii) reduced reliance on chemical pesticides (pest-resistant crops). (iii) helped to reduce post harvest losses. (iv) increased efficiency of mineral usage by plants (this prevents early exhaustion of fertility of soil). (v) enhanced nutritional value of food.
The beetles that infest Bt cotton plant are killed:

A bridge circuit is shown in the diagram. The equivalent resistance between A and B is

A concave lens and a convex mirror both are dipped in an oil of refractive index 1.2. The focal length of:

The diagonal elements of a skew symmetric matrix are all