A random variable X has the following probability distribution :

x	1	2	3	4	5
\(P ( X = x )\)	k	k	3k	\(2 k^2\)	\(4 k^2\)

Based on the above information, the value of ' \(K\) ' is :

For the given linear programming problem \(z=a x+b y, a, b>0\) subject to the constraint
\(2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0\). If the corner points are \((0,0),(5,0),(3,4)\) and \((0,5)\) and \(z\) is maximum at both \((3,4)\) and \((0,5)\), then the relationship between \(a\) and \(b\) is

The equation of tangent to the curve \(y=\sqrt{3 x-2}\) which is parallel to the line \(4 x-2 y+5=0\) is :

The area enclosed between the curves \(y=x^2\) and \(x=y^2\) is

If the given function \(f(x)\), defined as
\(f(x)=\left\{\begin{array}{ll} 5, & x \leq 2 \\ a x+b, & 2 < x<10 \\ 21, & x \geq 10 \end{array}\right.\)
is continuous, then value of \(2 a+b\) is:

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

The principal value of \(\operatorname{cosec}^{-1} \left(\frac{2}{\sqrt{3}}\right)\) is:

Read the passage carefully to answer the questions given below
Some strains of `Bacillus thuringiensis` produce proteins that kill certain insects such as lepidopterans (tobacco budworm, armyworm), coleopterans (beetles) and dipterans (flies, mosquitoes). In `Bacillus thuringiensis` the toxin is coded by a gene `crylAc` named cry. There are a number of them, for example, the proteins encoded by the genes `crylAc` and `cryllAb` control the cotton bollworms, that of `crylAb` controls corn borer. Using conventional methods of diagnosis (serum and urine analysis, etc.) early detection is not possible. Recombinant DNA technology, Polymerase Chain Reaction (PCR) and Enzyme Linked Immuno-sorbent Assay (ELISA) are some of the techniques that serve the purpose of early diagnosis. The first clinical gene therapy was given in 1990 to a 4-year old girl with adenosine deaminase (ADA) deficiency. This enzyme is crucial for the immune system to function. The disorder is caused due to the deletion of the gene for adenosine deaminase. `RNAi` takes place in all eukaryotic organisms as a method of cellular defense. This method involves silencing of a specific `mRNA` due to a complementary `dsRNA` molecule that binds to and prevents translation of the `mRNA`.
Which of the following Bt toxin gene controls the corn borer?

Benzene ring in phenol is expected to give the characteristic substitution or replacement reactions i.e. halogenation, nitration, sulphonation, alkylation etc. In these reactions, the attacking reagent is electrophile. The phenolic group activates the ortho and para positions in the ring due to conjugation. Disubstitution reactions are quite fast as the electron density in the ring has increased considerably. Phenols do not respond to nucleophilic substitution reactions as it is influenced by phenolic group.
Secondary alcohol among the following is:

(A)

(B) \(H _2 C = CH _2- CH _2 OH\)

(C) \(H _3 C - CH _2- CH _2 OH\)
(D)
(E)

Choose the correct answer from the options given below:

A letter is known to have come from either TATAPUR or from CHAKRATA. On the envelope, only two letters "TA" are visible consecutively. The probability that the letter has come from CHAKRATA is :

Read the paragraph given below and answer the question. Insulin used for diabetes was earlier extracted from slaughtered cattle and pigs. Insulin from an animal source, though caused some patients to develop allergy or other type of reactions. Insulin consists of two short polypeptide chains: chain A and chain B, that are linked bridges. In mammals, including human, insulin is synthesised as a pro-hormone. Patient with diabetes have:

The \(\Lambda_m^{\circ}\) for \(CaSO _4\) at 298 K , if \(\lambda_{ Ca ^{+2}}^{\circ}=119.0 Scm ^2 mol^{-1}\) and \(\lambda_{ SO _4^{2-}}^{\circ}=160.0 Scm ^2 mol^{-1}\) will be:

The value of the determinant \(\left|\begin{array}{ccc}85 & 15 & 30 \\ 1 & 3 & 4 \\ 17 & 3 & 6\end{array}\right|\) is: