The intervals in which the function \(f(x) = 2x^3 + 9x^2 + 12x + 20\) is increasing is given by:

"The probability distribution of a random variable \(x\) is :
\(P(X=x)=\left\{\begin{array}{ll}k x^2, & \text { for } x=1,2,3 \\2 k x, & \text { for } x=4,5,6 \text { where } k \text { is a constant. } \\0, & \text { otherwise }\end{array}\right.\)
Match List-I with List-II

List I	List II
(A) k	(I) 7/22
(B) \(P(X \geq 4)\)	(II) 1/44
(C) \(P(X<4)\)	(III) 95/22
(D) E[X]	(IV) 15/22

Choose the correct answer from the options given below:

\(\int(x-1) e^{-x} d x=\)

Match List-I with List-II

List-I (Definite integral)	List-II (Value)
(A) \(\int_0^1 \frac{2 x}{1+x^2} d x\)	(I) 2
(B) \(\int_{-1}^1 \sin ^3 x \cos ^4 x d x\)	(II) \(\log _e\left(\frac{3}{2}\right)\)
(C) \(\int_0^\pi \sin x d x\)	(III) \(\log _e 2\)
(D) \(\int_2^3 \frac{2}{x^2-1} d x\)	(IV) 0

Choose the correct answer from the options given below:

The function f defined by \((x)=\left\{\begin{array}{ll}a x+1, & x \leq 3 \\ b x+3, & x>3\end{array}\right.\) is continuous at x = 3 if

The area of the region bounded by the curve \(x=y^2\), y-axis and the line y = 3 and y = 4 is:

If the objective function \(Z=p x+q y, p>0, q>0\) of a linear programming problem attains its optimal value at the points \((4,7)\) and \((5,5)\) and \(pq =50\) then:

Find the value of \(\text{E}^{\circ}\) in the missing link in the following diagram.
\(Fe ^{3+} \xrightarrow{E_1^{\circ}=0.77 V} Fe ^{2+} \xrightarrow{E_2^{\circ}=-0.44 V} Fe\)
\(Fe ^{3+} \xrightarrow{E_3^{\circ}=?} Fe\)

Match List I with List II.

LIST I	LIST II
A. Down's Syndrome	I. Absence of a copy of X chromosome
B. Klinefelter's Syndrome	II. Presence of additional copy of X chromosome
C. Turner's Syndrome	III. Mutation in X chromosome
D. Colour blindness	IV. Trisomy of 21st chromosome

Choose the correct answer from the options given below:

Match List-I with List-II

List-I (Techniques)	List-II (Related features)
(A) ELISA	(I) Production of mature insulin
(B) PCR	(II) Antigen-antibody interaction
(C) Autoradiograph	(III) Amplification of DNA
(D) Recombinant DNA technology	(IV) Photographic film

Choose the correct answer from the options given below:

Which of the following postulates of Werner theory is incorrect?

Read the following paragraph and answer the given questions.
Repeated amplification is achieved by the use of a thermostable DNA polymerase (isolated from a bacterium, Thermus aquaticus), which remain active during the high temperature induced denaturation of double stranded DNA. The separated DNA fragments can be visualised only after staining the DNA with a compound known as ethidium bromide followed by exposure to UV radiation. The separated bands of DNA are cut out from the agarose gel and extracted from the gel piece. The convention for naming these enzymes is the first letter of the name comes from the genus and the second two letters come from the species of the prokaryotic cell from which they were isolated, e.g., EcoRI comes from Escherichia coli RY 13. In EcoRI, the letter 'R' is derived from the name of strain. Roman numbers following the names indicate the order in which the enzymes were isolated from that strain of bacteria. Stanley Cohen and Herbert Boyer accomplished this in 1972 by isolating the antibiotic resistance gene by cutting out a piece of DNA from a plasmid which was responsible for conferring antibiotic resistance. Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA, RNA, protein, polysaccharides etc. The RNA can be removed by treatment with ribonuclease whereas proteins can be removed by treatment with protease. Other molecules can be removed by appropriate treatments and purified DNA ultimately precipitates out after the addition of chilled ethanol.

Stanley Cohen and Herbert Boyer constructed the first recombinant DNA by isolating the antibiotic resistance gene by cutting out a piece of DNA from a plasmid of ____________ resistance. which was responsible for conferring antibiotic

Read the passage carefully and answer the questions
The chemical reactions proceed at different rates which depend on various factors like concentration, temperature, etc. Rate law is an expression which relates the rate of reaction with concentration of various reacting species in the reaction. For a first order reaction, the concentration of the reactant after time \(t\) is related with its initial concentration by the relation, \(C=C_0 e^{-k t}\). The rate constants for different orders have different expressions. For a first order reaction, \(k=\frac{2.303}{t} \log \frac{a}{a-x}\), where \(a\) is the initial concentration and \(x\) is the extent of reaction.
Which plot is linear with negative slope for a first order reaction