If \(x\sqrt{1+y} + y\sqrt{1+x} = 0\), then \(\frac{dy}{dx}\) is:

Given that \(\sigma p_{0} q_{0} = 1000, \sigma p_{0} q_{0} = 1200, \sigma p_{0} q_{0} = 1350, \sigma p_{0} q_{0} = 1500\), where subscripts 0 and and 1 used for base year and current year respectively. The Paasche's Index number is :

The function \(f(x)=\left\{\begin{array}{ll}\frac{x^2+2 x-3}{x-1}, & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{array}\right.\) is

For the given five values, \(16,25,19,34,43\), the three year moving averages are

Match List I with List II

LIST I	LIST II
A. \(\lim_{x \to 0} \frac{(1 - \cos 2x) \sin 5x}{x^2 \sin 3x}\)	I. 18
B. \(\lim_{x \to \infty} \frac{(3x-5)(2x-7)}{(4x-9)(5x-3)}\)	II. \(\frac{10}{3}\)
C. \(\lim_{x \to 0} \frac{2 \sin^2 3x}{x^2}\)	III. \(\frac{3}{4}\)
D. \(\lim_{x \to \frac{\pi}{4}} \frac{1 - \cos^3 x}{2 \cot x - \cot^3 x}\)	IV. \(\frac{3}{10}\)

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The area (in square units) of the region bounded by the curve \(x^2=y\) and the straight line \(y=4\) in the first quadrant is equal to

If \(x=t^{1 / 2}, y=t^{3 / 2}\), then \(\frac{d y}{d x}=\)

Let A and B be two events. Then which of the following statements are TRUE?
(A) \(P(B \mid A)=\frac{P(A \cap B)}{P(A)}\), provided \(P(A) \neq 0\)
(B) \(P\left(B^{\prime}\right)=1+P(B)\)
(C) \(P(A \cup B)=P(A)+P(B)+P(A \cap B)\)
(D) \(P(A \cap B)=P(A) \cdot P(B)\) if A and B are independent events
Choose the correct answer from the options given below:

Read the passage carefully and answer the questions based on the passage:
The number of reacting species (atoms, ions or molecules) taking part in an elementary reaction, which must collide simultaneously in order to bring about a chemical reaction is called molecularity of a reaction. In the rate equation Rate \(=k[\mathrm{~A}]^x[\mathrm{~B}]^y\)
\(x\) and \(y\) indicate how sensitive the rate is to the change in concentration of \(A\) and \(B\), respectively. Sum of these exponents, i.e., \(x +y\) gives the overall order of a reaction where \(x\) and \(y\) represent the order with respect to the reactants A and B respectively. Hence, the sum of powers of the concentration of the reactants in the rate law expression is called the order of that chemical reaction. For a first order reaction, the concentration of the reactant varies as \([\mathrm{R}]=[\mathrm{R}]_0 e^{-k t}\)
The initial concentration of \(\mathrm{N}_2 \mathrm{O}_5\) in the following first order reaction
\(\mathrm{N}_2 \mathrm{O}_5(g) \rightarrow 2 \mathrm{NO}_2(g)+1 / 2 \mathrm{O}_2(g)\)

The main causes of energy loss in transformers are
1. Hysteresis loss
2. Flux leakage
3. Eddy current
4. Damping
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Match List-l with List-II

List-l (Term)	List-II (Feature)
(A) Autogamy	(I) Flowers do not open
(B) Geitonogamy	(II) Transfer of pollen grains from anther to stigma of different plant
(C) Xenogamy	(III) Transfer of pollen grains from anther to stigma of another flower of same plant
(D) Cleistogamy	(IV) Transfer of pollen grains from anther to stigma of same flower

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The value of the integral \(I=\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x\) is :

Which of the statements below are correct?
A. Actinoid contraction is greater than lanthanoid contraction.
B. The magnetic property of the actinoids are more complex than those of the lanthanoids.
C. Actinoid contraction taking place due to poor shielding by 5f electrons.
D. Actinoid contraction taking place due to poor shielding by 6f electrons.
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