The area of the smaller portion enclosed by the circle \(x^2+y^2=4\) and the straight line \(x+y=2\) is:

Which of the following statements are correct in reference to the linear programming problem (LPP):
Maximize \(Z =5 x +2 y, Z =5 x +2 y, Z =5 x +2 y\)
subject to the following constraints
\(3 x+5 y \leq 15,5 x+2 y \leq 10, x \geq 0, y \geq 0\)
(A) The LPP has a unique optimal solution at \((2,0)\) only.
(B) The feasible region is bounded with corner points \((0,0),(2,0),(20 / 19,45 / 19)\) and \((0,3)\).
(C) The optimal value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is unbounded.
Choose the correct answer from the options given below:

The probability distribution of a random variable \(X\) is given by
\(\begin{array}{lcllll}x & 0 & 1 & 2 & 3 & 4 \\ p(x) & 0.1 & k & 2 k & 2 k & k \end{array} \)
The value of \(k\) is:

The derivative of \(f(\cot x)\) with respect to \(g(cosec x)\) at \(x=\frac{\pi}{4}\) (where \(\left.f^{\prime}(1)=2, g^{\prime}(\sqrt{2})=4\right)\) 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 solution set of the inequality \(\frac{2 x+3}{x-1}<0\) is:

The order and degree of differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^2=\frac{d^2 y}{d x^2}\) respectively, are

Phenol reacts with sodium hydroxide and carbon dioxide in acidic medium to produce:

Read the passage carefully and answer the questions based on the passage:
Species diversity on earth is not uniformly distributed. It is generally highest in the tropics and decreases towards the poles. Earth's fossil history reveals the incidences of mass extinctions in the past. Earth's rich biodiversity is vital for the very survival of mankind. It is believed that communities with high diversity tend to be less variable and more productive. The reasons of conserving biodiversity are narrowly utilitarian, broadly utilitarian and ethical. Biodiversity conservation may be in-situ as well as ex-situ.
How many episodes of mass extinction of species have occurred since the origin and diversification of life on earth?

With reference to D-glucose, which of the following cannot be explained by its open chain structure?
(A) D-Glucose does not give Schiff's test.
(B) The pentaacetate of D-glucose does not react with hydroxylamine.
(C) Glucose is found to exist in two different crystalline forms.
(D) D-glucose reacts with mild oxidizing agents like Br2-H2O and forms gluconic acid.
Choose the correct answer from the options given below:

Match List - I with List - II

List - I (Laws/properties)	List - II (Statements)
(A) Colligative properties	(I) for a solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in the solution.
(B) Henry's law	(II) the total pressure over the solution phase in the container will be the sum of the partial pressures of all the components of the solution.
(C) Raoult's law	(III) depends on the number of solute particles, irrespective of their nature.
(D) Dalton's law of partial pressures	(IV) at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas present above the surface of a liquid or solution.

Choose the correct answer from the options given below:

To find the focal length of a convex lens, a student obtains the real, inverted and magnified image on the screen as shown in the figure below.

Now the whole set up is immersed in water without disturbing the object and position of the screen. What will be student's observation on the screen?

The intensity of a plane monochromatic wave is \( 1380 \text{ Wm}^{-2} \). What is the amplitude of electric field in this wave ? (Take speed of light \( 3 \times 10^{8} \text{ ms}^{-1} \); \( \epsilon_{0} = 8.85 \times 10^{-12} \text{ C}^{2}\text{N}^{-1}\text{m}^{-2} \))