Given that \(\vec{a}=-3 \hat{i}-6 \hat{j}+4 \hat{k}, \quad \vec{b}=9 \hat{i}-\lambda \hat{j}-12 \hat{k}\)
If \(\vec{a} \times \vec{b}=\overrightarrow{0}\), then the value of \(\lambda\) is

For any real numbers x and y, we define x R y if and only if \(\operatorname{cosec}^2 x-\cot ^2 y=1\). The relation R is:

Match List-I with List-II
The function \(f(x)=2 x^3-15 x^2+36 x+5\) for \(x \in[2,5]\) has

List-I	List-II
(A) absolute maximum value	(I) 5
(B) absolute minimum value	(II) 60
(C) point of absolute maxima	(III) 3
(D) point of absolute minima	(IV) 32

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Match List-I with List-II

List-I (Lines)	List-II (Direction Ratios)
(A) \(\frac{x-1}{2}=\frac{2-y}{1}=z\)	(I) \((1,3,-1)\)
(B) \(\frac{2 x-1}{2}=\frac{y+1}{3}=\frac{1-z}{1}\)	(II) \((2,-2,0)\)
(C) \(\frac{x+1}{2}=\frac{3-y}{2}=z=2\)	(III) \((2,-1,1)\)
(D) \(\frac{2 x-3}{4}=\frac{1-2 y}{2}=\frac{z}{5}\)	(IV) (2, -1, 5)

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The solution of the differential equation \(\frac{d y}{d x}=\sqrt{\frac{y}{x}}\) is

\(\int_0^{\pi / 2} \frac{\sin ^8 x}{\sin ^8 x+\cos ^8 x} d x\) is equal to

Function \(f(x)=x^3-3 x+3\) is
(A) Increasing in the interval \((-1,1)\)
(B) Increasing in the interval \((1, \infty)\)
(C) Decreasing in the interval \((-1,1)\)
(D) Increasing in the interval \((-\infty,-1) \cup(1, \infty)\)
Choose the text answer from the options given below :

Arrange the following halocompounds in increasing order of their boiling points
(A) Chloromethane
(B) Bromomethane
(C) Bromoform
(D) Dibromomethane
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Consider the equation of the line \(\vec{r}=-\hat{i}+2 \hat{k}+\mu(4 \hat{i}-\hat{j}+2 \hat{k})\).
Match List-I with List-II

List-I	List-II
(A) It passes through the point	(I) \(4,-1,2\)
(B) Its direction ratios are	(II) \(\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}\)
(C) Its Cartesian form is	(III) \((-1,0,2)\)
(D) Its direction cosines are	(IV) \(\frac{x+1}{4}=\frac{y}{-1}=\frac{z-2}{2}\)

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Identify the incorrect statement(s) from the following in the context of an electric current flowing through a conductor.
(A) The electrons suffer collisions with heavy fixed ions in the conductor
(B) The resistance of the given conductor depends on its temperature
(C) Resistance and resistivity both depend on the dimensions of the conductor
(D) When a potential difference V is applied across the conductor then V= IR
Choose the correct answer from the options given below:

If \(y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\cdots \infty}}}\) then \(\frac{d y}{d x}\) equals to

Choose the correct statements, when a cell of emf E and internal resistance r connected to external resistance R having current I flowing through the circuit is:
A. Potential difference across the terminals of the cell in a closed circuit having resistance R is always less than the emf.
B. For practical calculations, the internal resistance of the cells in the circuits may be neglected when the current I is such that E >> I.
C. Terminal potential difference in a closed circuit is always more than emf E.
D. Terminal potential difference versus current graph for a cell with closed circuit having resistance R is a straight line with a negative slope.
Choose the correct answer from the options given below:

Read the following passage carefully and answer the given questions.
Productivity in the Ecosystem The flow of energy through a terrestrial ecosystem starts with the harnessing of sunlight by autotrophs. The radiant energy is converted by photosynthesis to organic compounds is known as productivity. The productivity is expressed in units of energy per unit of area per unit time. The amount of organic matter present at any given time is biomass. The productivity of any ecosystem is affected by many parameters like weather conditions, nutrient availability etc. The productivity of any ecosystem does vary seasonally and yearly.
The Net Primary Productivity of an ecosystem are represented by: