Let \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\), then which of the following statements are correct?
(A) \(|\vec{a}|=\sqrt{14}\)
(B) \(|\vec{b}|=29\)
(C) \(\vec{a} \cdot \vec{b}=8\)
(D) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\cos ^{-1}\left(\frac{-8}{\sqrt{406}}\right)\)
Choose the correct answert from the option given below :

If \(\left|\begin{array}{cc}2 x & 2 \\ 4 & x\end{array}\right|=10\), then \(x\) is:

The general solution of the differential equation \(\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0\) is :

Consider a line \(\vec{r}=(\hat{i}+4 \hat{j})+\lambda(2 \hat{i}-2 \hat{j}+3 \hat{k})\), then which of the following statements are correct?
(A) It passes through point \((9,-4,12)\)
(B) It passes through point \((1,4,-1)\)
(C) Its direction cosine's are \(\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}\)
(D) Its Cartesian equation is \(\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z}{3}\)
Choose the correct answer from the options given below:

The function, \(f(x)=x+\frac{a^2}{x}, a>0, x \neq 0\) has a local maxima at

If \(A\) is a square matrix such that \(A^2=A\) and \(I\) is the identify matrix of the same order as \(A\) then \((I+2 A)^3-6 A\) is equal to

The maximum value of \(\sin x \cdot \cos x\) is :

A thin infinite line charge has linear charge density of \( 2.0 \times 10^{-9} \text{ C/m} \). The electric field intensity at a distance of 3 m from the line charge is : (\( \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \))

If \(A=\left[a_{i j}\right]_{3 \times 3}\), where \(a_{i j}=\frac{(-1)^{i+j}-1, i=j}{(-1)^{i+j}, i \neq j}\) then the value of \(A+A^T\) is:

The least distance of distinct vision of a person is 50 cm. He wants to read a book placed at 25 cm. The focal length of the spectacles would be :

Match the wavelength of light absorbed given in List-l with the color of light absorbed in List-ll

List-l(Wavelength of light absorbed(nm))	List-lI(Colour of light absorbed)
(A) 535	(I) Yellow
(B) 475	(II) Red
(C) 600	(III) Blue
(D) 498	(IV) Blue Green

Choose the correct answer from the options given below:

Match List-I with List-II:

List-I (AC Circuit)	List-II (Phase Angle)
(A) Phase Difference between current and voltage in a pure inductive ac circuit	(I) zero
(B) Phase Difference between current and voltage in a pure resistive ac circuit	(II) \(\pi/2\), current lags voltage
(C) Phase Difference between current and voltage in a pure capacitive ac circuit	(III) \(\tan^{-1}(\frac{X_L - X_C}{R})\)
(D) Phase Difference between current and voltage in a series LCR circuit	(IV) \(\pi/2\), current leads voltage

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 biomass available for consumption by herbivores and decomposers is known as-