A person creates a sinking fund to accumulate 60,000 in 5 years to buy a bike.
The amount to be deposited at the end of every quarter with a 12\% rate of interest compounded quarterly is :
Use \(\frac{0.03}{(1.03)^{20}-1}=0.0372\)

If the radius of the spherical ball is increasing at the rate of 1 m/sec and the radius of the ball is 3 meter then the volume of the ball is increasing at the rate of

In a 10 km race, \(A, B\) and \(C\) each run at uniform speed and get the first, second and third positions respectively.
If \(A\) beats \(B\) by 1 km and \(B\) beats \(C\) by 1 km , then by how much distance does \(A\) beat \(C\) ?

Match the Matrix in List - I with its Adjoint's Determinant in List - II.
List - I (Matrix A) :

LIST 1	LIST 2
(A) \(\left[\begin{array}{ll}3 & 1 \\ 4 & 2\end{array}\right]\)	(I) 9
(B) \(\left[\begin{array}{cc}5 & -1 \\ 4 & 2\end{array}\right]\)	(II) 8
(C) \(\left[\begin{array}{cc}6 & -1 \\ 2 & 1\end{array}\right]\)	(III) 14
(D) \(\left[\begin{array}{ll}4 & 1 \\ 3 & 3\end{array}\right]\)	(IV) 2

Choose the correct answer from the options given below:

Let \(\theta\) be the angle between two vectors \(\vec{a}\) and \(\vec{b}\). Then match List-I with List-II

List-I	List-II
(A) \(\sin \theta\)	(I) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)
(B) \(\cos \theta\)	(II) \(|\vec{a} \times \vec{b}|\)
(C) Area of the parallelogram with adjacent sides represented by \(\vec{a}\) and \(\vec{b}\)	(III) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}\)
(D) Projection of \(\vec{a}\) on \(\vec{b}\)	(IV) \(\frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}\)

Choose the Correct answer from the options given below :

Minimum value of the function \(f(x) = \sin(3x) + 5\) is equal to :

Match List-I with List-II

List-I	List-II
(A) The degree of differential equation \(\frac{d^3 y}{d x^3}=e^{\frac{d y}{d x}}\)	(I) 2
(B) The order of differential equation \(\left(\frac{d y}{d x}\right)^2+\frac{d^3 y}{d x^3}=0\)	(II) 4
(C) The sum of order and degree of differential equation \(\frac{d}{d x}\left(\frac{d^2 y}{d x^2}\right)+\left(\frac{d y}{d x}\right)^5=x\)	(III) not defined
(D) The number of arbitrary constants in the general solution of a differential equation of order 2	(IV) 3

Choose the correct answer from the options given below :

Identify 'X' along with oxidation state of halogen in the given reaction :
\(3\text{Cl}_2 + 6\text{NaOH} \rightarrow 5\text{NaCl}+\text{X} + 3\text{H}_2\text{O}\) (hot and conc.)

n-Hexane is obtained when glucose is

\(\int \frac{\log _e x}{\left(1+\log _e x\right)^2} d x\)2 is equal to

Read the passage and answer the following
Microbes are organisms that cannot be seen with the naked eye. They include bacteria, archaea, protists, viruses, prions, and fungi. They account for most of the diversity of life on earth. Far more microbes inhabit a single person's body than there are people on the entire planet. Microbes also dominate in terms of their ability to live and thrive in extreme environments, including clouds, deep-sea volcanoes, springs, ice caps, and animal intestines. Most animals, including humans, exhibit numerous associations with microbes. Our "human microbiome" includes our skin, hair, mouth, and gut.
Select the chemical substances which are produced by some microbes and they can kill or prevent growth of other microbes :

Match List-I with List-II

List-I (Reaction property)	List-II (Formula)
(A) Half-life of decomposition of gaseous ammonia on hot platinum surface	(I) \(\frac{1}{t} \ln \left(\frac{[A]_0}{[A]_t}\right)\)
(B) Instantaneous rate	(II) \(\frac{[R]_0}{2 k}\)
(C) Rate constant of inversion of cane sugar	(III) \(-\frac{d x}{d t}\)
(D) Half-life of acid hydrolysis of methyl acetate	(IV) \(\frac{\ln (2)}{k}\)

Choose the correct answer from the options given below:

The instantaneous power supplied to a capacitor is given by\(P_c=i_m \cos (\omega t) v_m \sin (\omega t)\).
The average power over a complete cycle would be: