The equation of the normal to the curve \(y=4 x^3+2 \sin x\) at \((0,3)\) is :

A linear programming problem is as follows:
Maximize/minimize objective function \(z=2 x-y+5\) subject to constraints:
\(3 x+4 y \leq 60, \quad x+3 y \leq 30, \quad x \geq 0, y \geq 0 .\)
If the corner points of feasible region are \(A(0,10), B(12,6), C(20,0), O(0,0)\), then which of the following is true:

A stone is dropped into a lake and waves move in circles at a speed of \(8 cm / sec\). At the instant, when the radius of the circular wave is 10 cm , how fast is the enclosed area increasing?

The vector equation of the plane which is at a distance of 6 units from the origin and has \(2,-1,2\) as the direction ratios of normal to it, is :

The solution of the differential equation \(2 x \frac{d y}{d x}-y=3\) represents a family of :

The particular solution of the differential equation \(x d y=\left(2 x^2+1\right) d x, x \neq 0\), given that \(y=1\) when \(x=1\) is:

There is a Building which is tilted and equations of two different walls are \(2 x-y+z=4\) and \(x+y+2 z=3\). Angle between the walls is :

Read the passage and answer the question given below.
When a host is exposed to antigens, which may be in the form of living or dead microbes or other proteins, antibodies are produced in the host body. This type of immunity is called active immunity. Active immunity is slow and takes time to give its full effective response. Injecting the microbes deliberately during immunisation or infectious organisms gaining access into body during natural infection induce active immunity. When antibodies are directly given to protect the body against foreign agents, it is called passive immunity.
Production of antibodies against antigens in the body is:

An insulating rod of length \( L \) carries charge \( q \) distributed uniformly on it. The rod is pivoted at one end and is rotated at a frequency \( f \) about a fixed perpendicular axis. Then the magnetic dipole moment of the system is :

When a deuteron is bombarded on \(^{16}_{8}\text{O}\) nucleus, an α-particle is emitted. Other product nucleus is:

if a function \(f: Q \rightarrow R\) defined by \(f(x)=\frac{2 x-1}{2}\) and \(g: Q \rightarrow R\) is defined by \(g(x)=x+2\), then \((g \circ f)\left(\frac{3}{2}\right)\) is :

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
The alcohol which cannot be used to prepare carbonyl compound through oxidation is

On prolonged heating with HI , glucose forms