Identify the equations of lines through origin which intersect the line \(\frac{x-3}{2}=\frac{y-3}{2}=\frac{z}{1}\) at an angle \(\frac{\pi}{3}\).
(A) \(\frac{x}{1}=\frac{y}{2}=\frac{z}{-1}\)
(B) \(\frac{x}{1}=\frac{y}{-2}=\frac{z}{1}\)
(C) \(\frac{x}{-1}=\frac{y}{-1}=\frac{z}{-2}\)
(D) \(\frac{x}{-1}=\frac{y}{1}=\frac{z}{-2}\)
Choose the correct answer from the options given below:

Choose the correct statement about Sinking Fund?

If area of the triangle with vertices \(A(2, k), B(1,1), C(10,8)\) is \(\frac{47}{2}\) sq. units, then one of the values of \(k\) is:

The general solution of differential equation \(\frac{dy}{dx} - xy = e^{\frac{x^2}{2}}\) is:

Let \(f(x)=x^3-6 x^2+9 x-8\) be a function, then which of the following statements are TRUE?
(A) \(f^{\prime}(x)=3(x-1)(x-3)\)
(B) The critical points of the function are \(x=1\) and \(x=3\)
(C) \(x=1\) is the point of local minimum
(D) The local maximum value is -4
Choose the correct answer from the options below :

If \(|\vec{a}|=7\) then \(|3 \vec{a}|\) is:

A boat takes 1 hour 30 minutes less to travel 36 km downstream than to travel the same distance upstream.
If the speed of the boat in still water is 10 km/hr, then the speed of the stream is :

Ethanol cannot be prepared from :

If \(E _{ p }\) and \(E _{ k }\) represents the potential and kinetic energy respectively of an orbital electron, then according to Bohr's theory.

Read the passage carefully and answer the following questions
Aldehydes, carboxylic acids and ketones are widespread in the animal and plant kingdoms. They play a pivotal role in various biochemical processes of life. Their presence in nature adds fragrance and flavor. These compounds are widely used in food products, pharmaceuticals, paints, resins and other important product industries. These compounds are prepared by various laboratory methods, which mainly include oxidation, formylation, acylation and reduction. Due to the polar nature of the carbonyl group in aldehydes and ketones, they can exhibit different reactions like nucleophilic addition. They do exhibit redox and various condensation reactions which lead to the formation of various important compounds. On the other hand, the carboxylic acids are mainly prepared by oxidation and hydrolysis of different compounds. The carboxylic acid consists of a carbonyl group and the hydroxy group (attached to the carbonyl carbon atom). This makes it possible for the carboxylic acid to participate in various chemical reactions which involve cleavage of the C-OH bond and the O-H bond along with the reactions involving the complete - COOH group.
Arrange the following in the increasing order of their \(p K_{ a }\) values :
(A) \(CF _3- COOH\)
(B) \(O _2 N- CH _2- COOH\)
(C) \(NC - CH _2- COOH\)
(D) \(H - COOH\)
Choose the correct answer from the options given below :

To produce in large quantities, the development of bioreactors, where large volumes ______ of culture can be processed, is required.

The maximum value of the function \(f(x)=-x^2+2 x-1\) is:

Which of the following statements are correct?
A. The various lines in the atomic spectra are produced when electrons jump from higher energy state to a lower energy state.
B. An accelerated atomic electron must spiral into the nucleus as it losses energy.
C. Rutherford scattering determines an upper limit to the size of the nucleus.
D. Angular momentum of the electron orbiting around the nucleus is not quantised.