Let \(R\) be the set of all real numbers and \(f: R \rightarrow\) Range \(f\) be given by \(f(x)=3 x^2+1\). Then \(f^{-1}\{1,2\}\) is :

Let \(A\) and \(B\) are square matrices of order 3 such that \(A+B=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]\). If \(A\) is a symmetric matrix, then the value of \(|B|\) is:

The area (in sq.units) of the region bounded by the curves \(y = x\) and y = \(x^3\) is:

Area (in sq. units) of the region bounded by the curve \(y^2=4 x\) y-axis and the line y = 3 is

The area of the region enclosed between \(y^2=4 x\) and \(x=1\) and \(x=3\) in the first quadrant is:

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=7,|\vec{b}|=3\) and \(|\vec{c}|=5\), then angle between \(\vec{b}\) and \(\vec{c}\) is

Let \(A\) be a square matrix of order \(3 \times 3\), then \(|K A|\) is equal to :

Identify the correct sequence of occurrence of geological period from recent to oldest time?
(A) Permian
(B) Jurassic
(C) Silurian
(D) Devonian
Choose the correct answer from the options given below:

Let \(f: R \rightarrow R\) be a function defined as \(f(x)=x^4\). Which one of the following is true?

The largest interval, in which the function \(f(x)=x^3+2 x^2-1\) is increasing, is:

What will be the energy required to remove the electron from the first excited state of \( Li^{++} \), if the ground state energy of the electron in a hydrogen atom is -13.6 eV?

Read the passage carefully and answer the questions
Carboxylic acids are the organic compounds which comprise of carboxyl functional group, -COOH. These are generally found in nature, some higher members are known as fatty acids as they occur in natural fats as esters of glycerol. They can be prepared from oxidation of alcohols and aldehydes, hydrolysis of nitriles and amides and Grignard reagents. They are stronger acids than simple phenols and alcohols. but weaker than mineral acids. Substituents affect the acidity of aliphatic and aromatic carboxylic acids. They undergo reactions involving cleavage of -OH and C-OH bond. Carboxylic acids having an alpha hydrogen get halogenated in the presence of red phosphorus to form alpha halo carboxylic acids. The reaction is known as Hell Volhard Zelinsky reaction. Besides these aromatic carboxylic acids undergo electrophilic substitution reactions. They are versatile and are employed in the manufacture of various useful compounds. For example, methanoic acid is used in rubber, textile, dyeing, leather and electroplating industries. Higher fatty acids are used in the manufacture of soaps and detergents. Esters of Benzoic acid are used in perfumery.
The reagents needed for the preparation of Benzoic acid from ethyl benzene is :

The technique of using live organisms or derived enzymes to produce products and processes useful to humans is known as: