A and B are two students. Their chances of solving a problem correctly are \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively.
If the probability of their making a common error is \(\frac{1}{20}\) and they obtain the same answer, then the probability that their answer is correct is:

If A= \(\left[\begin{array}{cc}7 & 3 \\ 5 & -7\end{array}\right]\) be such that \(A^{-1}\) = kA, then k equals

If A is a matrix of order mn and B is a matrix such that AB' and B'A are both defined, the order of matrix B is :

If \(f(x)=\cos 3 x, 0 \leq x \leq \frac{\pi}{2}\), then :
(A) \(f\) is strictly increasing on \(\left[0, \frac{\pi}{3}\right]\)
(B) \(f\) is strictly decreasing on \(\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\)
(C)\(f\) is strictly increasing on \(\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\)
(D) \(f\) is strictly decreasing on \(\left[0, \frac{\pi}{3}\right]\)
(E) \(f\) is strictly increasing on \(\left[0, \frac{\pi}{2}\right]\)
Choose the correct answerfrom the ptions given below:

If a computer code is correctly programmed, it gives \(90\%\) acceptable results. But if it is not correctly programmed, it gives only \(40\%\) acceptable results. From previous experience, it is observed that only \(80 \%\) of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results then the approximate probability that the code is correctly programmed is

Let \(A\) be a matrix such that \(A=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]\) Then which of the following are TRUE?
(A) \(A\) is a non-singular matrix
(B) \(A^T=A\)
(C) \(A\) is not an invertible matrix
(D) \(A\) is not a skew-symmetric matrix
Choose the correct answer from the options given below :

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 biodiversity is ______ at lower altitudes and ______ with increasing higher latitudes.

A conducting circular loop is placed in a uniform magnetic field, B = 0.025 T with its plane perpendicular to the magnetic field. The radius of the loop is made to shrink at a constant rate of 2 mm \( s^{-1} \). The induced emf when the radius is 2 cm, is

A cell of emf \( \mathcal{E} = 10 \text{ V} \) and internal resistance \( r = 3 \Omega \) is connected to a uniform wire of length \( L = 500 \text{ cm} \) and resistance \( R = 3 \Omega \). The drop of potential with length in the wire is:

Ethanol cannot be prepared from :

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 :

Match the complex species given in List-l with the possible isomerism given in List-ll:

List-l Complex species	List-ll Isomerism
(A) \(\left[ Co \left( NH _3\right)_4 Cl _2\right]^{+}\)	(I) optical
(B) cis- \(\left[ Co ( en )_2 Cl _2\right]^{+}\)	(II) ionization
(C) \(\left[ Co \left( NH _3\right)_5\left( NO _2\right)\right] Cl _2\)	(III) coordination
(D) \(\left[ Co \left( NH _3\right)_6\right]\left[ Cr ( CN )_6\right]\)	(IV) geometrical

Choose the correct answer from the options given below: