If \(A\) and \(B\) are two square matrices of same order such that \(A B=A\) and \(B A=B\), then the value of \(A^{2024}+B^{2024}\) is equal to

The speed of a motorboat in still water and that of the current of water is in a ratio of \(27: 5\).
The boat goes along the current from point \(A\) to point \(B\) in 3 hours 40 minutes.
How much time will it take to come back from \(B\) to \(A\)?

The value of the integral \(\int \frac{d x}{e^x+e^{-x}}\) is equal to:

If \(A\) is a square matrix of order 2 , then \(|\operatorname{adj} A|\) is equal to:

Let \(x\) denote the number of doublets in three throws of a pair of dice with the following probability distribution.

\(x\)	\(0\)	1	2	3
\(P(x)\)	\(\frac{25}{72} k\)	\(\frac{15}{72} k\)	\(\frac{3}{72} k\)	\(\frac{1}{360} k\)

If value of \(k\) is equal to \(\frac{m}{n}, \operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to

Match List-I with List-II

Llst-I	Llst-(II)
(A) \(f(x)=[x]\)	(I) Continuous everywhere but not differentiable at \(x=-1\)
(B) \(f(x)=|x-1|\)	(II) Continuous everywhere except at all integral values
(C) \(f(x)=e^{|x|}\)	(III) Continuous everywhere but not differentiable at \(x=1\)
(D) \(f(x)=|x+1|\)	(IV) Continuous everywhere but not differentiable at \(x=0\)

Choose the correct answer from the options given below:

If \(y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}\), then

The corner points of the feasible region with the constraints \(x+y \leq 30, x+y \geq 15, y \leq 20, x \leq 15\), and \(x, y \geq 0\) are

The near point of a hypermetropic person is 80 cm from the eye. The power of lens, required to enable the person to read clearly a book held at 25 cm from the eye is:

Read the passage carefully and answer the questions
Ethers with the general formula \(C _n H _{2 n+2} O\) and functional group C-O-C may be divided as symmetrical and unsymmetrical.
Lower symmetrical ether may be obtained by the dehydration of alcohols at low temperatures in the presence of protic acids,
such as \(H _2 SO _4\) and \(H _3 PO _4\) whereas at high temperature intramolecular dehydration takes place.
Formation of ether from primary alcohols is also accompanied by the formation of small amount of alkenes.
Due to the steric hinderance, secondary and tertiary alcohols undergo elimination reaction rather than substitution reaction yielding alkene as major product. Ethers can also be obtained by heating alkyl halides with sodium or potassium alkoxides.
Substituted Ethers (secondary & tertiary) can also be prepared using this method.
If primary halides are replaced with secondary and tertiary halides, alkenes are the major products.
The major product in the following reaction is \(ROH + HOR \xrightarrow{ H _2 SO _4, 140^{\circ} C }\)

If \(\vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}, \vec{a} \times \vec{b}=\vec{a} \times \vec{c}\) and \(\vec{a} \neq \overrightarrow{0}\), then the vector \(\vec{b}\) is equal to:

Osmotic pressure measurements are less accurate and difficult to carry out. Therefore, this method is not useful for measuring molecular weights. But it is used to find out molecular weights of polymers eg. plastics, proteins and starch having many monomer units. Knowing the value of osmotic pressure of a given solution, molecular weight of the solute can be calculated using Van't Hoff equation.
An aqueous solution of salt contains 20 g of NaCl per liter. The pressure of solution at 35 \(^{\circ}\text{C}\) will be:

The resistance of 1N solution of four different electrolytes are given below. Arrange them in increasing order of their equivalent conductance values if the same conductivity cell was used to measure the resistance of all the electrolytic solutions:
(A) 250 ohm
(B) 150 ohm
(C) 50 ohm
(D) 280 ohm
Choose the correct answer from the options given below: