The general solution of the differential equation \(x\left(\frac{d y}{d x}\right)=y+x \tan \left(\frac{y}{x}\right)\) is

The point of local maxima of the function \(f(x)=(x-2)^5(x+2)^2\) is

A pipe can fill a tank in 2 hours. Because of a leak in the tank, the pipe takes 2 hours 40 minutes to fill the tank. The time the leakage takes to empty the full tank is :

Pipe A can fill the tank 3 times faster than pipe B. If both pipes A and B running together can fill the tank in 15 minutes, then the time taken by pipe B alone to fill the tank is :

Let \(\vec{a}=4 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+\hat{j}-2 \hat{k}\). Then
(A) \(\vec{a}\) is a unit vector
(B) \(\vec{a} \times \vec{b}=-\hat{i}+2 \hat{j}+2 \hat{k}\)
(C) \(\vec{a}\) and \(\vec{b}\) are parallel vectors
(D) \(\vec{a}\) and \(\vec{b}\) are neither parallel nor perpendicular vectors
Choose the \(\backslash\) textbf\{correct\} answer from the options given below :

If \(A\) be a square matrix of order 3 such that \(|A|=2\), then \(|\operatorname{adj}(2 A)|\) is equal to

If the two lines \(\frac{x-1}{2}=\frac{y-3}{4}=-z\) and \(\frac{x-4}{3}=\frac{y-1}{-2}=z-1\) are contained in a plane then the equation of the plane is :

In which ecosystem, a large fraction of energy flows more through DFC than through GFC?

The composition of Chromate ore is :

The least distance of distinct vision of a person is 50 cm. He wants to read a book placed at 25 cm. The focal length of the spectacles would be :

Read the passage carefully and answer the questions
Carbohydrates are naturally formed organic compounds from various plants. The common chemical formula, \(C _x\left( H _2 O \right)_y\), was considered as hydrates of carbon to derive the various carbohydrates. For instance, the general formula for glucose ( \(C _6 H _{12} O _6\) ) easily fits into the above formula. Carbohydrates also called saccharides are classified on the basis of their behavior towards hydrolysis. Monosaccharides cannot be hydrolyzed further to simpler units of polyhydroxy aldehyde or ketones. About 20 monosaccharides are known to occur in nature. Carbohydrates that yield large number of monosaccharide units are called polysaccharides. Amino acids contain ( \(- NH _2\) ) and ( - COOH ), which can be classified in various forms. Proteins are polymers of alpha amino acids which are connected to each other by a peptide bond or peptide linkage. When a native protein is subjected to physical change, its globules unfold and helix get uncoiled and the protein loses its biological activity. The protein is said to be denaturated.
The peptide linkage formed between the reaction of glycine and alanine is represented as

Rahul invested ₹ 20000 in a mutual fund in year 2018. If the value of mutual fund increased to ₹ 32000 in year 2023, then the compound annual growth rate of his investment is : [given that \((1.6)^{\frac{1}{5}}=1.098\) ]

Which type of isomerism is shown by the complex compounds?
\(\left[ Co \left( NH _3\right)_5 Br \right] SO _4\) and \(\left[ Co \left( NH _3\right)_3 SO _4\right]\)Br