If \(x \sqrt{1+y}+y \sqrt{1+x}=0\), where \(|x|<1,|y|<1\) and \(x \neq y\), then

If \(A=\left[\begin{array}{cc}1 & 0 \\ \frac{1}{25} & 1\end{array}\right]\), then the value of \(A^{125}\) is :

\(\int_{-1}^1\left(x^7+x^5+x^3+x+1\right) d x\) is equal to:

The probability distribution of number of aces when two cards are drawn without replacement from a pack of 52 cards is:

If \(x, y\) and \(z\) are non zero real numbers, the inverse of matrix \(A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]\) is:

The maximum value of \(\frac{\log x}{x}\) for \(x\) in \([2, \infty)\) is :

In a 500 m race, the ratio of speeds of two participants \(A\) and \(B\) is \(4: 5\).
If \(A\) has a start of 180 m , then the distance by which \(A\) wins is :

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\(KMnO _4\) is prepared by the fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidizing agent like \(KNO _3\) to give a dark-green manganate ion which disproportionate to give permanganate as follows.
\(\begin{array}{l}2 MnO _2+4 KOH + O _2 \rightarrow 2 K_2 MnO _4+2 H _2 O \\ 3 K_2 MnO _4+4 H ^{+} \rightarrow 2 KMnO _4+ MnO _2+2 H _2 O \end{array}\)
On heating \(K _2 MnO _4\) decomposes at 513 K to give \(K _2 MnO _4\). Permanganate ion is tetrahedral and diamagnetic
Acidified \(KMnO _4\) acts a strong oxidizing agent which oxidizes oxalic acid, ferrous ions, nitrite ion and iodides.
When one mole of \(KMnO _4\) is used for the oxidation of oxalic acid, how many moles of \(CO _2\) will be produced?

The greatest biodiversity on the earth is seen in :

For \(x>y>0\), if \(x^5 y^6=(x+y)^{11}\), then \(\frac{d^{2} y}{d x^2}\) is

The function \(y=f(x)\) has extreme values at the points where ___________ .

The area (in sq. units) of the region bounded by \(y=2 x+3\), the \(x\)-axis, and the ordinates \(x=-2, x=2\) is equal to

Read the passage carefully and answer the questions
Optically active polyhydroxy aldehydes or ketones or molecules which provide such units on hydrolysis are called carbohydrates. They can be broadly classified as monosaccharides, disaccharides and polysaccharides depending on their behavior towards hydrolysis. They are also classified as reducing or non-reducing sugars. Another class of biomolecules is proteins which are polymers of twenty different amino acids linked by peptide bonds. Amino acids can be classified as acidic, basic or neutral depending on the relative number of amino and carboxyl groups in their molecule. The essential amino acids cannot be synthesized by our body and hence must be provided through diet. Nucleic acids are the polymers of nucleotides which in turn consist of a base, pentose sugar and phosphate. There are two types of nucleic acids namely, DNA and RNA which differ in the composition of sugar and a base
Four bases present in RNA molecule are