The value of \(\lambda\), so that the vector \(\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}\) are perpendicular to each other, is :

The general solution of the differential equation \(x\left(1+y^2\right) d x+y\left(1+x^2\right) d y=0\) is

\(\int \frac{e^{2 x}-1}{e^{2 x}+1} d x\) =

The shortest distance between the lines \(\vec{r}_1=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})\) and \(\vec{r}_2=2 \hat{i}+\hat{j}-\hat{k}+\mu(4 \hat{i}-2 \hat{j}+2 \hat{k})\) is :

\(\int \sqrt{1+\frac{x^2}{9}} d x\) is equal to
(Where \(C\) is an arbitrary constant)

If \(17^{241} \equiv R(\bmod 13)\), then value of \(R\) is \(:\)

The mean of the number of heads in a simultaneous toss of three coins is :

If \(\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]\), then the value of \(a+2 b-3 c+4 d\) is equal to

The values of \(b\) for which \(y=2 x^2-b x+2\) touches the \(x\)-axis are:

The resistance of a wire is \(10 \Omega\). Its length is increased by \(10 \backslash \%\) by stretching. The new resistance of the wire will now be :

Let the degree and order of the differential equation \(2 x^3 \frac{d y}{d x}-5\left(\frac{d^2 y}{d x^2}\right)^2=6\left(\frac{d y}{d x}\right)^3\) be \(m\) and \(n\) respectively. Then
(A) \(m=2\)
(B) \(n=3\)
(C) \(m=3\)
(D) \(m n=4\)

Read the following passage carefully and answer the given questions
The earthworm is referred to as the farmer's friend. This is because they help in the breakdown of complex organic matter as well as in loosening of soil. Similarly, decomposers break down complex organic matter into inorganic substances like carbon dioxide, water and nutrients and the process is called decomposition. Dead plant remains such as leaves, bark, flowers and dead remains of animals, including fecal matter, constitute detritus, which is the raw material for decomposition. Some of the steps in the process of decomposition are fragmentation, catabolism, leaching, humification etc.
The degradation of detritus into simpler inorganic substances by the bacterial and fungal enzymes is called as-

Read the Passage carefully and answer the questions
When a ligand is bound to a metal ion through a single donor atom, it is said to be a unidentate ligand but when it bind through two donor atoms, then it is said to be didentate ligand and when several donor atoms are present in a single ligand, it is said to be a polydentate ligand. When a dior polydentate ligand uses its two or more donor atoms simultaneously to bind a single metal ion, it is said to be a chelate ligand. The number of such ligating groups is called the denticity of the ligand. ligand which has two different donor atoms and either of the two ligetes in the complex is called ambidentate ligands. The number of ligand donor atoms to which the metal is directly bonded, is called coordination number of the metal ion in the complex. Complexes in which a metal is bound to only one kind of donor groups are known as homoleptic and in which a metal is bound to more than one kind of donor groups are known as heteroleptic complexes.
The coordination number of Co and Al in \(\left[ CoCl ( en )_2\right] Cl\) and \(K _2\left[ Al \left( C _2 O _4\right)_3\right]\), respectively, are