The system of linear equations \(k x+5 y=5,2 x+3 y=5\) will be consistent if

Let \(a \leq \tan ^{-1} x+\cot ^{-1} x+\sin ^{-1} x \leq b\). If \(\alpha\) and \(\beta\) denote the minimum and maximum possible values of \(a\) and \(b\) respectively, then :

Which of the following statements is/are true?
(A) The vector sum of the three sides of a triangle in order is \(\vec{c}\)
b (B) The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector \(\vec{r}=a \hat{i}+b \hat{j}+c \hat{k}\) are related as \(l=\frac{a}{r}, m=\frac{b}{r}, n=\frac{c}{r}\)
(C) If θ is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta\)
(D) The cross product of two vectors is commutative
Choose the correct answert from the option given below :

The function \(f(x)=\tan x-x\) for \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) is :

The function \(f: Z \rightarrow Z\) defined by \(f(x)=x^2+x\), where \(Z\) is the set of integers, is:

If \(y=3^x+e^x+x^x+x^3\), then the value of \(\frac{d y}{d x}\) at \(x=3\) is:

If A and B are invertible matrices of the same order, then \(( AB )^{-1}\) is equal to

For a \(3 \times 3\) matrix \(A=\left[a_{i j}\right]\), whose element \(a_{i j}\) is given by \(a_{i j}=\sqrt{i^2+3 j}\), then what is element \(a_{31}\) ?

Which of the following is not the alien invasive species?

A screen is placed at a distance of 90 cm from an object. The image of the object on the screen is formed by a convex lens placed at two different locations separated by 20 cm. The focal length of the lens is approximately

Read the passage carefully and answer the Questions
In the formation of coordination complexes, if the inner \(d\) orbital \((n-1)d\) is used in hybridisation, the complex, is called an inner orbital or low spin or spin paired complex. And if it uses outer \(d\) orbital \((nd)\) in hybridisation (like \(sp^3d^2\)), it is called outer orbital or high spin or spin free complex. The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the complex. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting. For coordination complexes, the magnetic moment is determined by the number of unpaired electrons and is calculated by using the 'spin-only' formula, \(\mu = \sqrt{n(n+2)}\) where \(n\) is the number of unpaired electrons and \(\mu\) is the magnetic moment in units of Bohr magneton \((BM)\). The coordination compounds are of great importance. These compounds are widely present in the mineral, plant and animal worlds and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine.
\([Co(NH_3)_6]^{3+}\) and \([Ni(NH_3)_6]^{2+}\) respectively are

Choose the statements that are valid for the molecularity of a reaction
(A) Molecularity is equal to the number of reacting species involved in the overall reaction
(B) Decomposition of ammonium nitrite is an example of unimolecular reaction
(C) Reactions with the molecularity three are very rare
(D) Molecularity can only be determined experimentally
Choose the correct answer from the options given below:

Identify the incorrect statement about the \(H _2- O _2\) fuel cell from the following: