The magnitude of the projection of the vector \(\mathbf{a}=4 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}\) on the line which makes equal angles with the coordinate axes is

If \(N\) denotes the set of all positive integers and if \(f: N \rightarrow N\) is defined by \(f(n)=\) the sum of positive divisors of \(n\) then, \(f\left(2^k \cdot 3\right)\), where \(k\) is a positive integers, is

The range of a random variable $X$ is $\{1,2,3\dots ...\}$ and $P(X=x)=\frac{c^x}{x!}$, for $x=1,2,3,\dots \dots$ Then the value of $C$ is

\(\left|\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right|=\)

In any
\[
\triangle A B C, \frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4 b^2 c^2}
\]
equals to

For any value of \(\theta\), if the straight lines \(x \sin \theta+(1-\cos \theta) y=a \sin \theta\) and \(x \sin \theta-(1+\cos \theta) y+a \sin \theta=0\) intersect at \(P(\theta)\), then the locus of \(P(\theta)\) is a

The centroid of the triangle formed by the pair of straight lines \(12 x^2-20 x y+7 y^2=0\) and the line \(2 x-3 y+4=0\) is :

Let \(M=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\) and \(I\) be the identity matrix of order 3 . Then \(M^2-4 M=\)

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

If \(A+2 B=\left(\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right)\) and \(2 A-B=\left(\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right)\) then \(\operatorname{Tr}[\mathrm{A}]-\operatorname{Tr}[\mathrm{B}]=\)

Assertion (A) : It is more difficult to push a magnet into a coil with more number of turns.

Reason (R) : The emf induced in a coil opposes the motion of a magnet when it is moved towards the coil.

Dilute sodium hydroxide does not react with which of the following?

An electric charge \(10^{-3} \mu \mathrm{C}\) is placed at the origin of \(\mathrm{x}-\mathrm{y}\) plane. The potential difference between points \(A\) and \(B\) located at \((\sqrt{2} m, \sqrt{2} m)\) and \((2 m, 0 m)\) respectively is