\(\mathbf{x}, \mathbf{y}, \mathbf{z}\) are three vectors each of magnitude \(\sqrt{2}\) and each making an angle \(60^{\circ}\) with one another. If \(\mathbf{a}=\mathbf{x} \times(\mathbf{y} \times \mathbf{z}), \mathbf{b}=\mathbf{y} \times(\mathbf{z} \times \mathbf{x})\), \(\mathbf{c}=\mathbf{x} \times \mathbf{y}\), then \(\mathbf{x}=\)

If the circle \(S=x^2+y^2+2 g x+2 f y+c=0\) cuts each of the three circles \(x^2+y^2+4 x+4 y+7=0, x^2+y^2-4 x+\) \(4 y+7=0\) and \(x^2+y^2-4 x-4 y+7=0\) orthogonally, then the equation of the tangent drawn at the point \((\sqrt{3}, 2)\) to the circle \(\mathrm{S}=0\) is

If \(f: R /\{0\} \rightarrow R\) is such that \(2 f(x)+f\left(\frac{1}{x}\right)=4 x\) and \(S=\{x \in R: f(x)=f(-x)\}\), then the number of elements in \(S\) is

Let $A=\left[\begin{array}{ccc}a& 3& 5\\ 5& -1& 3\\ 2& 3& -4\end{array}\right]$ and $B=\left[\begin{array}{ccc}b& 1& 4\\ 4& c& 1\\ -3& 1& d\end{array}\right]$. If the trace of $A$ is $-4$ and $AB=\left[\begin{array}{ccc}-1& 0& 17\\ -3& 10& 25\\ 28& -8& 3\end{array}\right]$ then $a+b+c+d=$

The slope of a line $L$ is $2.$ If $m_1, m_2$ are slopes of two lines which are inclined at an angle of $\frac{\pi }{6}$ with $L,$ then $m_1+m_2=$

\(\int \frac{3^x(x \log 3-1)}{x^2} d x=\)

Consider the following statements: Assertion (A): The direction ratios of a line \(\mathrm{L}_1\) are \(2,5,7\) and the direction ratios of another line \(\mathrm{L}_2\) are \(\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}\). Then the lines \(\mathrm{L}_1, \mathrm{~L}_2\) are parallel Reason (R): If the direction ratios of a line \(L_1\) are \(a_1, b_1, c_1\), the direction ratios of a line \(\mathrm{L}_2\) are \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2\) and \(\mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2\) \(+c_1 c_2=0\), then the lines of \(L_1, L_2\) are parallel Which one of the following is True?

The order of basicity among the following nitrogen compounds is

If \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined by \(f(x)=x+2|x+1|+2|x-1|\), then the element in the co-domain, which has unique pre image in the domain is

A wire loop enclosing a semi-circle of radius $R$is located on the boundary of a uniform magnetic field of induction $\overrightarrow{B}.$At time $t=0$, the loop is set into rotation with velocity $\omega$? about its axis $0$, coinciding with a line of vector $\overrightarrow{B}$ on the boundary as shown in the figure. The emf induced in the loop is

If \(y=\tan ^{-1}(\sin \sqrt{x})+\operatorname{cosec}^{-1}\left(e^{2 x+1}\right)\), then \(\frac{d y}{d x}=\)

An empty tank has concave mirror as its bottom. When sunlight falls normally on the mirror, it is focussed at a height of \(32 \mathrm{~cm}\) from the mirror. If the tank is filled with water upto a height of \(20 \mathrm{~cm}\), then the sunlight focusses at (refractive index of water \(=\frac{4}{3}\) )

\(\int \frac{2 \sin x-3 \cos x}{4 \cos x-3 \sin x} d x=\)