\(\frac{d}{d x}\left[x^{\sin x}+(\sin x)^x\right]=\)

\(\cos \theta(\operatorname{cosec} \theta-\sec \theta)-\cot \theta=\)

Match the following
List - I
A) The equation of line passing through \((4,3)\) whose \(\mathrm{X}\)-intercept is twice its Y-intercept
B) The equation of the line passing through the centroid and circumcentre of \(\triangle A B C\) with vertices \(\mathrm{A}(1,1), \mathrm{B}(3,3), \mathrm{C}(6,-6)\)
C) The equation of the line whose \(\mathrm{X}\)-intercept is \((-3 / 5)\)
III) \(x+2 y+\sqrt{2}=0\) and is perpendicular to \(x-y+2=0\)
D) The equation of the line whose distance from the origin is 2 and the normal drawn from the origin makes an angle \(45^{\circ}\) with the positive direction of \(\mathrm{X}\)-axis
List - II
I) \(x+y-2 \sqrt{2}=0\)
II) \(7 x+23 y-8=0\)
IV) \(x+2 y-10=0\)
IV) \(x+2 y-10=0\)
V) \(5 x+5 y+3=0\)
The correct answer is

\(\frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\ldots\)
\(+\frac{1}{\sin 89^{\circ}+\sin 90^{\circ}}=\)

The vector \(\mathbf{c} .(\mathbf{b}+\mathbf{c}) \times(\mathbf{a}+\mathbf{b}+\mathbf{c})\) is equal to

The square of the distance from the origin to the point of intersection of the pair of lines \(a x^2+2 h x y-a y^2+2 g x+\) \(2 \mathrm{fy}+\mathrm{c}=0\) is

\(\int_{\frac{1}{25}}^3 \frac{e^{\frac{3}{x}}}{x^2} d x=\)

If \(c\) is a parameter, then the differential equation of the family of curves \(x^2=c(y+c)^2\) is

If a set of 30 games cards, 17 are white and rest are green. Out of 30,4 white and 5 green are marked IMPORTANT. If a card is chosen randomly from this set, then possibility of choosing a green card or an 'IMPORTANT' card is

In a potentiometer experiment, when a cell in the secondary circuit is shunted by a resistance ' \(R\) ', balancing length is ' \(L_1\) '. On doubling the shunt resistance, balancing length increases to ' \(L_2\) '. Internal resistance of the cell is

The kinetic energy of a body is increased by 4 times. Its momentum will

Identify the incorrect statements among the following?
(i) ${\mathrm{SF}}_6$ does not react with water
(ii) ${\mathrm{SF}}_6$ is ${\mathrm{sp}}^3\mathrm{d}$ hybridized
(iii) ${\mathrm{S}}_2{\mathrm{O}}_3^{2-}$ is a linear ion
(iv) There is no $\mathrm{\pi }$-bonding in ${\mathrm{SO}}_4^{2-}$ ion

If \(P\) divides the line segment joining the points \(A(1,2,-1)\) and \(B(-1,0,1)\) externally in the ratio \(1: 2\) and \(Q=(1,3,-1)\), then \(P Q=\)