If \(P\) is a point on the parabola \(y^2=8 x\) and \(A\) is the point \((1,0)\), then the locus of the mid-point of the line segment \(A P\) is

Normals are drawn from the point \(\mathrm{P}(8,0)\) to the parabola \(y^2=12 x\). If \(\theta\) is the acute angle between two nonhorizontal normals among them, then \(\tan \theta=\)

The term independent of \(x\) in the expansion of \(\left(1-3 x+2 x^3\right)\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9\) is

If \(A+B+C=\frac{\pi}{4}\), then \(4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}-\cos \frac{\pi}{8}=\)

The points whose position vectors are \(2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, 3 \mathbf{i}+4 \mathbf{j}+2 \mathbf{k}\) and \(4 \mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\) are the vertices of

If \(y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)+\tan ^{-1}\left(\frac{4 x-4 x^3}{1-6 x^2+x^4}\right)\) then \(\frac{d y}{d x}\) is equal to

If a balloon lying at an altitude of 30 m from an observer at a particular instant is moving horizontally at the rate of \(1 \mathrm{~m} / \mathrm{s}\) away from him, then the rate at which the balloon is moving away directly from the observer at the \(40^{\text {th }}\) second is (in \(\mathrm{m} / \mathrm{s}\) )

A long straight wire carries a current of \(18 \mathrm{~A}\). The magnitude of the magnetic field at a point \(12 \mathrm{~cm}\) from it is

The inverse point of \((1,2)\) with respect to the circle \(x^2+y^2-4 x-6 y+9=0\) is

' \(\mathrm{x}\) ' \(\mathrm{g}\) of urea (molar mass \(60 \mathrm{gmol}^{-1}\) ) is completely dissolved in ' \(\mathrm{y}\) ' \(\mathrm{g}\) of pure water and the solution boiled at \(373.202 \mathrm{~K}\). If the boiling point of pure water at \(1.013 \mathrm{bar}\) is \(373.15 \mathrm{~K}\), then \(\mathrm{x}: \mathrm{y}\) is \(\left(\mathrm{K}_{\mathrm{b}}\left(\mathrm{H}_2 \mathrm{O}\right)=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\)

A force separately produces accelerations of \(18 \mathrm{~ms}^{-2}, 9 \mathrm{~ms}^{-2}\) and \(6 \mathrm{~ms}^{-2}\) in three bodies of masses \(\mathrm{P}, \mathrm{Q}\) and R respectively. If the same force is applied on a body of mass \(\mathrm{P}+\mathrm{Q}+\mathrm{R}\), then the acceleration of that body is

\(\int \frac{\sin x \cdot \sec ^2 x-\tan x \cdot \sin x+\cos x}{(1-\cos 2 x)} d x=\)

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)