If the curves y² = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is

• 9/2

• 6

• 7/2

• 4

Let $\overrightarrow{u}$ be a vector coplanar with the vectors a $\overrightarrow{a} = 2\hat{i} + 3\hat{j} -\hat{k} and \overrightarrow{b} = \hat{j} +\hat{k}.$ $if \overrightarrow{u} is perpendicular to \overrightarrow{a} and \overrightarrow{u}.\overrightarrow{b} = 24, then \left|\overrightarrow{u}\right| is equal to:$

• 84

• 336

• 315

• 256

If L₁ is the line of intersection of the plane 2x – 2y + 3z – 2 = 0, x – y + z + 1 = 0 and L₂ is the line of intersection of the plane x + 2y – z – 3 = 0, 3x – y + 2z – 1 = 0, then the distance of the origin from the plane containing the lines L₁
and L₂ is :

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{4\sqrt{2}}$

• $\frac{1}{3\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{2}x}{1+2^x}dx is:$

• π/4

• π/8

• π/2

• 4π

25.

Let g(x) = cos x², f(x) = $\sqrt{x}$ and $\alpha , \beta (\alpha <\hspace{0.17em}\beta )$ be the roots of the quadrtic equation 18x² - 9πx + π² = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0 is

• $\frac{1}{2}\left(\sqrt{2-1}\right)$

• $\frac{1}{2}\left(\sqrt{3-1}\right)$

• $\frac{1}{2}\left(\sqrt{3+1}\right)$

• $\frac{1}{2}(\sqrt{3}-\sqrt{2})$

If sum of all the solutions of the equation $8 \cos x. \left(\cos \left(\frac{\pi }{6}+x\right).\cos \left(\frac{\pi }{6}-x\right)-\frac{1}{2}\right) = 1$ in [0,π] is kπ, then k is equal to:

• 20/9

• 2/3

• 13/9

• 8/9

The Integral$\int \frac{{\sin }^2 x {\cos }^2 x }{({\sin }^5 x + {\cos }^3 x {\sin }^2 x + {\sin }^3 x {\cos }^2 x + cos{ }^{5}x{)}^2}dx$ is equal to

(where C is a constant of integration)

• $\frac{-1}{1+ co{t}^3 x } + C$

• $\frac{1}{3(1 + {\tan }^3 x) } + C$

• $\frac{-1}{3(1 + {\tan }^3 x )} +C$

• $\frac{1}{1+ co{t}^3 x } + C$

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:

• 3/4

• 3/10

• 2/5

• 1/5

Let the orthocentre and centroid of a triangle be A(–3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is

• $\frac{3\sqrt{5}}{2}$

• $\sqrt{10}$

• $2\sqrt{10}$

• $3\sqrt{\frac{5}{2}}$

The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane, x + y + z = 7 is:

• $\sqrt{\frac{2}{3}}$

• $\frac{2}{\sqrt{3}}$

• 2/3

• 1/3