31.

If 64, 27, 36 are the Pth Qth and Rth terms of a GP, then P + 2Q is equal to

• R

• 2R

• 3R

• 4R

If ${\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}{\sin }^{-1}\left(\mathrm{y}\right) + {\sin }^{-1}\left(\mathrm{z}\right) = \frac{3\mathrm{\pi }}{2}$, then the value of x⁹ + y⁹ + z⁹ - $\frac{1}{{\mathrm{x}}^9{\mathrm{y}}^9{\mathrm{z}}^9}$ is equal to

• 0

• 1

• 2

• 3

Let p, q and r be the sides opposite to the angles P, Q and R, respectively in a $∆$PQR. If r²sin(P)sin(Q) = pq, then the triangle is 

• equilateral

• acute angled but not equilateral

• obtuse angled

• right angled

Let p, q and r be the side4s opposite to the angles P, Q and R, respectively in a $∆$PQR. Then, 2pr$\sin \left(\frac{\mathrm{P} - \mathrm{Q} + \mathrm{R}}{2}\right)$ equals

• p² + q² + r²

• p² + r² - q²

• q² + r² - p²

• p² + q² - r²

Let P (2,-3) , Q -2 1) be the vertices of the $∆$PQR. If the centroid of $∆$PQR lies on the line 2x + 3y = 1, then the locus of R is

• 2x + 3y = 9

• 2x - 3y = 7

• 3x + 2y = 5

• 3x - 2y = 5

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{\pi }}^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}} - 1}$

• does not exist

• equals log_e(${\mathrm{\pi }}^2$)

• equals 1

• lies between 10 and 11

Let P be the mid-point of a chord joining the vertex of the parabola y² = 8x to another point on it. Then, the locus of P is

• y² = 2x

• y² = 4x

• $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$

• ${\mathrm{x}}^2 +\hspace{0.17em}\frac{{\mathrm{y}}^2}{4}$ = 1

The line x = 2y intersects the ellipse $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$ at the points P and Q. The equation of the circle with PQ as diameter is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \frac{1}{2}$

• x² + y² = 1

• x² + y² = 2

• x² + y² = $\frac{5}{2}$

The eccentric angle in the first quadrant of a point on the ellipse $\frac{{\mathrm{x}}^2}{10} + \frac{{\mathrm{y}}^2}{8} = 1$  at a distance units from the centre of the ellipse is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The transverse axis of a hyperbola is along the x - axis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

• 6x² - y² = 3a²

• x² - 3y² = 3a²

• x² - 6y² = 3a²

• 3x² - y² = 3a²