Part-II Graduation Standard: MCQ — 268 Practice Questions with Answers

Explanation

For a right circular cone with vertex at the origin and axis along the x-axis, the perpendicular distance from the axis satisfies y^2 + z^2 = x^2 tan^2(alpha), where alpha is the semi-vertical angle. With alpha = 30 degrees, tan^2(alpha) = 1/3. Thus 3(y^2 + z^2) = x^2, or x^2 - 3y^2 - 3z^2 = 0.

Explanation

Acceleration is the time derivative of velocity. Since v = t^2 + 3t, a = dv/dt = 2t + 3; putting t = 2 gives a = 4 + 3 = 7 m/s^2.

Explanation

For a projectile landing at the same level, the horizontal range is R = u^2 sin 2theta/g. Here u = 20 m/s and theta = 30 degrees, so R = 400 sin 60 degrees/10 = 40(sqrt(3)/2) = 20 sqrt(3) m.

Explanation

By Gauss' divergence theorem, outward flux through the closed sphere equals triple integral over the ball of div F. Since div(xi+yj+zk)=3 and the unit ball has volume 4pi/3, the flux is 4pi.

Explanation

For x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0, the centre is (-u, -v, -w) and r^2 = u^2 + v^2 + w^2 - d. Here u = -2, v = 3, w = -4, d = -11. Hence C = (2, -3, 4) and r^2 = 4 + 9 + 16 + 11 = 40.