MCQ
Part-II Graduation Standard: MCQ - Practice Questions with Answers
Solve 268 Part-II Graduation Standard: questions for RAS/RPSC preparation.
Practice questions
Q1A right circular cone has vertex at the origin, axis along the x-axis, and semi-vertical angle 30 degrees. Its equation is:
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.
Q2The velocity of a particle moving along a straight line is v = t^2 + 3t, where v is in m/s and t is in seconds. What is its acceleration at t = 2 s?
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.
Q3A projectile is fired with speed 20 m/s at an angle of 30 degrees above the horizontal from level ground. Neglecting air resistance and taking g = 10 m/s^2, what is its horizontal range?
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.
Q4For F=(x)i+(y)j+(z)k, the outward flux through the unit sphere x^2+y^2+z^2=1 is
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.
Q5The sphere x^2 + y^2 + z^2 - 4x + 6y - 8z - 11 = 0 has centre C and radius r. Which pair (C, r) is correct?
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.
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More questions
6Which condition is equivalent to a subgroup N of a group G being normal in G?
7A projectile is fired with speed 20 m/s at an angle of 30 degrees to the horizontal. Taking g = 10 m/s^2, its horizontal range on level ground is
8Which statement about convex sets is always true and is most directly used in linear programming theory?
9The polar curves r = a sin theta and r = a cos theta meet at a non-pole point. The angle between the two curves at that point is:
10The value of the integral integral_0^infinity x^(3/2)/(1 + x)^5 dx is:
11For a subset A of a group G, the normaliser N_G(A) is best described as which set?
12Let E = (0,1) union {2}. Which statement is correct in the usual topology of R?
13Let u = (x^2 + y^2 + z^2)^(3/2). The value of x(du/dx) + y(du/dy) + z(du/dz) is:
14For a subgroup N of a group G, which condition is necessary and sufficient for the set of cosets G/N to form a quotient group under (aN)(bN) = abN?
15The enveloping cylinder of the sphere x^2 + y^2 + z^2 = 25 whose generators are parallel to the z-axis is: