Let f and g be functions from the interval [0, ∞) to the interval [0, ∞), f being an increasing and g being a decreasing function. If f{g(0)} = 0 then (a) f{g(x)} ≥ f{g(0)} (b) g{f (x)} ≤ g{f (0)} (c) f {g(2)} = 7 (d) None of these Solution: f '(x) > 0 if x ≥ 0... View Article
The largest area of a trapezium inscribed in a semicircle of radius R, if the lower base is on the diameter, is (a) 3√3R2/4 (b) (√3/2)R2 (c) (3√3/8)R2 (d) R2 Solution: ABCD is the trapezium. Centre of semicircle is O. ∠ADB = 900 (angle in a... View Article
If at any instant t, for a sphere, r denotes the radius, S denotes the surface area and V denotes the volume, then what is dV/dt equal to? (a) ½ S (dr/dt) (b) ½ r (dS/dt) (c) r dS/dt (d) ½ r2 dS/dt Solution: Surface area of sphere S = 4πr2... View Article
The function f(x) = 2 log (x – 2) – x2 + 4x + 1 increases on the interval (a) (1, 2) (b) (2, 3) (c) (5/2, 3) (d) (2, 4) Solution: Given f(x) = 2 log (x - 2) - x2 + 4x + 1 Differentiate... View Article
The straight line x/a + y/b = 2 touches the curve (x/a)n + (y/b)n = 2 at the point (a, b) for (a) n = 1, 2 (b) n = 3, 4, -5 (c) n = 1, 2, 3 (d) any value of n Solution: Given curve is (x/a)n + (y/b)n = 2. nxn-1/an +... View Article
The number of solutions of the equation 3 tan x + x3 = 2 in (0, π/4) is (a) 0 (b) 1 (c) 2 (d) 3 Solution: Let f(x) = 3 tan x + x3 - 2 Differentiate w.r.t.x => f’(x) = 3 sec2x + 3x2 > 0 f(x)... View Article
What is the minimum value of px + qy (p > 0, q > 0) when xy = r2 ? (a) 2r√(pq) (b) 2pq√r (c) - 2r√(pq) (d) 2 rpq Solution:Given xy = r2 => y = r2/x Let S = px +qy = px +... View Article
The curve y = xex has minimum value equal to (a) -1/e (b) 1/e (c) -e (d) e Solution: Given y = xex Differentiate w.r.t.x => dy/dx = ex + xex = ex(1+x) Put dy/dx = 0... View Article
The number of tangents to the curve x3/2 + y3/2 = 2a3/2, a > 0, which are equally inclined to the axes, is (a) 2 (b) 1 (c) 0 (d) 4 Solution: Given curve x3/2 + y3/2 = 2a3/2 ..(i) Differentiate w.r.t.x => (3/2)... View Article
The motion of a particle is described as s = 2 – 3t + 4t3. What is the acceleration of the particle at the point where its velocity is zero? (a) 0 (b) 4 unit (c) 8 unit (d) 12 unit Solution: Given s = 2 - 3t + 4t3 Velocity, V = ds/dt = -3 + 12t2... View Article
The function f (x) = x2/ex monotonically increasing if (a) x < 0 only (b) x > 2 only (c) 0 < x < 2 (d) x ∈ (-∞, 0) ⋃ (2, ∞) Solution: Given f(x) =... View Article
The cost of running a bus from A to B is Rs. (av + b/v) where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/ h) of the bus is : (a) 45 (b) 50 (c) 60 (d) 40 Solution: Given that cost, C = av+b/v When the bus travels at 30 km/h, the cost comes... View Article
If the curve y = ax2 – 6x + b passes through (0, 2) and has its tangent parallel to the x-axis at x = 3/2 then (a) a = b = 0 (b) a = b = 1 (c) a = b = 2 (d) a = b = -1 Solution: y = ax2 - 6x + b passes through (0, 2). => 2 = 0 -... View Article
