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If OT is the perpendicular drawn from the origin to the tangent at any point t to the curve x = a cos³t, y = a sin³ t, then OT is equal to

(a) a sin 2t (b) (a/2) sin 2t (c) 2a sin 2t (d) 2a Solution: Given x = a cos3 t y = a sin3 t dx/dt = -3a cos2t sin t dy/dt... View Article

The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs Rs. 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to Rs. 300 per hour is

(a) 10 (b) 20 (c) 30 (d) 40 Solution: Let the speed of the train be v and distance to be covered be s so that the... View Article

The profit function, in rupees, of a firm selling x items (x≥ 0) per week is given by P(x) = – 3500 + (400 – x)x. How many items should the firm sell so that the firm has maximum profit?

(a) 400 (b) 300 (c) 200 (d) 100 Solution: Given P(x) = - 3500 + (400 - x)x = -3500 + 400x - x2 Differentiate... View Article

If f(x) = x ln x, then f(x) attains minimum value at which one of the following points?

(a) x = e–2 (b) x = e (c) x = e–1 (d) x = 2e–1 Solution: Given f(x) = x ln x Differentiate w.r.t.x... View Article

Area of the triangle formed by the normal to the curve x = e^{sin y} at (1, 0) with the coordinate axes is :

(a) 1/4 (b) 1/2 (c) 3/4 (d) 1 Solution: Given x = esin y Take log on both sides log x = sin y log e log x = sin... View Article

If the function y = (ax+b)/(x-1)(x-4) has turning point at P(2, -1), then

(a) a = b = 1 (b) a = b = 0 (c) a = 1, b = 0 (d) a = b = 2 Solution: Given y = (ax+b)/(x-1)(x-4) has turning point at P(2,... View Article

A curve is represented by the equation x = sec²t and y = cot t, where t is a parameter. If the tangent at the point P on the curve where t = π/4 meets the curve again at the point Q, then |PQ| is equal to

(a) 5√3/2 (b) 5√5/2 (c) 2√5/3 (d) 3√5/2 Solution: Given x = sec2t y = cot t dx/dt = 2 sec t sec t tan... View Article

A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is 90⁰. What is the maximum area which can be enclosed inside the quadrilateral?

(a) 68 cm2 (b) 70 cm2 (c) 71.25 cm2 (d) 72. 25 cm2 Solution: Let the length of the quadrilateral be x and breadth be y. So... View Article

How many tangents are parallel to x-axis for the curve y = x² – 4x + 3?

(a) 1 (b) 2 (c) 3 (d) No tangent is parallel to x-axis Solution: Given y = x2 - 4x + 3 dy/dx = 2x-4 Tangent is parallel to x axis.... View Article

The total number of parallel tangents of f₁(x)= x² – x + 1 and f₂(x) = x³ – x² -2x + 1 is

(a) 2 (b) 0 (c) 1 (d) infinite Solution: Given f1(x)= x2 - x + 1 ..(i) f2(x) = x3 - x2 -2x + 1 ..(ii) Differentiate (i)... View Article

f(x) = log(π+ x)/log (e+x) is

(a) increasing on [0, ∞) (b) decreasing on [0, ∞) (c) increasing on [0, π/e], decreasing on [π/e,... View Article

Find the minimum value of the function 40/(3x⁴ + 8x³ -18x² +60).

(a) 1/3 (b) 2/3 (c) 4/3 (d) 5/3 Solution: Let y = 40/(3x4 +8x3 -18x2 +60) For y to be minimum, the denominator should be maximum,... View Article

If the rate of change in volume of a spherical soap bubble is uniform, then the rate of change of the surface area varies as

(a) square of radius (b) square root of radius (c) inversely proportional to radius (d) cube of the radius Solution: Let r be the... View Article

A stone thrown vertically upward satisfies the equation s = 64t – 16t², where s is in meter and t is in second. What is the time required to reach the maximum height?

(a) 1s (b) 2s (c) 3s (d) 4s Solution: Given s = 64t - 16t2 Differentiate w.r.t.t ds/dt = 64 - 32t At maximum... View Article

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