The length of the longest size rectangle of maximum area that can

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 Multiple Choice QuestionsMultiple Choice Questions

131.

The length of the subtangent to the curve x2 + xy + y2 = 7 at (1, - 3) is :

  • 3

  • 5

  • 35

  • 15


132.

Twenty two metres are available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, the radius of the circle must be :

  • 4 m

  • 3 m

  • 6 m

  • 5 m


133.

The value of x for which the polynomial 2x3 - 9x2 + 12x + 4 is a decreasing function of x, is :

  • - 1 < x < 1

  • 0 < x < 2

  • x > 3

  • 1 < x < 2


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134.

The length of the longest size rectangle of maximum area that can be inscribed in a semicircle of radius 1, so that 2 vertices lie on the diameter, is :

  • 2

  • 2

  • 3

  • 23


A.

2

Let the length and breadth of the rectangle be l and b.

In OAB,      b = sinθand l = 2cosθ A = Area of rectangle        = lb = 2sinθcosθ        = sin2θ

On differentiating w.r.t. θ, we get

dA = 2cos2θ

Put dA = 0 for maxima or minima

 cos2θ = 0          θ = π4

               d2A2 = - 4sin2θd2A2θ = π4 < 0

  Function is maximum at θ = π4 Length of rectangle = 2cosθ = 2cosπ4                                    = 212 = 2


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135.

If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is :

  • proportional to s2

  • proportional to 1s2

  • proportional to s

  • a constant


136.

The maximum value of xy when x + 2y = 8 is :

  • 20

  • 16

  • 24

  • 8


137.

The function f(x) = tan-1(sin(x) + cos(x)), x > 0 is always an increasing function on the interval :

  • 0, π

  • 0, π2

  • 0, π4

  • 0, 3π4


138.

The radius of a cylinder is increasing at the rate of 3 m/s and its altitude is decreasing at the rate of 4 m/s. The rate of change of volume when radius is 4 m and altitude is 6 m, is :

  • 80 π cu m/s

  • 144 π cu m/s

  • - 80 π cu m/s

  • 64 π cu m/s


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139.

A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 emfs. The height of the upper end while it is descending at the rate of 4 emfs, is :

  • 43

  • 53

  • 52

  • 6 m


140.

The equation of the tangent to the curve y = 1 + xy + sin-1sin2x

  • x - y + 1 = 0

  • x + y + 1 = 0

  • 2x - y + 1 = 0

  • x + 2y + 2 = 0


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