Let exp (x) denote the exponential function ex. If f (x) = expx1x

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 Multiple Choice QuestionsMultiple Choice Questions

21.

let y = y(x) be the solution of the differential equation sin x dydx + y cos x = 4x, x (0, π). If y = π2 = 0, then yπ6 is equal to:

  • -49π2

  • 493π2

  • -893π2

  • -89π2


22.

If the line ax + by + c = 0, ab  0, is a tangent to the curve xy = 1- 2x, then

  • a> 0, b < 0

  • a>0, b> 0

  • a< 0, b > 0

  • a< 0, b < 0


23.

Time period T of a simple pendulum of length l is given by T = 2π1g. If the length is increased by 2%, then an approximate change in the time period is

  • 2 %

  • 1 %

  • 12 %

  • None of these


24.

The number of values of k, for which the equation x2 - 3x + k=0 has two distinct roots lying in the interval (0, 1), are

  • three

  • two

  • infinitely many

  • no value of k satisfies the requirement


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25.

The smallest positive root of the equation tan(x) - x = 0 lies in

  • 0, π2

  • π2, π

  • π, 3π2

  • 3π2, 2π


26.

Let y =  ex2 and y = ex2sinx be two given curves. Then, angle between the tangents to the curves at any point of their intersection is

  • 0

  • π

  • π2

  • π4


27.

Suppose that the equation f (x) = x2 + bx + c = 0 has two distinct real roots α and β. The angle between the tangent to the curve y = f (x) at the point α + β2, fα + β2 and the positive direction of the x-axis is

  • 30°

  • 60°

  • 90°


28.

The angle of intersection between the curves y = sinx + cosx and x2 + y2 = 10, where [x] denotes the greatest integer  x, is

  • tan-13

  • tan-1- 3

  • tan-13

  • tan-11/3


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29.

For the curve x2 + 4xy + 8y = 64 the tangents are parallel to the x-axis only at the points

  • 0, 22 and 0, - 22

  • (8, - 4) and (- 8, 4)

  • 82, - 22 and - 82, 22

  • (9, 0) and (- 8, 0)


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30.

Let exp (x) denote the exponential function ex. If f (x) = expx1x, x > 0, then the minimum value off in the interval [2, 5] is

  • expe1e

  • exp212

  • exp515

  • exp313


C.

exp515

Given that,

f(x) = ex1x, x > 0

Taking log on both sides, we get

       logf(x) = x1x = g(x)                ...(i)   Here, gx = x1x loggx = 1xlogxOn differentiating w.r.t. x, we get1gx . g'(x) = x . 1x - logxx2                  = 1 - logxx2     g'(x) = x1x - 21 - logxFor maximum or minimum of g(x) put         g'(x) = 0 x1x - 21 - logx = 0               logx = 1 = loge                              x = e

So, g(x) is minimum at z = e

 g(x)increases in (0, e)and decreases in e, , it will be minimum at either 2 or 5.

 212 > 515   Minimum value of f(x) = e515


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