The angle at which the curve y = x2 and the curve x = 53cost

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 Multiple Choice QuestionsMultiple Choice Questions

181.

The maximum value of fx = logxxx  0, x  1 is

  • e

  • 1e

  • e2

  • 1e2


182.

If the volume of spherical ball is increasing at the rate of 4π cm3/s, then the rate of change of its surface area when the volume is 288 π cm3, is

  • 43π cm2/s

  • 23π cm2/s

  • 4π cm2/s

  • 2π cm2/s


183.

The equation of displacement of a particle is x(t) = 5t2 - 7t + 3. The acceleration at the moment when its velocity becomes 5 m/sec is

  • 3 m/sec2

  • 7 m/sec2

  • 10 m/sec2

  • 8 m/sec2


184.

The mean value of the function fx = 2ex + 1 on the interval [0, 2] is

  • 2 - loge2e2 + 1

  • 2 + loge2e2 + 1

  • 2 + loge2e2 - 1

  • - 2 + loge2e2 - 1


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185.

The function y = 2x - x2

  • increases in (0, 1) but decreases in (1, 2)

  • decreases in (0, 2)

  • increases m (1, 2) but decreases in (0, 1)

  • increases in (0, 2)


186.

The interval in which the function y = x - 2sinx0  x  2π increases throughout is

  • 5π3, 2π

  • 0, π3

  • π3, 5π3

  • 0, π4


187.

The points of the curve y = x3 + x - 2 at which its tangent are parallel to the straight line y = 4x - 1 are

  • (2, 7), (- 2, - 11)

  • (0, 2), (21/3, 21/3)

  • (- 21/3, - 21/3), (0, - 4)

  • (1, 0), (- 1, - 4)


188.

The equation of the normal to the curve y = - x + 2 at the point of its intersection with the bisector of the first quadrant is

  • 4x - y + 16 = 0

  • 4x - y = 16

  • 2x - y - 1 = 0

  • 2x - y + 1 = 0


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189.

The angle at which the curve y = x2 and the curve x = 53cost, y = 54sint intersect is

  • tan-1241

  • tan-1412

  • - tan-1241

  • 2tan-1412


B.

tan-1412

Given, y = x2     ...ix = 53cost,y = 54sint        ...ii

Which is parametnc equation, we change this equation is cartesian equation as follows

cost = 35x, sint = 45y

On squaring and adding both i.e. cos(t) and sin(t), we get

925x2 + 1625y2 = cos2t + sin2t 9x2 + 16y2 = 25             ...iii           cos2θ+ sin2θ = 1

 The intersection points at Eq. (i) and (iii) are (1, 1) and (- 1, 1)

Now, slope of tangent of Eq. (i) at point (1, 1) is

     m1 = dydx=2x m1 = dydx1, 1 = 2

And slope of tangent of Eq (iii), at point (1, 1) is

m2 = dydx = - 916

 Angle at point of intersection of Eqs. (i) and (iii), we get

θ1 = tan-1m1 - m21 + m1m2θ1 = tan-12 + 9161 - 2 × 916 = tan-1412

Similarly, slope of tangent of Eq. (i) at point (- 1, 1)

m1 = dydx- 1, 1 =-2

And slope of tangent of Eq (iii) at point (-1, 1)

m2 = dydx = - 916

 Angle at point of intersection of Eqs. (i) and (iii), we get

θ2 = tan-1- 2 - 9161 - 1816 = tan-1412

 


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190.

The maximum value of the function y = 2tanx - tan2x over 0, π2 is

  • 1

  • 3

  • 2


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