If f(x) =xn, n being a non-negative integer, then the values of n

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21.

The differential equation for the family of curves,x2 +y2 -2ay = 0  where a is an arbitrary constant is

  • 2(x2-y2)y' = xy

  • (x2+y2)y' = xy

  • 2(x2+y2)y' = xy

  • 2(x2+y2)y' = xy

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22.

The solution of the differential equation ydx + (x + x2y) dy = 0 is

  • -1/ XY  =C

  • -1/XY + log y = C

  • 1/XY + log y = C

  • 1/XY + log y = C

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23.

If i=19 (xi -5) = 9 and i = 19(xi - 5)2 = 45 then the
standard deviation of the 9 items x1, x2, ...., x9 is

  • 3

  • 9

  • 4

  • 2


24.

Let f(x) = xpsinx4, if 0 < x  π20              , if x = 0

(p, q  R). Then,  Lagrange's mean value theorem is applicable tof(x) in closed interval [ 0, x]

  • for all p, q

  • only when p > q

  • only when p < q

  • for no value of p, q


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25.

limx0sinx2tanx is equal to

  • 2

  • 1

  • 0

  • does not exist


26.

Let for all x> 0, f(x)=limnnx1/n - 1, then

  • f(x) + f(1x) = 1

  • f(xy) = f(x) +f(y)

  • f(xy) = xf(y) +yf(x)

  • f(xy) = xf(x) + yf(y)


27.

The value of K in order that f (x) = sin(x) - cos(x) - Kx + 5 decreases for all positive real values of x is given by

  • K<1

  • K  1

  • K > 2

  • K < 2


28.

Let f : R ➔ R be twice continuously differentiable. Let f(0) = f(D) = f'(0) = 0. Then,

  • f''(x)  0 for all x

  • f''(c) = 0 for some c  R

  • f''(x)  0 if x  0

  • f'(x) > 0 for all x


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29.

If f(x) =xn, n being a non-negative integer, then the values of n for which f'(α + β) = f'(α) + f'(β) for all α, β > 0 is

  • 1

  • 2

  • 0

  • 5


B.

2

We have,

f(x) = xn

 f'(x) = nxn - 1

Now, f'(α + β) = f'(α) + f'(β)

 nα + βn - 1 = n - 1 + n - 1

 α + β = αn - 1 + βn - 1

From Given options we see that n = 2, satisfy the above equation.

 n = 2


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30.

If y = (1 + x)(1 + x2)(1 + x4)...(1 + x2n), then the value of dydx at x = 0 is

  • 0

  • - 1

  • 1

  • 2


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