If f(x) and g(x) are twice differentiable functions on (0, 3) sat

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.
Advertisement

 Multiple Choice QuestionsMultiple Choice Questions

51.

Let f(x) = 0x1 - tdt,    x > 0x - 12,         x  1. Then

  • f(x) is continuous at x = 1

  • f(x) is not continuous at x = 1

  • f(x) is differentiable at x = 1

  • f(x) is not differentiable at x = 1


52.

If f(x) = x3 - 3x + 2,                      x < 2,x3 - 6x2 + 9x + 2,           x  2

then

  • limx2f(x) does not exist

  • f is not continuous at x = 2

  • f is continuous but not differentiable at x = 2

  • f is continuous and differentiable at x = 2


53.

Let y = 3x - 13x + 1sinx + loge1 + x, x > - 1. Then, at x = 0, dydx equals

  • 1

  • 0

  • - 1

  • - 2


54.

If f is a real-valued differentiable function such that f(x)f' (x) < 0 for all real x, then

  • f(x) must be an increasing function

  • f(x) must be a decreasing function

  • f(x) must be an increasing function

  • f(x) must be a decreasing function


Advertisement
55.

Rolle's theorem is applicable in the interval [- 2, 2] for the function

  • f(x) = x3

  • f(x) = 4x4

  • f(x) = 2x3 + 3

  • f(x) = πx


56.

The solution of 25d2ydx2 - 10dydx + y = 0, y(0) = 1, y(1) = 2e15 is

  • y = e5x + e- 5x

  • y = 1 + xe5x

  • y = 1 + xex5

  • y = 1 + xe- x5


Advertisement

57.

If f(x) and g(x) are twice differentiable functions on (0, 3) satisfying f"(x) = g''(c), f'(1) = 4g'(D) = 6, f(2) = 3, g(2) = 9, then f(1) - g(1) is

  • 4

  • - 4

  • 0

  • - 2


B.

- 4

According to question,

  f"(x) = g"(x)

Integrating w.r.t. x, we get

   f'(x) = g'(x) + C1

Put   x = 1 ⇒ f'(1) = g'(1) + C1

    4 = 6 + C,

   C1 = - 2

 f'(x) = g'(x) - 2

Again, integrating w.r.t.x, we get

     fx = gx - 2x + C2    3 = 9 - 4 + C2  C2 = - 2 fx = gx - 2x - 2Put   x = 1, we getf1 - g1 = - 21 - 2 = - 4


Advertisement
58.

For function f(x) = ecosx, Rolle's theorem is

  • applicable, when π2  x  3π2

  • applicable, when 0  x  π2

  • applicable, when 0  x  π

  • applicable, when π4  x  π2


Advertisement
59.

f(x) = 0,            x = 0x - 3,     x > 0 the function f(x) is

  • increasing when x  0

  • strictly increasing when x > 0

  • strictly increasing at x = 0

  • not continuous at x = 0 and so it is not increasing when x > 0


60.

The function f(ax) = ax + b is strictly increasing for all real x, if

  • a > 0

  • a < 0

  • a = 0

  •  0


Advertisement