Differential Calculus Model Question Paper

11th Standard

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Business Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. Let \(f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}\), then f(5) is

    (a)

    -1

    (b)

    2

    (c)

    5

    (d)

    7

  2. The graph of y = ex intersect the y-axis at

    (a)

    (0,0)

    (b)

    (1,0)

    (c)

    (0,1)

    (d)

    (1,1)

  3. Which one of the following functions has the property f (x) = \(f\left( \frac { 1 }{ x } \right) \)

    (a)

    \(f\left( x \right) =\frac { { x }^{ 2 }-1 }{ x } \)

    (b)

    \(f\left( x \right) =\frac { 1-{ x }^{ 2 } }{ x } \)

    (c)

    f(x) = x

    (d)

    \(f\left( x \right) =\frac { { x }^{ 2 }+1 }{ x } \)

  4. The graph of f(x) = ex is identical to that of

    (a)

    f(x) = ax, a > 1

    (b)

    f(x) = ax, a < 1

    (c)

    f(x) = ax, 0 < a < 1

    (d)

    y = ax +b, a \(\ne\) 0

  5. \(\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =\)

    (a)

    e

    (b)

    nx(n-1)

    (c)

    1

    (d)

    0

  6. 5 x 2 = 10
  7. Determine whether the following functions are odd or even?
    f(x)=x+x2

  8. Let f be defined by f(x) =x3-kx2+2x,x ∈ R,Find k,if 'f' is an odd function.

  9. For \(f(x)=\frac { x-1 }{ 3x+1 } \) ,write the expression of \(f\left( \frac { 1 }{ x } \right) \) and \(\frac { 1 }{ f(x) } \)

  10. Evaluate \(\underset { h\rightarrow 0 }{ lim } \frac { \sqrt { x+h } -\sqrt { x } }{ h } \)

  11. Differentiate (sec x -1) (sec x +1)

  12. 5 x 3 = 15
  13. Differentiate sin2 x with respect to x2.

  14. Evaluate the following \(\lim _{ x\rightarrow \infty }{ \frac { 2x+5 }{ { x }^{ 2 }+3x+9 } } \)

  15. Find y2 of the following function x = a cos \(\theta\), y = a sin \(\theta\)

  16. Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

  17. If ey (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

  18. 4 x 5 = 20
  19. Draw the graph of the following function f(x)=e2x

  20. Draw the graph of the following function f(x)=e-2x

  21. If \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 4 }-1 }{ x-1 } \end{matrix}=\begin{matrix} \underset { x\rightarrow k }{ lim } & \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \end{matrix}\),then find the value of K

  22. If \(x=a\left( t+\frac { 1 }{ t } \right) ;y=a\left( t-\frac { 1 }{ t } \right) \) show that \(\frac { dy }{ dx } =\frac { x }{ y } \)

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