Model Questions-Differential Calculus

11th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 60

    Part A 

    Answer all the questions

    10 x 1 = 10
  1. \(lim_{x\rightarrow\infty}{sin \ x \over x} \)

    (a)

    1

    (b)

    0

    (c)

    \(\infty\)

    (d)

    -\(\infty\)

  2. \(lim_{x\rightarrow {\pi/2}}{2x-\pi\over cosx} \)

    (a)

    2

    (b)

    1

    (c)

    -2

    (d)

    0

  3. \(lim_{x \rightarrow \infty}{\sqrt{x^2-1}\over 2x+1}=\)

    (a)

    1

    (b)

    0

    (c)

    -1

    (d)

    \(1\over 2\)

  4. If f(x)=x(-1)\(\left\lfloor 1\over x \right\rfloor \),\(x\le0\),then the value of \(lim_{x\rightarrow 0}f(x)\) is equal to

    (a)

    -1

    (b)

    0

    (c)

    2

    (d)

    4

  5. \(lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

    (a)

    \(1\over 2\)

    (b)

    0

    (c)

    1

    (d)

    \(\infty\)

  6. \(lim_{x \rightarrow 0}{e^{tan \ x}-e^x\over tan x-x}=\)

    (a)

    1

    (b)

    e

    (c)

    \({1\over2}\)

    (d)

    0

  7. The value of \(lim_{x \rightarrow 0}{sin x\over \sqrt{x^2}}\) is

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

    \(\infty\)

  8. The value of \(lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor \)where k is an integer is

    (a)

    -1

    (b)

    1

    (c)

    0

    (d)

    2

  9. At x\(={3\over 2}\) the function \(f(x)={|2x-3|\over 2x-3}\) is

    (a)

    continuous

    (b)

    discontinuous

    (c)

    differentiable

    (d)

    non-zero

  10. Let a function f be defined by \(f(x)={x-|x|\over x}\) for x \(\neq\)0 and f(0)=2. Then f is

    (a)

    continuous nowhere

    (b)

    continuous everywhere

    (c)

    continuous for all x except x = 1

    (d)

    continuous for all x except x = 0

  11. Part B 

    Answer all the questions

    10 x 2 = 20
  12. Consider the function f(x) = \(\sqrt{x},x\ge0.\)
    Does\(lim_{x\rightarrow0}f(x)\) exist?

  13. Complete the table using calculator and use the result to estimate the limit.
    \(lim_{x\rightarrow 2}{x-2\over x^2-x-2}\)

    x 1.9 1.99 1.999 2.001 2.01 2.1
    f(x)            
  14. Complete the table using calculator and use the result to estimate the limit.
    \(lim_{x\rightarrow{-3}}{\sqrt{1-x}-2\over x+3}\)

    x -3.1 -3.01 -3.00 -2.999 -2.99 -2.9
    f(x)            
  15. Complete the table using calculator and use the result to estimate the limit.
    \(lim_{x\rightarrow0}{cos x-1\over x}\)

    x -0.1 -0.01 -0.001 0.001 0.01 0.1
    f(x)            
  16. Suppose that the diameter of an animal’s pupils is given by \(f(x)={160x^{-0.4}+90\over 4x^{-0.4}+15},\) where x is the intensity of light and f(x) is in mm. Find the diameter of the pupils with minimum light.

  17. Find the left and right limits of \(f(x)={x^2-4\over (x^2+4x+4)(x+3)}at \ x=-2\) .

  18. Evaluate the following limits :\(lim_{x\rightarrow \infty}(1+{k\over x})^{m\over x} \)

  19. Determine if f defined by 

  20. Examine the continuity of the following: e2x + x2

  21. Find the points of discontinuity of the function f, where f(x)={\(\begin{matrix} { x }^{ 3 }-3, & if\quad x\le 2 \\ { x }^{ 2 }+1, & if\quad x>2 \end{matrix}\)

  22. Part C 

    Answer all the questions

    5 x 3 = 15
  23. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow3}(4-x)\).

  24. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow1}(x^2+2)\)

  25. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow2}f(x)\)

  26. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow1}f(x)\)

  27. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow3}{1\over x-3}\)

  28. Part D 

    Answer all the questions

    3 x 5 = 15
  29. Sketch the graph of f, then identify the values of x0 for which \(lim_{x\rightarrow{x_o}}f(x)\) exists.

  30. Calculate \(lim_{x\rightarrow0}{1\over (x^2+x^3)}\)

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