Important 5m questions

11th Standard

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Maths

Time : 00:50:00 Hrs
Total Marks : 70

    Part A

    Answer all the questions

    5 x 1 = 5
  1. If \(\lambda \hat{i}+2\lambda \hat{j}+2\lambda \hat{k}\) is a unit vector, then the value of \(\lambda\)is

    (a)

    \({1\over3}\)

    (b)

    \({1\over4}\)

    (c)

    \({1\over9}\)

    (d)

    \({1\over2}\)

  2. If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

    (a)

    \(7\over4\)

    (b)

    \(15\over4\)

    (c)

    \(3\over4\)

    (d)

    \(17\over4\)

  3. \(lim_{x\rightarrow o}{8^x-4^x-2^x+1^x\over x^2}=\)

    (a)

    2 log 2

    (b)

    2( log2)2

    (c)

    log 2

    (d)

    3 log 2

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

    (a)

    \(1\over 2\)

    (b)

    0

    (c)

    1

    (d)

    \(\infty\)

  5. The function is not defined for x = −1. The value of ( 1) f − so that the function extended by this value is continuous is

    (a)

    \({2\over3}\)

    (b)

    -\({2\over3}\)

    (c)

    1

    (d)

    0

  6. Part B

    Answer all the questions

    5 x 2 = 10
  7. Let \(\overrightarrow{a}\) and \(\overrightarrow{b}\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \({\overrightarrow{a}+2\overrightarrow{b}\over 3} and \ {\overrightarrow{b}+2\overrightarrow{a}\over3}.\)

  8. Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\)-3\(\hat{k}\)+4\(\hat{j}\)

  9. Find \(\overrightarrow{a}\).\(\overrightarrow{b}\)when \(\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\) and \(\overrightarrow{b}=3\hat{i}-4\hat{j}-2\hat{k}\)

  10. Let 
    Verify the existence of limit as x\(\rightarrow\)0.

  11. Evaluate the following limits :
    \(lim_{x\rightarrow1}{x^m-1\over x^n-1}\) ,m and n are integers.

  12. Part C

    Answer all the questions

    10 x 3 = 30
  13. If \(\overrightarrow{PO}\) +\(\overrightarrow{OQ}\) = \(\overrightarrow{QO}\) +\(\overrightarrow{OR}\), prove that the points P, Q, R are collinear.

  14. Show that the vectors 2\(\hat{i}\)\(\hat{j}\)+3\(\hat{k}\),3 \(\hat{i}\) −4\(\hat{j}\) -4\(\hat{k}\),  \(\hat{i}\) −3\(\hat{j}\) -5 \(\hat{k}\) form a right angled triangle.

  15. If \(\overrightarrow{a}=2\hat{i}+3\hat{j}-4\hat{k},\) \(\overrightarrow{b}=3\hat{i}-4\hat{j}-5\hat{k},\) and \(\overrightarrow{c}=-3\hat{i}+2\hat{j}+3\hat{k},\)find the magnitude and direction cosines of  \(\overrightarrow{a}\)+ \(\overrightarrow{b}\) + \(\overrightarrow{c}\) 

  16. Calculate \(\lim _{ x\rightarrow0}{|x| } \).

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

  18. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow{0}}sec \ x\)

  19. Evaluate the following limits :
    \(lim_{x-1}{3\sqrt{7+x^3}-\sqrt{3+x^2}\over x-1}\)

  20. Evaluate the following limits :\(lim_{x\rightarrow 0}{2^x-3^x\over x}\)

  21. At the given point xo discover whether the given function is continuous or discontinuous citing the reasons for your answer :

  22. Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f. f(x)={\(\begin{matrix} { (x-1) }^{ 3 }, & if\quad x<0 \\ { (x+1) }^{ 3 }, & if\quad x\ge 0 \end{matrix}\)

  23. Part D

    Answer all the questions

    5 x 5 = 25
  24. Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that \(\overrightarrow{AD}\) + \(\overrightarrow{BE}\) +\(\overrightarrow{CF}\) = \(\overrightarrow{0}\).

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

  26. Evaluate : \(lim_{x \rightarrow {\pi\over 4}}{4\sqrt{2}-(cos \ x+sin \ x)^5\over 1-sin 2x}\)

  27. State how continuity is destroyed at x= x o for each of the following graphs.

  28. State how continuity is destroyed at x= x o for each of the following graphs.

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