#### Important 5m questions

11th Standard

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Maths

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

Part A

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

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

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

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.