#### Revision Model Question Paper 2

11th Standard

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Maths

Time : 03:00:00 Hrs
Total Marks : 90

Part I

Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

20 x 1 = 20
1. For any four sets A, B, C and D, which of the following is not true?

(a)

A x C   B x D

(b)

(A x B) ∩ (C x D) = (A ∩ C) x (B ∩ D)

(c)

A x (B U,C) = (A x B) U (A x C)

(d)

A x (B ∩ C) = (A x B) ∩ (A x C)

2. The number of roots of (x+3)4+(x+5)4=16 is

(a)

4

(b)

2

(c)

3

(d)

0

3. If tanx=$\frac { -1 }{ \sqrt { 5 } }$ and x lies in the IV quadrant, then the value of cosx is

(a)

$\sqrt { \frac { 5 }{ 6 } }$

(b)

$\frac { 2 }{ \sqrt { 6 } }$

(c)

$\frac { 1 }{ 2 }$

(d)

$\frac { 1 }{ \sqrt { 6 } }$

4. In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

(a)

125

(b)

124

(c)

64

(d)

63

5. The coefficient of x5 in the series e-2x is

(a)

$\frac { 2 }{ 3 }$

(b)

$\frac { 2 }{ 3 }$

(c)

$\frac { -4 }{ 15 }$

(d)

$\frac { 4 }{ 15 }$

6. $\left(1+\frac{1}{\lfloor2}+\frac{1}{\lfloor4}+\frac{1}{\lfloor6}+...\right)^2-\left(1+\frac{1}{\lfloor3}+\frac{1}{\lfloor5}+\frac{1}{\lfloor7}+...\right)^2=$

(a)

1

(b)

2

(c)

e

(d)

2e

7. AB = 12 cm. AB slides with A on x-axis, B on y-axis respectively. Then the radius of the circle which is the locus of ΔAOB, where O is origin is:

(a)

36

(b)

4

(c)

16

(d)

9

8. The co-ordinates of a point on x+y+3=0 whose distance from x+2y+2=0 is $\sqrt 5$, is

(a)

(9,6)

(b)

(-9,6)

(c)

(6,-9)

(d)

(-9,-6)

9. If aij =${1\over2}(3i-2j)$ and A=[aij]2x2 is

(a)

$\begin{bmatrix} {1\over 2}& 2 \\ -{1\over2} & 1 \end{bmatrix}$

(b)

$\begin{bmatrix} {1\over 2}& -{1\over2} \\ 2& 1 \end{bmatrix}$

(c)

$\begin{bmatrix} 2& 2\\ {1\over 2}& -{1\over2} \end{bmatrix}$

(d)

$\begin{bmatrix} -{1\over 2}& {1\over2} \\ 1& 2 \end{bmatrix}$

10. If A=$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ then A2 is equal to

(a)

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

(b)

$\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$

(c)

$\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}$

(d)

$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

11. If (1, 2, 4) and (2, - 3$\lambda$, - 3) are the initial and terminal points of the vector $\hat{i}+\hat{j}+7\hat{k}$then the value of $\lambda$ is equal to

(a)

$7\over 3$

(b)

-$7\over 3$

(c)

-$5\over 3$

(d)

$5\over 3$

12. If m $\left( \overset { \rightarrow }{ 2 } +\overset { \rightarrow }{ j } +\overset { \rightarrow }{ k } \right)$ is a unit vector then the value of m is

(a)

$\pm \frac { 1 }{ \sqrt { 3 } }$

(b)

$\pm \frac { 1 }{ \sqrt { 5 } }$

(c)

$\pm \frac { 1 }{ \sqrt { 6 } }$

(d)

$\pm \frac { 1 }{ {2 } }$

13. $lim_{x \rightarrow 0}{e^{sin \ x}-1\over x}=$

(a)

1

(b)

e

(c)

${1\over e}$

(d)

0

14. Find the odd one out of the following

(a)

|x|

(b)

sin x

(c)

cos x

(d)

$\frac{1}{x}$

15. If y=cos (sin x2),then ${dy\over dx}$ at x= $\sqrt{\pi\over 2}$ is

(a)

-2

(b)

2

(c)

$-2\sqrt{\pi\over 2}$

(d)

0

16. Choose the correct or the most suitable answer from the given four alternatives.
$If\quad y=\log { \left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) } then\quad \frac { dy }{ dx } \quad is$

(a)

$\frac { 4{ x }^{ 3 } }{ 1-{ x }^{ 4 } }$

(b)

$-\frac { 4x }{ 1-{ x }^{ 4 } }$

(c)

$\frac { 1 }{ 4-{ x }^{ 4 } }$

(d)

$\frac { -4{ x }^{ 3 } }{ 1-{ x }^{ 4 } }$

17. The gradient (slope) of a curve at any point (x, y) is${x^2-4\over x^2}$If the curve passes through the point (2, 7), then the equation of the curve is

(a)

$y=x+{4\over x}+3$

(b)

$y=x+{4\over x}+4$

(c)

y=x2+3x+4

(d)

y=x2-3x+6

18. $\int { { 3 }^{ x+2 } }$ dx = __________+c.

(a)

$\frac { { 3 }^{ x } }{ log3 }$

(b)

$9\left( \frac { { 3 }^{ x } }{ log3 } \right)$

(c)

$\frac { { 3 }^{ x } }{ 9log3 }$

(d)

3x.9

19. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. The probability that the second ball drawn is red will be

(a)

$5\over 12$

(b)

$1\over 2$

(c)

$7\over 12$

(d)

$1\over 4$

20. Assertion (A) : In rolling die, getting number
Reason (R) : In a die contains only numbers 1,2,3,4,5,6

(a)

Both (A) and (R) are true and (R) is the correct rexplanation of (A)

(b)

Both (A) and (R) are true but (R) is not the correct explantion of (A)

(c)

(A) is true (R) is false

(d)

(A) is false (R) is true

21. Part II

Answer any 7 questions. Question no. 30 is compulsory.

7 x 2 = 14
22. Simplify $\sqrt{x^2-10x+25}$

23. Find n if (n+1)! = 12$\times$(n-1)!

24. Evaluate: 8P4.

25. Find a negative value of m if the Co-efficient of x2 in the expansion of (1+x)m ,|x|<1 is 6

26. If 9x2 + 12xy + 4y2 + 6x + 4y - 3 = 0 represents two parallel lines, find the distance between them.

27. Using properties of determinant, show that$\triangle =\left| \begin{matrix} { cosec }^{ 2 }\theta & -{ cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & -cose{ c }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0$

28. Evaluate $\lim _{ x\rightarrow a }{ \frac { \sqrt { x } +\sqrt { a } }{ x+a } }$

29. Differentiate $\sin { (\sqrt { 3 } \sin { x } +\cos { x } ) }$ with respect to x.

30. Integrate the function with respect to x
x11

31. Events A and B are such that P(A) = $\frac { 1 }{ 2 }$ , P(B) = $\frac { 7 }{ 12 }$ and P(not A or not B) = $\frac { 1 }{ 4 }$. State whether A and B are independent?

32. Part III

Answer any 7 questions. Question no. 40 is compulsory.

7 x 3 = 21
33. Consider the functions:(i) y = ex; (ii) y = logeX.

34. Find the value of log2 $\left({{\sqrt [ 3 ]{4 } }\over{4^2\sqrt{8}}} \right).$

35. Prove that$1+cos2x+cos4x+cos6x=4cosx\quad cos2x\quad cos3x$

36. A question paper has two parts A and B, each containing 10 questions. If a student has to choose 8 from part A, 5 from Part B, in how many ways can he choose the questions?

37. Expand ${\left( 2x-{1\over 2x} \right)}^{4}.$

38. If $\theta$  is the parameter, find the equation of the locus of a moving point, whose co-ordinates are x=a cos3 $\theta$ ; y = without$\theta$ .

39. Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$be unit vectors such that$\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}=0$ and the angle between $\overrightarrow{b} \ and \ \overrightarrow{c}$ is  ${\pi\over 3}.$ Prove that $\overrightarrow{a}=\pm{2\over \sqrt{3}}(\overrightarrow{b}\times \overrightarrow{c}).$

40. Suppose $f(x)=\{ \begin{matrix} a+bx, & x<1 \\ 4, & x=1 \\ b-ax & x>1 \end{matrix}$ and,if $\underset { x\rightarrow 1 }{ lim } f(x)=f(1)$ .What are possible values of a and b?

41. Evaluate $\int { \frac { { e }^{ -x } }{ 16+9{ e }^{ -2x } } }$dx

42. In a box containing 10 bulbs, 2 ae defective. What is the probability that among 5 bulbs chosen at random, none is defective?

43. Part IV

7 x 5 = 35
1. Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

2. Evaluate the integral $\int {5x-7\over \sqrt{3x-x^2-2}}dx$

1. Evaluate $\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \left( \frac { \pi }{ 2 } -x \right) \tan { x } }$

2. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver?

1. Using the mathematical induction, show that for any natural number n,$\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ n+1 }$.

2. If $A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right]$ and $B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right]$ and (A + B)2=A2 + B2, find X and.

1. Find the second order derivative if x and y are given by
x = a cos t
y = a sin t.

2. Prove that ap + q = 0 if f(x) = x3 - 3px + 2q is divisible by g(x) = x2 + 2ax + a2.

1. Show that the following vectors are coplanar 5$\hat{i}$ +6$\hat{j}$ +7$\hat{k}$ ,7 $\hat{i}$ -8$\hat{j}$ +9 $\hat{k}$,3$\hat{i}$+20$\hat{j}$ +5$\hat{k}$ .

2. From the curve y=sin x, graph the functions.
(i) y=sin(-x)
(ii) y=-sin(-x)
(iii) $y=sin\left( {\pi\over 2}+x\right)$ which is cos x
(iv) $y=sin\left({\pi\over 2}-x \right)$​ which is also cos x (refer trigonometry)

1. Find p and q, if the following equation represents a pair of perpendicular lines 6x2 + 5xy - py2 + 7x + qy - 5 = 0.

2. If x = 1 is one root of the equation x3 - 6x2 + 11x - 6 = 0, find the other roots.

1. If n is a postive integer, show that 9n+1 - 8n - 9 is always divisible by 64