#### 11th First Revision Test

11th Standard

Reg.No. :
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Maths

Time : 02:30:00 Hrs
Total Marks : 90

I. Choose the best suitable answer:

20 x 1 = 20
1. If n((A x B) ∩(A x C)) = 8 and n(B ∩ C) = 2, then n(A) is

(a)

6

(b)

4

(c)

8

(d)

16

2. $n(A\cap B)=4$ and $(A\cup B)=11$ then $n(p(A\triangle B))$ is:

(a)

44

(b)

256

(c)

64

(d)

128

3. The value of ${ log }_{ 3 }\frac { 1 }{ 81 }$ is

(a)

-2

(b)

-8

(c)

-4

(d)

-9

4. The quadratic equation whose roots are tan75° and cot75° is:

(a)

x2+4x+ 1 =0

(b)

4x2-x+ 1 =0

(c)

4x2+ 4x - 1 = 0

(d)

x2 - 4x + 1 = 0

5. The numerical value of tan-11+tan-12+tan-13=

(a)

$\pi$

(b)

$\frac{\pi}{2}$

(c)

0

(d)

$\frac{\pi}{4}$

6. If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

(a)

45

(b)

40

(c)

10!

(d)

210

7. There is a letter lock with 3 rings each marked with 5 letters and do not know the keyword. The total number of attempts can be made to know the keyword is

(a)

35

(b)

53

(c)

124

(d)

5

8. The sum up to n terms of the series $\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 1 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +$....is

(a)

$\sqrt { 2n+1 }$

(b)

$\frac { \sqrt { 2n+1 } }{ 2 }$

(c)

$\sqrt { 2n+1 } -1$

(d)

$\frac { \sqrt { 2n+1 } -1 }{ 2 }$

9. Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2$\sqrt{2}$ is

(a)

x+y+2=0

(b)

x+y-2=0

(c)

$x+y-\sqrt{2}=0$

(d)

$x+y+\sqrt{2}=0$

10. The equation of the bisectors of the angle between the co-ordinate axes are

(a)

x+y=0

(b)

x-y=0

(c)

x$\pm$y=0

(d)

x=0

11. If $\left\lfloor . \right\rfloor$ denotes the greatest integer less than or equal to the real number under consideration and −1$\le$x < 0, 0 $\le$ y <1, 1$\le$ z <2, then the value of the determinant $\begin{vmatrix} \left\lfloor x \right\rfloor +1& \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor +1& \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor +1\end{vmatrix}$ is

(a)

$\left\lfloor z \right\rfloor$

(b)

$\left\lfloor y \right\rfloor$

(c)

$\left\lfloor x \right\rfloor$

(d)

$\left\lfloor x \right\rfloor$+1

12. The value of $\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right|$=0, where a, b, c are in AP is

(a)

(x+1)(x+2)(x+3)

(b)

I

(c)

0

(d)

(x+a)(x+b)(x+c)

13. If$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are the position vectors of three collinear points, then which of the following is true?

(a)

$\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$

(b)

$2\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$

(c)

$\overrightarrow{b}=\overrightarrow{c}+\overrightarrow{a}$

(d)

$4\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

14. $lim_{x\rightarrow {\pi/2}}{2x-\pi\over cosx}$

(a)

2

(b)

1

(c)

-2

(d)

0

15. The points of discontinuity of the function $\frac { { x }^{ 2 }+6x+8\quad }{ { x }^{ 2 }-5x+6\quad } is$

(a)

3,2

(b)

3,-2

(c)

-3,2

(d)

-3,-2

16. If is differentiable at x = 1, then

(a)

$a={1\over2},b={-3\over 2}$

(b)

$a={-1\over2},b={3\over 2}$

(c)

$a=-{1\over2},b=-{3\over 2}$

(d)

$a={1\over2},b={3\over 2}$

17. If $\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c$ ,then the value of k is

(a)

log 3

(b)

-log 3

(c)

$-{1\over log3}$

(d)

${1\over log3}$

18. $\int { \frac { \left( log{ x } \right) ^{ 3 } }{ x } }$ dx = _________+c.

(a)

$\frac { \left( log{ x } \right) ^{ 4 } }{ 4 }$

(b)

(sin-1 x)4

(c)

(log x)4

(d)

$\frac { 1 }{ 3logx }$

19. If A and B are any two events, then the probability that exactly one of them occur is

(a)

$P(A\cup\overline{B})+P(\overline{A}\cup B)$

(b)

$P(A\cap\overline{B})+P(\overline{A}\cap B)$

(c)

$P(A)+P({B})-P({A}\cap B)$

(d)

$P(A)+P({B})+2P({A}\cap B)$

20. Three integers are chosen at random from the first 20 integers. The probability that their product is even is

(a)

$\frac { 2 }{ 19 }$

(b)

$\frac { 3 }{ 19 }$

(c)

$\frac { 17 }{ 19 }$

(d)

$\frac { 4 }{ 19 }$

21. II. Answer any 7 questions. Q.no 30 is compulsory:

7 x 2 = 14
22. Write the following in roster form.
The set of all positive roots of the equation (x-1)(x+1)(x2-1)=0.

23. Determine whether the following functions are even, odd or neither.sin2x - 2 cos2x - cos x.

24. If nP4 = 20 $\times$ 3 nP2, then find n.

25. If a, b, c are in A.P., show that (a-c)2 = 4(b2 - ac).

26. Find the combined equation of the straight lines through the origin one of which is parallel to and the other is perpendicular to the straight line 3x + y + 5 = 0.

27. Find $\overrightarrow{a}.\overrightarrow{b}$ when $\overrightarrow{a}$=$\hat{i}-\hat{j}+5\hat{k}$ and $\overrightarrow{b}=3\hat{i}-2\hat{k}$

28. Evaluate $\lim _{ x\rightarrow 1 }{ \frac { 1+(x-1{ ) }^{ 2 } }{ 1+{ x }^{ 2 } } }$

29. Differentiate the following with respect to x :$y={log x \ x \over e^x}$

30. Integrate the following functions with respect to x:${1\over (2-3x)^4}$

31. Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

32. III. Answer any 7 questions. Q.no 40 is compulsory:

7 x 3 = 21
33. Prove that the relation "friendship" is not an equivalence relation on the set of all people in Chennai.

34. Solve the quadratic equation 52x-5x+3+125=5x.

35. Express the following angles in radian measure
300

36. The Sum of infinite number of terms of G.P is 23 and the sum of their sequence is 69.Find the G.P

37. If the slope of one of the lines given by ax2+2hxy+by2 = 0 is k times the other, prove that 4Kh2 = ab (HK)2

38. Identify the singular and non-singular matrices:$\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}$

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

40. If ${ x }^{ 2 }+2xy+{ y }^{ 3 }=42,\quad find\quad \frac { dy }{ dx }$

41. Integrate the following functions with respect to x :${sin^2x\over 1+cos \ x}$

42. The chances of A, B, and C becoming manager of a certain company are 5 : 3: 2. The probabilities that the office canteen will be improved if A, B, and C become managers are 0.4, 0.5 and 0.3 respectively. If the office canteen has been improved, what is the probability that B was appointed as the manager?

43. IV. Answer all in detail:

7 x 5 = 35
44. If f,g,h are real valued functions defined on R, then prove that (f+g) oh=foh+goh. What can you say about fo(g+h)? Justify your answer.

45. Solve the equation x3+5x2-16x-14=0.Given x+7 is a root

46. In a $\triangle$ABC, if $\frac { sin\quad A }{ sin\quad C } =\frac { sin(A-B) }{ sin(B-c) }$,prove that a2,b2,c2are in arithmetic progression

47. Find the sum of all 4-digit numbers that can be formed using the digits 1, 2, 4, 6, 8.

48. Prove that $\sqrt [ 3 ]{ x^3+7 } -\sqrt [ 3 ]{ x^3+4 }$ is approximately equal to ${1\over x^2}$ when x is large.

49. Find the equation of the lines passing through the point of intersection lines 4x - y + 3 = 0 and 5x + 2y + 7 = 0
(i) through the point (-1, 2)
(ii) Parallel to x - y + 5 = 0
(iii) Perpendicular to x - 2y + 1 = 0.

50. Locus of the mid points of the portion of the line $x\sin\theta+y\cos\theta=p$ intercepted between the axis is ............

51. A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins, and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds.
Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds.
Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds.
The cost of 50 gm of cashew nuts is Rs.50, 50 gm of raisins is Rs.10, and 50 gm of almonds is Rs.60. What is the cost of each gift pack?

52. 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 $\overrightarrow{a}=\pm{2\over \sqrt{3}}(\overrightarrow{b}\times \overrightarrow{c}).$

53. Evaluate the following limits :$lim_{x\rightarrow 0}{e^{ax}-e^{bx}\over x}$

54. Discuss the differentiability of $f\left( x \right) =\begin{cases} x{ e }^{ -\left( \frac { 1 }{ \left| x \right| } +\frac { 1 }{ x } \right) } \\ 0,\quad \quad \quad \quad x=0 \end{cases},\quad x\neq 0$ at x=0

55. Integrate the following with respect to x: ex

56. Evaluate $\int { \frac { { x }^{ 3 }dx }{ { x }^{ 4 }+{ 3x }^{ 2 }+2 } }$

57. A purse contains 3 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled out at random from one of the two purses, what is the probability that it is a silver coin?