#### Class 11 Revision Test 3

11th Standard

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Maths

Time : 02:30:00 Hrs
Total Marks : 90
20 x 1 = 20
1. The function f:[0,2π]➝[-1,1] defined by f(x)=sin x is

(a)

one-to-one

(b)

on to

(c)

bijection

(d)

cannot be defined

2. Which one of the following is false?

(a)

A⋂(BΔ\C) = (A ⋂ B)Δ(A ∩ C)

(b)

A∩(B - C) = (A ∩B) \ (A∩C)

(c)

(A U B), = A' ∩ B'

(d)

(A \ B) U B = A ⋂ B

3. If a and b are the roots of the equation x2-kx+c = 0 then the distance between the points (a, 0) and (b, 0)

(a)

$\sqrt { { 4k }^{ 2 }-c }$

(b)

$\sqrt { { k }^{ 2 }-4c }$

(c)

$\sqrt { 4c-{ k }^{ 2 } }$

(d)

$\sqrt { k-8c }$

4. If tan400=λ, then $\frac { tan{ 140 }^{ 0 }-tan{ 130 }^{ 0 } }{ 1+tan{ 140 }^{ 0 }.tan{ 130 }^{ 0 } }$=

(a)

$\frac { 1-\lambda ^{ 2 } }{ \lambda }$

(b)

$\frac { 1+{ \lambda }^{ 2 } }{ \lambda }$

(c)

$\frac { 1+{ \lambda }^{ 2 } }{ 2\lambda }$

(d)

$\frac { 1-{ \lambda }^{ 2 } }{ 2\lambda }$

5. If tanθ=$\frac{-4}{3}$, then sinθ is

(a)

$\frac{-4}{5}$

(b)

$\frac{4}{5}$

(c)

$\frac{-4}{5}\quad or\quad \frac{4}{5}$

(d)

None

6. The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

(a)

$\times$ 11 C7+10C8

(b)

11C7+10C8

(c)

12C8-10C6

(d)

10C6+2!

7. There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

(a)

45

(b)

40

(c)

39

(d)

38

8. The sum up to n terms of the series $\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +$.....is

(a)

$\frac { n(n+1) }{ 2 }$

(b)

2n(n+)

(c)

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

(d)

1

9. The distance between the line 12x - 5y + 9 = 0 and the point (2, 1) is

(a)

$\pm\frac{28}{13}$

(b)

$\frac{28}{13}$

(c)

$-\frac{28}{13}$

(d)

none of these

10. The image of the point (1,2) with respect to the line y=x is

(a)

(-1,-2)

(b)

(2,1)

(c)

(2,-1)

(d)

(2,1)

11. If the square of the matrix $\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is the unit matrix of order 2, then$\alpha ,\beta$ and $\gamma$ should satisfy the relation.

(a)

1+$\alpha ^2+\beta \gamma=0$

(b)

1-$\alpha ^2-\beta \gamma=0$

(c)

1-$\alpha ^2+\beta \gamma=0$

(d)

1+$\alpha ^2-\beta \gamma=0$

12. The value of$\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1+sin\theta & 1 \\ 1 & 1 & 1+cos\theta \end{matrix} \right|$ is

(a)

3

(b)

1

(c)

2

(d)

$\frac{1}{2}$

13. The angle between two vectors$\vec{a}$ and$\vec{b}$ with magnitudes$\sqrt{3}$ and 4 respectively and $\vec{a}.\vec{b}=2\sqrt{3}$ is

(a)

$\frac{\pi}{6}$

(b)

$\frac { \pi }{ 3 }$

(c)

$\frac { \pi }{ 2 }$

(d)

$\frac { 5\pi }{ 2 }$

14. 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

15. If y= 6x -x3 and x increases at the ratio of 5 units per second, the rate of change of slope when x = 3 is ______ units/sec.

(a)

-90

(b)

90

(c)

180

(d)

-180

16. If f(x)=x2-3x, then the points at which f(x)=f'(x) are

(a)

both positive integers

(b)

both negative integers

(c)

both irrational

(d)

one rational and another irrational

17. $\int e^{\sqrt{x}}dx$ is

(a)

$2\sqrt{x}(1-e^{\sqrt{x}})+c$

(b)

$2\sqrt{x}(e^{\sqrt{x}}-1)+c$

(c)

$2e^{\sqrt{x}}(1-\sqrt{x})+c$

(d)

$2e^{\sqrt{x}}(\sqrt{x}-1)+c$

18. $\int { \left| x \right| ^{ 3 } }$ dx is equal to ________+c.

(a)

$\frac { -{ x }^{ 4 } }{ 4 } +c$

(b)

$\frac { \left| x \right| ^{ 4 } }{ 4 }$

(c)

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

(d)

none of these

19. If a and b are chosen randomly from the set {1,2,3,4} with replacement, then the probability of the real roots of the equation $x^2+ax+b=0$ is

(a)

${3\over 16}$

(b)

${5\over 16}$

(c)

${7\over 16}$

(d)

${11\over 16}$

20. If P(B)=$\frac { 3 }{ 5 }$, P(A/B)=$\frac { 1 }{ 2 }$ and P(AUB)=$\frac { 4 }{ 5 }$, then P(B/$\bar { A }$) =

(a)

$\frac { 1 }{ 5 }$

(b)

$\frac { 3 }{ 10 }$

(c)

$\frac { 1 }{ 2 }$

(d)

$\frac { 3 }{ 5 }$

21. 7 x 2 = 14
22. On the set of natural number let R be the relation defined by aRb if a + b $\le$ 6. Write down the relation by listing all the pairs. Check whether it is symmetric

23. If the arcs of same lengths in two circles subtend central angles 30° and 800, find the ratio of their radii.

24. Let p(n) be the statement "7 divides 23n-1" What is p(n+1) =?

25. Find the sum of first n terms of the series 12+32+52+...

26. Find the locus of a point P that moves at a constant distant of
(i) two units from the X-axis
(ii) three units from the Y-axis

27. Represent graphically the displacement of 60 km 50° south of east.

28. $If\lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\quad and\quad n\in N,\quad find\quad n.$

29. Find the derivatives of the following tan (x + y) + tan (x - y) = x

30. Integrate the following with respect to x: x sin 3x

31. The probability that student selected at random from a class will pass in Mathematics is $\frac { 2 }{ 3 }$ and the probability that he passes in Mathematics and English is $\frac { 1 }{ 3 }$. What is the probability that he will pass in English if it is known that he has passed in Mathematics?

32. 7 x 3 = 21
33. Consider the functions:
(i) f(x) = |x|
(ii) f(x) = |x| − 1
(iii) f(x) = |x| + 1

34. Simplify: $\sqrt { 98 } +\sqrt { 50 } -\sqrt { 18 } +\sqrt { 75 } -\sqrt { 27 }$

35. Solve the equation sin 9$\theta$ = sin $\theta$.

36. An A.P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the series.

37. Find $\sqrt [ 3 ]{ 65 }$.

38. Find the value of $\begin{vmatrix} 1 & log_xy & log_x z \\ log_y x & 1 & log_yz \\ log_z x & log_zy & 1 \end{vmatrix}$  if x, y, z $\neq$ 1.

39. Evaluate the following limits :$lim_{x\rightarrow 0}{1-cosx\over x^2}$

40. Show  that$f\left( x \right) ={ x }^{ 2 }$ is differentiable at x = 1 and find $f^{ ' }\left( 1 \right)$

41. Evaluate : $\int{x^3+2\over x-1}dx$

42. A firm manufactures PVC pipes in three plants viz, X, Y, and Z. The daily production volumes from the three firms X, Y and Z are respectively 2000 units, 3000 units, and 5000 units. It is known from the past experience that 3% of the output from plant X, 4% from plant Y and 2% from plant Z are defective. A pipe is selected at random from a day’s total production,
(i) find the probability that the selected pipe is a defective one.
(ii) if the selected pipe is a defective, then what is the probability that it was produced by plant Y?

43. 7 x 5 = 35
44. If $f:R-\{ -1,1\}\rightarrow R$ is defined by $f(x)={x \over x^2-1},$ verify whether f is one-to-one or not.

45. Resolve into partial fractions $\frac { { x }^{ 2 }-2x-9 }{ (x+1)({ x }^{ 2 }+x+6) }$

46. Using Heron's formula, show that the equilateral triangle has the maximum area for any fixed perimeter. [Hint: In xyz$\le$  k, maximum occurs when x = y = z]

47. Using the Mathematical induction, show that for any integer
n$\ge$ 2, 3n2 > (n + 1)2

48. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

49. Find all the equations of the straight lines in the family of the lines y = mx - 3, for which m and the x - coordinate of the point of intersection of the lines with x - y = 6 are integers.

50. Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos $\alpha$ + y sin $\alpha$ = p is $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$ where p is a constant.

51. For what value of x, the matrix A=$\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & x^3 \\ 2 & -3 & 0 \end{bmatrix}$ is skew-symmetric

52. Find $\lambda$, when the projection of $\overrightarrow{a}=\lambda \hat{i}+\hat{j}+4\hat{k}$ on $\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}$ is 4 units.

53. Sketch the graph of a function f that satisfies the given values :
f(0) is undefined
$lim_{x\rightarrow0}f(x)=4$
f(2) = 6
$lim_{x\rightarrow2}f(x)=3$

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${1\over xlog \ xlog(log \ x)}$

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

57. The probability of simultaneous occurrence of atleast one of two events A and B is p. if the probability that exactly one A, B occurs is q then prove that P($\bar { A }$) + P($\bar { B }$) = 2-2 p+q.