#### 11th Second Revision Test 2019

11th Standard

Reg.No. :
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Maths

Time : 02:30:00 Hrs
Total Marks : 90
20 x 1 = 20
1. For real numbers x and y, define xRy if x-y+√2 is an irrational number. Then the relation R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

none of these

2. The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and 14% of the enrollment in Chemistry. The number of students take at least one of these two subjects, is

(a)

1120

(b)

1130

(c)

1100

(d)

insufficient data

3. If $\frac { |x-2| }{ x-2 } \ge 0$, then x belongs to

(a)

$[2,\infty]$

(b)

$(2,\infty )$

(c)

$(-\infty,2)$

(d)

$(-2,\infty )$

4. If sin(45 ° + 10°) - sin(45° -10°) =$\sqrt{2}$sin x then x is

(a)

0o

(b)

(c)

10°

(d)

15°

5. If $\alpha$ and $\beta$ are two values of θ obtained from the equation a cosθ+b sinθ=c then the value of $tan(\frac{\alpha+\beta}{2})$ is

(a)

$\frac{a}{b}$

(b)

$\frac{b}{a}$

(c)

$\frac{c}{a}$

(d)

$\frac{c}{b}$

6. The number of ways to average the letters of the word CHEESE are

(a)

120

(b)

240

(c)

720

(d)

6

7. If nPr = 720, nCr =120 then r is:

(a)

2

(b)

4

(c)

3

(d)

5

8. The nth term of the sequence $\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 }$.+.....is

(a)

2n-n-1

(b)

1-2n

(c)

2-n+n-1

(d)

2n-1

9. If 7x2 - 8xy +A = 0 represents a pair of perpendicular lines, the A is

(a)

7

(b)

-7

(c)

-8

(d)

8

10. If the points (2k,k)(k,2k) and (k,k)enclose a triangle of area 18 sq units, then the centroid of the triangle is

(a)

(8,8)

(b)

(4,4)

(c)

(3,3)

(d)

(2,2)

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. If A is a square matrix of order 3, then the number of minors in determinant of A are

(a)

3

(b)

21

(c)

9

(d)

27

13. If $\overrightarrow{a},\overrightarrow{b}$ are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

(a)

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

(b)

${2\overrightarrow{a}-\overrightarrow{b}\over 2}$

(c)

${2\overrightarrow{a}+\overrightarrow{b}\over 2}$

(d)

${\overrightarrow{a}-\overrightarrow{b}\over 3}$

14. $lim_{\alpha \rightarrow {\pi/4}}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}$ is

(a)

$\sqrt{2}$

(b)

$1\over \sqrt{2}$

(c)

1

(d)

2

15. For what values of x is the rate of increase of x3 - 2x2 + 3x + 8 is twice the rate of increase of x?

(a)

$\left( -\frac { 1 }{ 3 } ,-3 \right)$

(b)

$\left( \frac { 1 }{ 3 } ,3 \right)$

(c)

$\left( -\frac { 1 }{ 3 } ,3 \right)$

(d)

$\left( \frac { 1 }{ 3 } ,1 \right)$

16. If y=${1\over4}u^4,u={2\over 3}x^3+5,$ then ${dy\over dx}$ is

(a)

${1\over 27}x^2(2 x^3+15)^3$

(b)

${2\over 27}x(2 x^3+5)^3$

(c)

${2\over 27}x^2(2 x^3+15)^3$

(d)

$-{2\over 27}x(2 x^3+5)^3$

17. $\int{x^2+cos^2x\over x^2+1}cosec^2xdx$ is

(a)

cot x+sin -1x+c

(b)

-cot x+tan-1x+c

(c)

-tan x+cot-1x+c

(d)

-cot x-tan-1x+c

18. $\int { x }$ sin x dx = -x cos x + a, then a =

(a)

sin x + c

(b)

cos x + c

(c)

c

(d)

none of these

19. A number x is chosen at random from the first 100 natural numbers. Let A be the event of numbers which satisfies${(x-10)(x-50)\over x-30}\ge0$, then P(A) is

(a)

0.20

(b)

0.51

(c)

0.71

(d)

0.70

20. If P(A$\cup$B) = 0.8 and P(A$\cap$B) = 0.3 then $P(\bar { A } )+P(\bar { B } )$ =

(a)

0.3

(b)

0.5

(c)

0.7

(d)

0.9

21. 7 x 2 = 14
22. If X = {1, 2, 3, .. 10} and A = {1, 2, 3, 4, 5}, find the number of sets $B\subseteq X$ such that A - B = {4}.

23. Find the value of $cot(\frac{-15\pi}{4})$.

24. Write down all the permutations of the vowels A, E, I, O, U in English alphabets taking there at a time starting with A.

25. Find the general term in the expansion of ${ \left( \frac { 4x }{ 5 } -\frac { 5 }{ 2x } \right) }^{ 9 }$

26. The sum of the distance of a moving point from the points (4,0) and (-4, 0) is always 10 units. Find the equation to the locus of the moving point.

27. Find a direction ratio and direction cosines of the following vectors $3\hat{i}+4\hat{j}-6\hat{k}$

28. Evaluate $\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 2 }-3x+2 }{ { x }^{ 2 }-x-2 } }$

29. Find the derivatives of the following functions using first principle.f(x)=-x2+2

30. Integrate the following with respect to x:${1\over \sqrt{1-25 x^2}}$

31. The ratio of the number of boys to the number of girls in a class is 1:2. It is known that the probability of a girl and a boy getting a first class are 0.25 and 0.28 respectively. Find the probability that a student chosen  at random will get first class?

32. 7 x 3 = 21
33. Consider the functions:(i) y = ex; (ii) y = logeX.

34. Evaluate log ${{(\sqrt{\sqrt{625}+11})(\sqrt{64})}\over{\sqrt{\sqrt [ 5 ]{ 3125 }+\sqrt{\sqrt [ 3 ]{ 343 } } }}}$

35. What is the length of the arc intercepted by a central angle of measure 410 in a circle radius 10 ft ?

36. Write the first six terms of the sequences given by a1 = a2 = 1,an = an-1+ an-2 (n≥3)

37. If the angle between two lines is $\frac{\pi}{4}$ and slope of one of the line is $\frac{1}{2}$ find the slope of the other line

38. Give your own examples of matrices satisfying the following conditions in each case
A and B such that AB=O=BA, A$\neq$ O, and B $\neq$ O.

39. Calculate $lim_{x \rightarrow \infty}{1-x^3\over 3x+2}$

40. If x =$a\sec ^{ 3 }{ \theta } and\quad y=a\tan ^{ 3 }{ \theta } find\quad \frac { dy }{ dx } at\quad \theta =\frac { \pi }{ 3 } .$

41. Evaluate the following :$\int \sqrt{(x-3)(5-x)}dx$

42. A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96.
(i) What is the probability that a fire engine is available when needed?
(ii) What is the probability that neither is available when needed?

43. 7 x 5 = 35
1. Check the following functions for one-to-oneness and ontoness.
$f:N\rightarrow N$ defined by f(n)=n2.

2. Find the values of k so that the equation x2 =-2x(1+3k)+7(3+2k)=0 has real and equal roots.

1. The Government plans to have a circular zoological park of diameter 8 km. A separate area in the form of a segment formed by a chord of length 4 km is to be allotted exclusively for a veterinary hospital in the park. Find the area of the segment to be allotted for the veterinary hospital.

2. By the principle of mathematical induction, prove that, for all integers n $\ge$1,1+2+3+....+n=${n(n+1)\over2}$ .

1. Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

2. Show that the lines are 3x + 2y + 9 = 0 and 12x + 8y - 15 = 0 are paralle llines.

1. Find the equation of the line passing through the point of intersection 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x +4y = 7.

2. Find the value of the product $\begin{vmatrix} log_364 &log_43 \\ log_38 & log_49 \end{vmatrix}\times \begin{vmatrix} log_23 & log_83 \\ log_34 & log_34 \end{vmatrix}$

1. Let A, Band C represent the angles of a $\triangle$ABC and a, band c represent the lengths of the sides opposite to them, then prove that a2 = b2 + c2 - 2bc cos A (Law of cosines)

2. Evaluate the following limits :$lim_{x\rightarrow0}{e^x-e^{-x}\over sin x}$

1. Differentiate ${ \left( \sin { x } \right) }^{ { \cos { ^{ -1x } } } }$ with respect to 'x'.

2. Integrate the following functions with respect to x:${1\over \sqrt{1-81x^2}}$

1. Evaluate $\int { \frac { { x }^{ 2 }{ tan }^{ -1 }\left( { x }^{ 3 } \right) }{ 1+{ x }^{ 6 } } }$dx

2. for a loaded die, the probabilities of outcomes are given as under
P(1) = P(2) = $\frac { 2 }{ 10 }$, P(3) = P(5) = P(6) = $\frac { 1 }{ 10 }$ and P(4) = $\frac { 3 }{ 10 }$
The die is thrown 2 times. Let A and B be the events as defined below
A: Getting same number each time
B: Getting a total score of 10 or more
Discuss the independency of the events A and B