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#### REVISION TEST

12th Standard EM

Reg.No. :
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Maths

Time : 03:00:00 Hrs
Total Marks : 90
PART-A
20 x 1 = 20
1. If A$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, then A =

(a)

$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right]$

2. If A is a non-singular matrix such that A-1 = $\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right]$, then (AT)−1 =

(a)

$\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right]$

3. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

(a)

0

(b)

2n

(c)

2n

(d)

n2

4. The conjugate of a complex number is $\cfrac { 1 }{ i-2 }$/Then the complex number is

(a)

$\cfrac { 1 }{ i+2 }$

(b)

$\cfrac { -1 }{ i+2 }$

(c)

$\cfrac { -1 }{ i-2 }$

(d)

$\cfrac { 1 }{ i-2 }$

5. If z is a non zero complex number, such that 2iz2=$\bar { z }$ then |z| is

(a)

$\cfrac { 1 }{ 2 }$

(b)

1

(c)

2

(d)

3

6. The principal value of the amplitude of (1+i) is

(a)

$\frac { \pi }{ 4 }$

(b)

$\frac { \pi }{ 12 }$

(c)

$\frac { 3\pi }{ 4 }$

(d)

$\pi$

7. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

(a)

-1

(b)

$\frac { 5 }{ 4 }$

(c)

$\frac { 4 }{ 5 }$

(d)

5

8. If sin-1 x+sin-1 y+sin-1 z=$\frac{3\pi}{2}$, the value of x2017+y2018+z2019$-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } }$is

(a)

0

(b)

1

(c)

2

(d)

3

9. If cot-1 2 and cot-1 3 are two angles of a triangle, then the third angle is

(a)

$\frac{\pi}{4}$

(b)

$\frac{3\pi}{4}$

(c)

$\frac{\pi}{6}$

(d)

$\frac{\pi}{3}$

10. The number of real solutions of the equation $\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi <x<\pi$ is

(a)

0

(b)

1

(c)

2

(d)

infinte

11. The circle x2+y2=4x+8y+5intersects the line3x−4y=m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85<m < −35

(d)

−35<m <15

12. If $[\vec { a } ,\vec { b } ,\vec { c } ]=1$$\frac { \vec { a } .(\vec { b } \times \vec { c } ) }{ (\vec { c } \times \vec { a } ).\vec { b } ) } +\frac { \vec { b } .(\vec { c } \times \vec { a } ) }{ (\vec { a } \times \vec { b } ).\vec { c } } +\frac { \vec { c } .(\vec { a } \times \vec { b } ) }{ (\vec { c } \times \vec { b } ).\vec { a } }$ is

(a)

1

(b)

-1

(c)

2

(d)

3

13. The position of a particle moving along a horizontal line of any time t is given by set) = 3t2 -2t- 8. The time at which the particle is at rest is

(a)

t= 0

(b)

$\\ \\ \\ t=\cfrac { 1 }{ 3 }$

(c)

t =1

(d)

t = 3

14. If u (x, y) = ex2+y2, then $\frac { \partial u }{ \partial x }$ is equal to

(a)

ex2+y2

(b)

2xu

(c)

x2u

(d)

y2u

15. The value of $\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \\$ is

(a)

$\frac{7}{27}$

(b)

$\frac{5}{27}$

(c)

$\frac{4}{27}$

(d)

$\frac{2}{27}$

16. The order and degree of the differential equation $\sqrt { sin\quad x } (dx+dy)=\sqrt { cos\quad x } (dx-dy)$

(a)

1,2

(b)

2,2

(c)

1,1

(d)

2,1

17. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

6

(b)

4

(c)

3

(d)

2

18. Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

19. The operation * defined by a*b =$\frac{ab}{7}$ is not a binary operation on

(a)

Q+

(b)

Z

(c)

R

(d)

C

20. If a*b=$\sqrt { { a }^{ 2 }+{ b }^{ 2 } }$ on the real numbers then * is

(a)

commutative but not associative

(b)

associative but not commutative

(c)

both commutative and associative

(d)

neither commutative nor associative

21. PART-B

WRITE ANY 7 QUESTIONS

7 x 2 = 14
22. Prove that $\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right]$ is orthogonal

23. If z1=3-2i and z2=6+4i, find $\cfrac { { z }_{ 1 } }{ z_{ 2 } }$

24. Discuss the nature of the roots of the following polynomials:
x2018+1947x1950+15x8+26x6+2019

25. If $\hat { a } =\hat { -3i } -\hat { j } +\hat { 5k }$$\hat{b}=\hat{i}-\hat{2j}+\hat{k}$$\hat{c}=\hat{4i}-\hat{4k}$and $\hat { a } .(\hat { b } \times \hat { c } )$

26. Find the slope of the tangent to the curves at the respective given points.
y = x4 + 2x2 − x at x =1

27. If w(x, y, z) = x2 y + y2z + z2x, x, y, z∈R, find the differential dw .

28. Evaluate the following integrals using properties of integration:
$\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx }$

29. A differential equation, determine its order, degree (if exists)
$y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } }$

30. An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

31. Determine whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)

32. PART-C

WRITE ANY 7 QUESTIONS

7 x 3 = 21
33. Verify the property (AT)-1 = (A-1) with A = $\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right]$.

34. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are, 2α, 2β, 2γ

35. Find the domain of sin−1(2−3x2)

36. Find the equation of the tangent and normal to the circle x2+y2−6x+6y−8=0 at (2,2) .

37. Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is $\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right|$.

38. Find the points on the curve y2 - 4xy = x2 + 5 for which the tangent is horizontal.

39. Let F(x, y) = x3 y + y2x + 7 for all (x, y)∈ R2. Calculate $\frac { \partial F }{ \partial x }$(-1,3) and $\frac { \partial F }{ \partial y }$(-2,1).

40. Solve $\frac { dy }{ dx } +2y={ e }^{ -x }$

41. If X- B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6 • Find the distribution, mean and standard deviation of X.

42. Construct the truth table for $(p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)$

43. PART-D

WRITE ANY 7 QUESTIONS

7 x 5 = 35
44. If F($\alpha$) = $\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right]$, show that [F($\alpha$)]-1 = F(-$\alpha$).

45. If $2cosa=x+\cfrac { 1 }{ x }$ and $2cos\beta =y+\cfrac { 1 }{ y }$, show that
i) $\cfrac { x }{ y } +\cfrac { y }{ x } =2cos\left( \alpha -\beta \right)$.
ii) $xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right)$
iii)
$\cfrac { { x }^{ m } }{ { y }^{ n } } -\cfrac { { y }^{ n } }{ { x }^{ m } } =2isin\left( m\alpha -n\beta \right)$
iv)
${ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )$

46. Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

47. If $\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k }$ find $(\vec { a } \times \vec { b } )\times \vec { c }$ and $(\vec { a } \times \vec { b } )\times \vec { c }$. State whether they are equal.

48. Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ R at any point on the curve.

49. For each of the following functions find the fx, fy, and show that fxy =fyx
f(x,y) = $\frac { 3x }{ y+sinx \ }$

50. Evaluate: $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ (\sqrt { tan\quad x } +\sqrt { cot\quad x } )dx }$

51. Solve the Linear differential equation:
$\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } }$, given that y=2 when x=1

52. Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function
$\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and 500 litres?

53. Let M=$\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\}$ and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.