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Public Exam Model Question Paper II 2019 - 2020

12th Standard EM

Reg.No. :
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Maths

Time : 02:45:00 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. If A = $\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right]$ and AT = A−1 , then the value of x is

(a)

$\frac { -4 }{ 5 }$

(b)

$\frac { -3 }{ 5 }$

(c)

$\frac { 3 }{ 5 }$

(d)

$\frac { 4 }{ 5 }$

2. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ $\in$ R) is consistent with unique solution if

(a)

λ = 8

(b)

λ = 8, μ ≠ 36

(c)

λ ≠ 8

(d)

none

3. $\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } }$ = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

$cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 }$

4. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

(a)

mn

(b)

m+n

(c)

mn

(d)

nm

5. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

(a)

0

(b)

1

(c)

4

(d)

3

6. ${ sin }^{ -1 }\frac { 3 }{ 5 } -{ cos }^{ -1 }\frac { 12 }{ 13 } +{ sec }^{ -1 }\frac { 5 }{ 3 } { -cosec }^{ 1- }\frac { 13 }{ 2 }$is equal to

(a)

2$\pi$

(b)

$\pi$

(c)

0

(d)

tan-1$\frac{12}{65}$

7. If ${ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha$ then x2 =

(a)

$sin2\alpha$

(b)

$sin\alpha$

(c)

$cos2\alpha$

(d)

$cos\alpha$

8. The equation of the circle passing through(1,5) and (4,1) and touching y -axis is x2+y2−5x−6y+9+(4x+3y−19)=0 whereλ is equal to

(a)

$0,-\frac { 40 }{ 9 }$

(b)

0

(c)

$\frac { 40 }{ 9 }$

(d)

$\frac { -40 }{ 9 }$

9. The volume of the parallelepiped with its edges represented by the vectors $\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k }$ is

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { \pi }{ 3}$

(c)

${ \pi }$

(d)

$\frac { \pi }{ 4 }$

10. The maximum value of the product of two positive numbers, when their sum' of the squares is 200, is

(a)

100

(b)

$25\sqrt { 7 }$

(c)

28

(d)

$24\sqrt { 14 }$

11. The angle made by any tangent to the curve y = x5 + 8x + 1 with the X-axis is a

(a)

obtuse

(b)

right angle

(c)

acute angle

(d)

no angle

12. If u(x, y) = x2+ 3xy + y - 2019, then $\frac { \partial u }{ \partial x }$(4, -5) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13

13. If f(x,y) = 2x2 - 3xy + 5y + 7 then f(0, 0) and f(1, 1) is

(a)

7,11

(b)

11,7

(c)

0,7

(d)

1,0

14. For any value of n∈Z, $\int _{ 0 }^{ \pi }{ e{ cos }^{ 2x }{ cos }^{ 3 } } [(2n+1)x]$ is

(a)

$\frac{\pi}{2}$

(b)

$\pi$

(c)

0

(d)

2

15. $\int _{ 0 }^{ \frac { \pi }{ 4 } }{ { cos }^{ 3 }2x \ dx= }$

(a)

$\frac { 2 }{ 3 }$

(b)

$\frac { 1 }{ 3 }$

(c)

0

(d)

$\frac { 2\pi }{ 3 }$

16. The degree of the differential equation y $y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....$ is

(a)

2

(b)

3

(c)

1

(d)

4

17. Using y = vx, the differential equation $\frac { dy }{ dx } =\frac { y }{ x+\sqrt { xy } }$ is reduced to ________.

(a)

x(1+$\sqrt{v}$)dv=v$\sqrt{v}$dx

(b)

x(1-$\sqrt{v}$)dv=v$\sqrt{v}$dx

(c)

x(1+$\sqrt{v}$)dv=-v$\sqrt{v}$dx

(d)

v(1+$\sqrt{v}$)dx-v$\sqrt{v}$dv=0

18. A computer salesperson knows from his past experience that he seUs computers to one in every twenty customers who enter the showroom. What is the probability that he will seU a computer to exactly two of the next three customers?

(a)

$\cfrac { 57 }{ { 20 }^{ 3 } }$

(b)

$\cfrac { 57 }{ { 20 }^{ 2 } }$

(c)

$\cfrac { { 19 }^{ 3 } }{ { 20 }^{ 3 } }$

(d)

$\cfrac { 57 }{ 20 }$

19. The proposition p ∧ (¬p ∨ q) is

(a)

a tautology

(b)

(c)

logically equivalent to p ∧ q

(d)

logically equivalent to p ∨ q

20. Define * on Z by a*b = a+b+1 ∀ a,b $\in$ Z. Then the identity element of z is

(a)

1

(b)

0

(c)

1

(d)

-1

21. Part II

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

10 x 2 = 20
22. Find the rank of the matrix A =$\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right]$.

23. Find the number of positive and negative roots of the equation x7 - 6x6 + 7x5 + 5x2+2x+2

24. Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis

25. The volume of the parallelepiped whose coterminus edges are $7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k }$$-3\hat { i } +7\hat { j } +5\hat { k }$ is 90 cubic units. Find the value of λ.

26. A particle moves in a line so that x=$\sqrt { t }$. Show that the acceleration is negative and proportional to the cube of the velocity.

27. Evaluate $\begin{matrix} lim \\ (x,y)\rightarrow (1,2) \end{matrix}$, if the limit exists, where $(x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 }$

28. Evaluate $\int { \sum _{ r=0 }^{ \infty }{ \cfrac { { x }^{ r }{ 2 }^{ r } }{ r! } } dx }$

29. Determine the order and degree (if exists) of the following differential equations:
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right)$

30. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function $f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}$
Find the expected life of this electronic equipment.

31. An operation * is defined as mXn=mn-nm.Is it binary on N?

32. Part III

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

10 x 3 = 30
33. For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?

34. If $\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 }$, find the complex number z

35. If α and β are the roots of the quadratic equation 2x2−7x+13 = 0 , construct a quadratic equation whose roots are α2 and β2.

36. Solve: 2x+2x-1+2x-2=7x+7x-1+7x-2

37. If $sin\left( { sin }^{ -1 }\cfrac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1$ then find the value ofx.

38. Find the equations of the tangent and normal to hyperbola 12x2−9y2=108 at $\theta =\frac { \pi }{ 3 }$ (Hint: use parametric form)

39. Prove that $\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right]$=$\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right]$

()

0

40. Find the local extremum of the function f (x) = x4 + x 32x

41. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \cfrac { dx }{ 1+{ tan }^{ 3 }x } }$

42. Construct the truth table for (-p) v (q ∧ r)

43. Part IV

14 x 5 = 70
44. The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

45. Show that $\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }$=-1

46. Determine k and solve the equation 2x3-6x2+3x+k=0 if one of its roots is twice the sum of the other two roots.

47. Find the principal value of
sec−1(−2).

48. Provethat ${ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\$

49. The foci of a hyperbola coincides with the foci of the ellipse $\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1$. Find the equation of the hyperbola if its eccentricity is 2.

50. Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

51. Evaluate: $\underset{x \rightarrow1}{lim} \ x^{\frac{1}{1-x}}$

52. Let f(x, y) = sin(xy2) + ex3+5y for all ∈ R2. Calculate $\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } }$and $\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } }$

53. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = $\frac { \pi }{ 3 }$ and x axis are as 2:3

1. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours; I
(iii) at least 2 will not have a useful life of at : least 600 hours.

2. Construct the truth table for (p ∧ q) v r.

1. Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

2. Sovle : (x+y+1)2dy=dx,y(-1)=0