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#### Public Exam Model Question Paper III 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 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

(a)

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

(b)

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

(c)

$\frac { 5\pi }{ 6 }$

(d)

$\frac { \pi }{ 4 }$

2. If A = [2 0 1] then the rank of AAT is ______

(a)

1

(b)

2

(c)

3

(d)

0

3. If a=cosθ + i sinθ, then $\frac { 1+a }{ 1-a }$ =

(a)

cot $\frac { \theta }{ 2 }$

(b)

cot θ

(c)

i cot $\frac { \theta }{ 2 }$

(d)

i tan$\frac { \theta }{ 2 }$

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 ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

6. The equation tan-1 x-cot-1 x=tan-1$\left( \frac { 1 }{ \sqrt { 3 } } \right)$has

(a)

no solution

(b)

unique solution

(c)

two solutions

(d)

infinite number of solutions

7. If ${ sin }^{ -1 }x-cos^{ -1 }x=\cfrac { \pi }{ 6 }$ then

(a)

$\cfrac { 1 }{ 2 }$

(b)

$\cfrac { \sqrt { 3 } }{ 2 }$

(c)

$\cfrac { -1 }{ 2 }$

(d)

none of these

8. The eccentricity of the ellipse (x−3)2 +(y−4)2 =$\frac { { y }^{ 2 } }{ 9 }$ is

(a)

$\frac { \sqrt { 3 } }{ 2 }$

(b)

$\frac { 1 }{ 3 }$

(c)

$\frac { 1 }{ 3\sqrt { 2 } }$

(d)

$\frac { 1 }{ \sqrt { 3 } }$

9. The angle between the lines $\frac { x-2 }{ 3 } =\frac { y+1 }{ -2 }$, z=2 and $\frac { x-1 }{ 1 } =\frac { 2y+3 }{ 3 } =\frac { z+5 }{ 2 }$

(a)

$\frac{\pi}{6}$

(b)

$\frac{\pi}{4}$

(c)

$\frac{\pi}{3}$

(d)

$\frac{\pi}{2}$

10. The point of inflection of the curve y = (x - 1)3 is

(a)

(0,0)

(b)

(0,1)

(c)

(1,0)

(d)

(1,1)

11. $\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx }$ is _________

(a)

1

(b)

-1

(c)

0

(d)

12. Linear approximation for g(x) = cos x at x=$\frac{-\pi}{2}$ is

(a)

x + $\frac{-\pi}{2}$

(b)

- x + $\frac{\pi}{2}$

(c)

x - $\frac{\pi}{2}$

(d)

- x + $\frac{\pi}{2}$

13. If the radius of the sphere is measured as 9 em with an error of 0.03 cm, the approximate error in calculating its volume is

(a)

9.72 cm3

(b)

0.972 cm3

(c)

0.972π cm3

(d)

9.72π cm3

14. The value of $\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx }$ is

(a)

$\pi$

(b)

$2\pi$

(c)

$3\pi$

(d)

$4\pi$

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 integrating factor of the differential equation $\frac{dy}{dx}$+P(x)y=Q(x)is x, then P(x)

(a)

x

(b)

$\frac { { x }^{ 2 } }{ 2 }$

(c)

$\frac{1}{x}$

(d)

$\frac{1}{x^2}$

17. The differential equation associated with the family of concentric circles having their centres at the origin is _________.

(a)

$\frac { dy }{ dx } =\frac { -x }{ y }$

(b)

$\frac { dy }{ dx } =\frac { -y }{ x }$

(c)

$\frac { dy }{ dx } =\frac { x }{ y }$

(d)

$\frac { dy }{ dx } =\frac { y }{ x }$

18. If the function  $f(x)=\cfrac { 1 }{ 12 }$ for. a < x < b, represents a probability density function of a continuous random variable X, then which of the followingcannot be the value of a and b?

(a)

0 and 12

(b)

5 and 17

(c)

7 and 19

(d)

16 and 24

19. Determine the truth value of each of the following statements:
(a) 4+2=5 and 6+3=9
(b) 3+2=5 and 6+1=7
(c) 4+5=9 and1+2= 4
(d) 3+2=5 and 4+7=11

(a)
 (a) (b) (c) (d) F T T T
(b)
 (a) (b) (c) (d) T F T F
(c)
 (a) (b) (c) (d) T T F F
(d)
 (a) (b) (c) (d) F F T T
20. The identity element in the group {R - {1},x} where a * b = a + b - ab is

(a)

0

(b)

1

(c)

$\frac { 1 }{ a-1 }$

(d)

$\frac { a }{ a-1 }$

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 th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

24. Find the locus of a point which divides so that the sum of its.distances from (-4, 0) and (4, 0) is 10 units.

25. Find the angle between the following lines.
2x = 3y =  −z and 6x = − y = −4z.

26. Determine the domain of convexity of the function y=ex

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

28. Find the area of the region enclosed by the curve y = $\sqrt x$ + 1, the axis of x and the lines x=0, x=4.

29. Solve the Linear differential equation:
$\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y$

30. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is
$f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 3 } & 0<x<30 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}$
Obtain and interpret the expected value of the random variable X .

31. A and B are Boolean matrices of order 2X2.If AVB=A, is it necessary that $B=\left( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \right)$

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. Simplify the following:
i 1729

35. 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

36. Solve:(x-1)4+(x-5)4=82

37. Solve: ${ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 }$

38. Identify the type of the conic for the following equations:
(1) 16y2=−4x2+64
(2) x2+y2=−4x−y+4
(3) x2−2y=x+3
(4) 4x2−9y2−16x+18y−29 = 0

39. If $\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i }$ then find $\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right]$

()

parametric form od vector equation

40. Compute the value of 'c' satisfied by Rolle’s theorem for the function $f(x)=log(\frac{x^{2}+6}{5x})$ in the interval [2,3]

41. Find the area bounded by the curve y=x|x|,x-axis and the ordinates x=-1,x=1

42. Give an example (S,*), where a2=e for all a∈S.Given that e is identity element.

43. Part IV

14 x 5 = 70
44. Write thefunction$f(x)={ tan }^{ -1 }\sqrt { \cfrac { a-x }{ a+x } } -a<x<a$ in the simplest form

45. 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.

46. If $\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k }$ are two given vector, then find a vector B satisfying the equations $\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B }$$\overset { \rightarrow }{ C }$ and $\overset { \rightarrow }{ A }$.$\overset { \rightarrow }{ B }$=3

47. Using the l’Hôpital Rule prove that, $\underset{x\rightarrow 0^{+}}{lim}(1+x)^{\frac{1}{x}}=e$

48. Prove that f(x, y) = x3 - 2x2y + 3xy2 + y3 is taomogeneous; what is the degree? Verify fuler's Theorem for f.

49. Find the area of the region common to the circle x2+y2=16 and the parabola y2=6x.

50. The equation of electromotive force for an electric circuit containing resistance and self inductance is E=Ri L$\frac{di}{dt},$ Where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0.

51. Sovle $\left( x+2 \right) \cfrac { dy }{ dx } =x2+4x-9$ .Also find the domain of the function.

1. If X is the random variable with distribution function F(x) given by,
$F(x)=\begin{cases} \begin{matrix} 0 & -\infty then find (i) the probability density function f{x) (ii) P(0.3 ≤ X ≤ 0.6) 2. Show that Z7-{[0]} satisfies the closure, associative, identity, inverse and commutative axioms under multiplication modulo 5. 1. For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2 is consistent. 2. 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

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

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