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Half Yearly Question Paper-2019

12th Standard EM

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Maths

Time : 02:30:00 Hrs
Total Marks : 90

    Part A

    Answer All Questions

    Choose the most appropriate answer from the four given alternatives and write the option code with the corresponding answer.

    20 x 1 = 20
  1. If |adj(adj A)| = |A|9, then the order of the square matrix A is

    (a)

    3

    (b)

    4

    (c)

    2

    (d)

    5

  2. The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    infinitely many

  3. The principal argument of \(\cfrac { 3 }{ -1+i } \)

    (a)

    \(\cfrac { -5\pi }{ 6 } \)

    (b)

    \(\cfrac { -2\pi }{ 3 } \)

    (c)

    \(\cfrac { -3\pi }{ 4 } \)

    (d)

    \(\cfrac { -\pi }{ 2 } \)

  4. If z=cos\(\frac { \pi }{ 4 } \)+i sin\(\frac { \pi }{ 6 } \), then

    (a)

    |z| =1, arg(z) =\(\frac { \pi }{ 4 } \)

    (b)

    |z| =1, arg(z) =\(\frac { \pi }{ 6 } \)

    (c)

    |z|=\(\frac { \sqrt { 3 } }{ 2 } \), arg(z)=\(\frac { 5\pi }{ 24 } \)

    (d)

    |z| =\(\frac { \sqrt { 3 } }{ 2 } \), arg (z) =tan-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

  5. 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

  6. lf the root of the equation x3 +bx2+cx-1=0 form an lncreasing G.P, then

    (a)

    one of the roots is 2

    (b)

    one of the rots is 1

    (c)

    one of the rots is -1

    (d)

    one of the rots is -2

  7. 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

  8. The domain of cos-1(x2 - 4) is______

    (a)

    [3, 5]

    (b)

    [-1, 1]

    (c)

    \(\left[ -\sqrt { 5 } ,-\sqrt { 3 } \right] \cup \left[ \sqrt { 3 } ,\sqrt { 5 } \right] \)

    (d)

    [0, 1]

  9. If x < 0, y < 0 such that xy = 1, then tan--1(x) + tan-l(y) =_____

    (a)

    \(\cfrac { \pi }{ 2 } \)

    (b)

    \(\cfrac { -\pi }{ 2 } \)

    (c)

    \(-\pi \)

    (d)

    none

  10. If x+y=k is a normal to the parabola y2 =12x, then the value of k is

    (a)

    3

    (b)

    -1

    (c)

    1

    (d)

    9

  11. If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is

    (a)

    6x2 + 10y2 = 5

    (b)

    6x2 + 10y2 = 15

    (c)

    x2 + 3y2 = 10

    (d)

    none

  12. If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

    (a)

    \(\vec { a } \)

    (b)

    \(\vec { b} \)

    (c)

    \(\vec { c } \)

    (d)

    \(\vec { 0 } \)

  13. If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to

    (a)

    81

    (b)

    9

    (c)

    27

    (d)

    18

  14. If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k } \)\(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \)\(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) then \(\overset { \rightarrow }{ a } +\left( -\overset { \rightarrow }{ b } \right) \)will be perpendiculur to \(\overset { \rightarrow }{ c } \) only when t =

    (a)

    5

    (b)

    4

    (c)

    3

    (d)

    \(\frac { 7 }{ 3 } \)

  15. Angle between y2 = x and.x2= y at the origin is

    (a)

    \({ tan }^{ -1 }\cfrac { 3 }{ 4 } \)

    (b)

    \({ tan }^{ -1 }\left( \cfrac { 4 }{ 3 } \right) \)

    (c)

    \(\cfrac { \pi }{ 2 } \)

    (d)

    \(\cfrac { \pi }{ 4 } \)

  16. The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

    (a)

    \(\frac{1}{31}\)

    (b)

    \(\frac15\)

    (c)

    5

    (d)

    31

  17. The value of \(\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx\)

    (a)

    \(\frac { { \pi }^{ 2 } }{ 4 } -1\)

    (b)

    \(\frac { { \pi }^{ 2 } }{ 4 } +2\)

    (c)

    \(\frac { { \pi }^{ 2 } }{ 4 } +1\)

    (d)

    \(\frac { { \pi }^{ 2 } }{ 4 } -2\)

  18. 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)

    \(\frac{1}{x^2}\)

  19. A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
    \(f(x)=\begin{cases} \begin{matrix} \frac { 2 }{ { x }^{ 3 } } & 0<x>l \end{matrix} \\ \begin{matrix} 0 & 1\le x<2l \end{matrix} \end{cases}\)

    (a)

    \(\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 3 } \)

    (b)

    \(\\ \cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 6 } \)

    (c)

    \(1,\cfrac { { l }^{ 2 } }{ 12 } \)

    (d)

    \(\cfrac { 1 }{ 2 } ,\cfrac { { l }^{ 2 } }{ 12 } \)

  20. Which one of the following statements has truth value F?

    (a)

    Chennai is in India or \(\sqrt 2\) is an integer

    (b)

    Chennai is in India or \(\sqrt 2\) is an irrational number

    (c)

    Chennai is in China or \(\sqrt 2\) is an integer

    (d)

    Chennai is in China or \(\sqrt 2\) is an irrational number

  21. Part B

    Answer any SEVEN questions. Question number 30 is compulsory.

    7 x 2 = 14
  22. If A is symmetric, prove that then adj Ais also symmetric.

  23. If (cosθ + i sinθ)2 = x + iy, then show that x2+y2 =1

  24. Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

  25. Find the principal value of cos-1\((\frac{1}{2})\).

  26. Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

    ()

    p=-1

  27. Find the local extrema for the following function using second derivative test:
    f(x) = -3x5 +5x3

  28. Let g(x, y) = \(\frac { { e }^{ y }sinx }{ x } \), for x ≠ 0 and g(0, 0) = 1. Show that g is continuous at (0,0).

  29. Show that y = ae-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)

  30. Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X.

  31. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
    a*b = a + 3ab − 5b2;∀a,b∈Z

  32. Part C

    Answer any SEVEN questions. Question number 40 is compulsory

    7 x 3 = 21
  33. Verify that (A-1)T = (AT)-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

  34. Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) =0.

  35. Prove that 
    \({ sin }^{ -1 }(\frac { 3 }{ 5 } )-{ cos }^{ -1 (}\frac { 12 }{ 13 } )={ sin }^{ -1 }(\frac { 16 }{ 65 }) \)

  36. A circle of area 9π square units has two of its diameters along the lines x+y=5 and x−y=1.
    Find the equation of the circle.

  37. Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)

    ()

    s, t

  38. Sketch the curve \(y=\frac { { x }^{ 2 }-3x }{ (x-1) } \)

  39. Consider g(x,y) = \(\frac { 2{ x }^{ 2 }y }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x,y) ≠ (0,0) and g(0,0) = 0 Show that g is continuous on R2

  40. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 4{ sin }^{ 2 }x+5{ cos }^{ 2 }x } } \)

  41. Find the constant C such that the function \(f(x)=\begin{cases} \begin{matrix} { Cx }^{ 2 } & 1 is a density function, and compute (i) P(1.5 < X < 3.5)
    (ii) P(X ≤2)
    (iii) P(3 < X ) .

  42. Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
    Find AVB

  43. Part D

    Answer All Questions 

    7 x 5 = 35
    1. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

    2. Let z1,z2, and z3 be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z1+z2+z3 \(\neq \) 0 prove that \(\left| \cfrac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) =r

    1. Solve: \(\frac{dv}{dx}+2y\ cot\ x=3x^2 cosec^2x\)

    2. Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.

    1. ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If. the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

      ()

      plane

    2. A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
      (i) Find the probability mass function.
      (ii) Find the cumulative distribution function.
      (iii) Find P(3 ≤ X< 6) (iv) Find P(X ≥ 4) .

    1. Show that the straight lines \(\vec { r } =(5\hat { i } +7\hat { j } -3\hat { k } )+s(-4\hat { i } +4\hat { j } -5\hat { k } )\) and \(\vec { r } =(8\hat { i } +4\hat { j } +5\hat { k } )+t(7\hat { i } +\hat { j } +3\hat { k } )\) are coplanar. Find the vector equation of the plane in which they lie.

    2. Find the acute angle between y = x2 and y = (x − 3)2.

    1. If the sum of the roots of the quadratic equation ax2+ bx + c = 0 (abe≠ 0)  is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \)  are H.P.

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

    1. Solve the equation 3x2-16x2+23x-6=0 if the product of two roots is 1.

    2. For the ellipse4x2+y2+24x−2y+21 = 0 , find the centre, vertices, and the foci. Also prove that the length of latus rectum is 2 .

    1. Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

    2. Find (i) cos-1 \((-\frac{1}{\sqrt2})\)
      ii) cos-1\((cos(-\frac{\pi}{3}))\)
      iii) cos-1\((cos(-\frac{7\pi}{6}))\)

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