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12th Standard Maths English Medium Reduced Syllabus Model Question paper - 2021 Part - 2

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 90

      Part I

      Answer all the questions.

      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 = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \)

    (a)

    \(\frac { 1 }{ 3 } \)

    (b)

    \(\frac { 1 }{ 9 } \)

    (c)

    \(\frac { 1 }{ 4 } \)

    (d)

    1

  2. The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

    (a)

    1

    (b)

    2

    (c)

    4

    (d)

    3

  3. If xayb = em, xcyd = en, Δ1 = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ2 = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ3 = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

    (a)

    e21), e31)

    (b)

    log (Δ13), log (Δ23)

    (c)

    log (Δ21), log(Δ31)

    (d)

    e(Δ13),e(Δ23)

  4. If |z-2+i|≤2, then the greatest value of |z| is

    (a)

    \(\sqrt { 3 } -2\)

    (b)

    \(\sqrt { 3 } +2\)

    (c)

    \(\sqrt { 5 } -2\)

    (d)

    \(\sqrt { 5 } +2\)

  5. If \(\left| z-\cfrac { 3 }{ z } \right| =2\) then the least value |z| is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    5

  6. The polynomial x3-kx2+9x has three real zeros if and only if, k satisfies

    (a)

    |k|≤6

    (b)

    k=0

    (c)

    |k|>6

    (d)

    |k|≥6

  7. 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)

    \(\frac { -40 }{ 9 } \)

  8. If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

    (a)

    \(\frac { \pi }{ 2 } \)

    (b)

    \(\frac { 3\pi }{ 6 } \)

    (c)

    \(\frac { \pi }{ 4 } \)

    (d)

    \( { \pi }\)

  9. If the planes \(\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } =(4+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are

    (a)

    \(\frac { 1 }{ 2 } ,-2\)

    (b)

    \(-\frac { 1 }{ 2 } ,2\)

    (c)

    \(-\frac { 1 }{ 2 } ,-2\)

    (d)

    \(\frac { 1 }{ 2 } ,2\)

  10. If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\) then the value of λ is

    (a)

    \(2\sqrt { 3 } \)

    (b)

    \(3\sqrt { 2 } \)

    (c)

    0

    (d)

    1

  11. The number given by the Rolle's theorem for the functlon x3-3x2, x∈[0,3] is

    (a)

    1

    (b)

    \(\\ \\ \\ \sqrt { 2 } \)

    (c)

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

    (d)

    2

  12. The number given by the Mean value theorem for the function \(\cfrac { 1 }{ x } \),x∈[1,9] is

    (a)

    2

    (b)

    2.5

    (c)

    3

    (d)

    3.5

  13. If w (x, y, z) = x2 (v - z) + y2 (z - x) + z2(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

    (a)

    xy + yz + zx

    (b)

    x(y + z)

    (c)

    y(z + x)

    (d)

    0

  14. If (x,y, z) = xy +yz +zx, then fx - fz is equal to

    (a)

    z - x

    (b)

    y - z

    (c)

    x - z

    (d)

    y - x

  15. If f(x)\(f(x)=\int _{ 1 }^{ x }{ \frac { { e }^{ { sin }^{ u } } }{ u } } du,x>1\quad and\quad \int _{ 1 }^{ 3 }{ \frac { { e }^{ { sinx }^{ 2 } } }{ x } } dx=\frac { 1 }{ 2 } [f(a)-f(1)]\), then one of the possible value of a is

    (a)

    3

    (b)

    6

    (c)

    9

    (d)

    5

  16. 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)

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

  17. The number of arbitrary constants in the general solutions of order n and n +1are respectively

    (a)

    n-1,n

    (b)

    n,n+1

    (c)

    n+1,n+2

    (d)

    n+1,n

  18. The number of arbitrary constants in the particular solution of a differential equation of third order is

    (a)

    3

    (b)

    2

    (c)

    1

    (d)

    0

  19. Subtraction is not a binary operation in

    (a)

    R

    (b)

    Z

    (c)

    N

    (d)

    Q

  20. If a compound statement involves 3 simple statements, then the number of rows in the truth table is

    (a)

    9

    (b)

    8

    (c)

    6

    (d)

    3

    1. Part II

      Answer any 7 questions. Question no. 27 is compulsory.

    7 x 2 = 14
  21. Find the rank of each of the following matrices:
    \(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

  22. Write \(\cfrac { 3+4i }{ 5-12i } \) in the x+iy form, hence find its real and imaginary parts.

  23. Find the modulus of the following complex numbers
    \(\cfrac { 2i }{ 3+4i } \)

  24. If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)

  25. Evaluate the following if z=5−2i and w= −1+3i
    2z+3w

  26. Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3 \right| \)

  27. Show that the equation x9-5x5+4x4+2x2+1=0 has atleast 6 imaginary solutions.

    1. Part III

      Answer any 7 questions. Question no. 34 is compulsory.

    7 x 3 = 21
  28. If w (x,y, z) =x2 +y2 +y2, x= et, y = esin t, z = et cos t, find \(\frac{dw}{dt}\)

  29. Let z(x, y) = x3 - 3x2y3, where x = set, y = se-t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \)

  30. Evaluate the following
    \(\int _{ 0 }^{ \pi /4 }{ { sin}^{ 6}x\quad dx } \)

  31. Solve the differential equation:
    x cos y dy = ex(x log x + 1)dx

  32. The probability density function of X is 

    Find P(0.2≤X<0.6)

  33. If the probability mass function f (x) of a random variable X isx

    x 1 2 3 4
    f (x) \(\cfrac { 1 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 1 }{ 12 } \)

    find (i) its cumulative distribution function, hence find
    (ii) P(X ≤ 3) and,
    (iii) P(X ≥ 2)

  34. Verify the
    (i) closure property,
    (ii) commutative property,
    (iii) associative property
    (iv) existence of identity and
    (v) existence of inverse for the arithmetic operation + on Z.

    1. Part IV

      Answer all the questions.

    7 x 5 = 35
  35. Verify Euler’s theorem for the function \(f(x,y)=\cfrac { 1 }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 } } } \)

  36. Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\cfrac { \pi }{ 3 } \) and x- axis are as 2 : 3.

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

  38. A thermometer reading 80°

  39. The probability distribution of a random variable X is given by

    X 0 1 2 3
    P(X) 0.1 0.3 0.5 0.1

    If Y=X2+3X, find the mean and the variance of Y.

  40. Show that (2018)2017+(2020)2017≡0(mod2019).

  41. Prove without using the truth table \(\sim \left( pV\left( qVr \right) \right) \equiv \left( \sim pV\sim q \right) V\left( -r \right) \)

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