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12th Standard Maths English Medium Reduced Syllabus Model Question paper with answer key - 2021 Part - 1

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 is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = 

    (a)

    A

    (b)

    B

    (c)

    I

    (d)

    BT

  2. If (AB)-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B-1 = 

    (a)

    \(\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right] \)

  3. If ATA−1 is symmetric, then A2 =

    (a)

    A-1

    (b)

    (AT)2

    (c)

    AT

    (d)

    (A-1)2

  4. If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

    (a)

    \(\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right] \)

  5. Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

    (a)

    2

    (b)

    4

    (c)

    3

    (d)

    1

  6. If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

    (a)

    \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right] \)

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

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    5

  8. If |z|=1, then the value of \(\cfrac { 1+z }{ 1+\overline { z } }\) is

    (a)

    z

    (b)

    \(\bar { z } \)

    (c)

    \(\cfrac { 1 }{ z } \)

    (d)

    1

  9. If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \) ,then (A,B) equals

    (a)

    (1,0)

    (b)

    (−1,1)

    (c)

    (0,1)

    (d)

    (1,1)

  10. The principal argument of the complex number \(\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

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

    (a)

    2\(\pi\)

    (b)

    \(\pi\)

    (c)

    0

    (d)

    tan-1\(\frac{12}{65}\)

  12. The equation of the normal to the circle x2+y2−2x−2y+1=0 which is parallel to the line
    2x+4y=3 is

    (a)

    x+2y=3

    (b)

    x+2y+3= 0

    (c)

    2x+4y+3=0

    (d)

    x−2y+3= 0

  13. If P(x, y) be any point on 16x2+25y2=400 with foci F1 (3,0) and F2 (-3,0) then PF1 PF2 +
    is

    (a)

    8

    (b)

    6

    (c)

    10

    (d)

    12

  14. 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

  15. 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)

    \( { \pi }\)

  16. If the distance of the point (1,1,1) from the origin is half of its distance from the plane x + y + z + k =0, then the values of k are

    (a)

    \(\pm 3\)

    (b)

    \(\pm 6\)

    (c)

    -3, 9

    (d)

    3, 9

  17. The function sin4 x + cos4X is increasing in the interval

    (a)

    \(\left[ \cfrac { 5\pi }{ 8 } ,\cfrac { 3\pi }{ 4 } \right] \)

    (b)

    \(\left[ \cfrac { \pi }{ 2 } ,\cfrac { 5\pi }{ 8 } \right] \)

    (c)

    \(\left[ \cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 2 } \right] \)

    (d)

    \(\left[ 0,\cfrac { \pi }{ 4 } \right] \)

  18. 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

  19. 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

  20. 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 + \(\frac{\pi}{2}\)

    1. Part II

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


    7 x 2 = 14
  21. Show that (1+x)n=1+nx approximately if x is close to zero.

  22. Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

  23. Find the area bounded by the curve y=sin2x between the ordinates x=0.x=π and x-axis.

  24. Find the order and degree of \(\\ y+\cfrac { dy }{ dx } =\cfrac { 1 }{ 4 } \int { ydx } \)

  25. Form the D.E of family of parabolas having vertex at the origin and axis along positive y-axis.

  26. Find the variance of the binomial distribution with parameters 8 and \(\cfrac { 1 }{ 4 } \)

  27. If a≡b(mod n) and b≡c(mod n),check whether a≡c(mod n).

    1. Part III

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


    7 x 3 = 21
  28. Show that \(\left( \cfrac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \cfrac { 20-5i }{ 7-6i } \right) ^{ 12 }\) is real

  29. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\cfrac { 3\pi }{ 2 } \).

  30. Obtain the Cartesian form of the locus of z in
    |2z-3-i|=3

  31. If p and q are the roots of the equation lx2+nx+n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \)=0.

  32. How many rows are needed for following statement formulae?
    p ∨ ¬ t ( p ∨ ¬s)

  33. How many rows are needed for following statement formulae?
    (( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

  34. 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 AΛB

    1. Part IV

      Answer all the questions.


    7 x 5 = 35
  35. Solve the Linear differential equation:
    \((x+a)\frac { dy }{ dx } -2y={ (x+a) }^{ 4 }\)

  36. The probability density function of X is given
    \(f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}\)
    Find
    (i) the value of k
    (ii) the distribution function.
    (iii) P(X <3)
    (iv) P(5 ≤X)
    (v) P(X ≤ 4)

  37. A retailer purchases a certain kind of' electronic device from a manufacturer. The manufacturer ,indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
    (i) at least one defective item
    (ii) exactly two defective items.

  38. 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 ) .

  39. Verify
    (i) closure property,
    (ii) commutative property,
    (iii) associative property,
    (iv) existence of identity, and
    (v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5.

  40. 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.

  41. Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.

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