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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 \(\times\) 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = 

    (a)

    A

    (b)

    B

    (c)

    I3

    (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 \(\left| z-\frac { 3 }{ z } \right| =2\)then the least value |z| is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    5

  8. If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) 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 \(\frac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  11. \(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) 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 x+ y− 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 16x+ 25y= 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 \(\vec { a } \ and \ \vec { b } \) is

    (a)

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

    (b)

    \(\frac { 3\pi }{ 4 } \)

    (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 + cos4 x is increasing in the interval

    (a)

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

    (b)

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

    (c)

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

    (d)

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

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

    (a)

    1

    (b)

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

    (c)

    \(\frac { 3 }{ 2 } \)

    (d)

    2

  19. If u(x, y) = x2+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) 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. Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

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

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

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

    1. Part III

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


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

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

  27. Obtain the Cartesian form of the locus of z in in each of the following cases.
    |2z - 3 - i| = 3

  28. If p and q are the roots of the equation 1x2+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

  29. How many rows are needed for following statement formulae?
    \(p \vee \neg t \wedge(p \vee \neg s)\)

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

  31. 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
  32. Solve the Linear differential equation:
    \((x+a)\frac { dy }{ dx } -2y={ (x+a) }^{ 4 }\)

  33. 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)

  34. 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.

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

  36. 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.

  37. 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.

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

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