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12th Standard English medium Maths Reduced Syllabus Two Mark Important Questions - 2021(Public Exam )

12th Standard

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Maths

Time : 02:45:00 Hrs
Total Marks : 100

    2 Marks

    50 x 2 = 100
  1. Find the inverse (if it exists) of the following:
    \(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

  2. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)

  3. Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

  4. Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

  5. If z= 1 - 3i, z= - 4i, and z3 = 5 , show that (z+ z2) + z= z1+ (z+ z3)

  6. Which one of the points i, −2 + i, and 3 is farthest from the origin?

  7. Find the principal argument Arg z, when z = \(\frac { -2 }{ 1+i\sqrt { 3 } } \)

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

  9. Find the modulus of the following complex numbers
    (1-i)10

  10. Find the modulus of the following complex numbers
     2i(3−4i)(4−3i).

  11. Find the modulus and principal argument of the following complex numbers.
    \(\sqrt { 3 } \)-i

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

  13. Show that if p, q, r  are rational the roots of the equation x− 2px + p− q+ 2qr − r= 0 are rational.

  14. Discuss the nature of the roots of the following polynomials:
    x5-19x4+ 2x3+ 5x2+11

  15. Find the value of
    \(2{ cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) \)

  16. If tan-1 x + tan-1y + tan-1 z = \(\pi\), show that x + y + z = xyz

  17. Find the value of 
    tan(tan-1(-0.2021)).

  18. Simplify \({ sec }^{ -1 }\left( sec\left( \frac { 5\pi }{ 3 } \right) \right) \)

  19. Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) \right) \)

  20. Identify the type of the conic for the following equations :
    11x2−25y2−44x+50y−256 = 0

  21. Find centre and radius of the following circles.
    x2+y2−x+2y−3 = 0

  22. Show that the lines \(\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 } \) and \(\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 } \) are parallel.

  23. Find the angle between the straight line \(\vec { r } =(2\hat { i } +\hat { j } +\hat { k } )+t(\hat { i } -\hat { j } +\hat { k } )\) and the plane 2x-y+z = 5

  24. Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

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    x = -1 is one root

  25. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
    f(x) = x2 − x, x ∈ [0, 1]

  26. Evaluate the following limit, if necessary use l ’Hôpital Rule
    \(\underset { x\rightarrow \infty }{ lim } \frac { x }{ logx } \) 

  27. Verify Lagrange’s Mean Value theorem for \(f(x)=\sqrt { x-2 } \) in the interva [2,6]

  28. Prove that the function f(x)=e-x is strictky increasing on [0,1]

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

  30. If U(x, y, z) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y\), find \(\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y } \) and \(\frac { \partial U }{ \partial z } \)

  31. Evaluate the following definite integrals:
    \(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)

  32. Evaluate the following \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)

  33. Evaluate \(\int _{ 1 }^{ 2 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

  34. For each of the following differential equations, determine its order, degree (if exists)
    \(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

  35. Show that y = e−x + mx + n is a solution of the differential equation ex \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

  36. Determine the order and degree (if exists) of the following differential equations: 
    dy + (xy − cos x)dx = 0

  37. solve: x dy + y dx = xy dx

  38. The cumulative distribution function of a discrete random variable is given by

    Find
    (i) the probability mass function
    (ii) P(X < 3) and
    (iii) P(X \(\ge \)2).

  39. Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
    (i) ¬p
    (ii) p ∧ ¬q
    (iii) ¬p ∨ q
    (iv) p➝ ¬q
    (v) p↔q

  40. Construct the truth table for the following statements.
    (¬p ⟶ r) ∧ ( p ↔️ q)

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