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12th Standard English Medium Maths Reduced Syllabus Creative Two Mark Questions with Answer key - 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. For the matrix A, if A3 = I, then find A-1.

  2. Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

  3. Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

  4. Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.

  5. Find Re (z) and im (z) if z = 5i11 + 7i3

  6. If z=\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\) , then show that Im (z) =0

  7. Find the argument of -2

  8. Find value of a for which the sum of the squares of the equation x2 - (a- 2) x - a-1=0 assumes the least value.

  9. Find the number of positive and negative roots of the equation x7 - 6x6 + 7x5 + 5x2+2x+2

  10. If \({ cot }^{ -1 }\left( \cfrac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

  11. Ecalute \(sin\left( { cos }^{ -1 }\left( \cfrac { 3 }{ 5 } \right) \right) \)
     

  12. Find the equation of tangent to the circle x2 +y2 + 2x - 3y - 8 = 0 at (2, 3).

  13. If the line y = 3x + 1, touches the parabola y2 = 4ax, find the length of the latus rectum?

  14. For the ellipse x2 + 3y2 = a2, find the length of major and minor axis.

  15. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

    ()

    -1

  16. Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3.as direction ratios of normal to the plane.

    ()

    2

  17. Flnd 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)

    ()

    x = -1 is one root

  18. Find the equation of the tangent to the curve y2=4x+5 and which is parallel to y=2x+7

  19. Verify Rolle ’s Theorem for \(f(x)=\left| x-1 \right| ,O\le x\le 2\) 

  20. Obtain Maclaurin’s Series expansion for e2x.

  21. Evaluate the following limits, if necessary using L’Hopitalrule
    (i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
    (ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
    (iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
    (iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

  22. Find the absolute extreme of the function f(x)=x2-2x+2 on the closed interval [0,3]

  23. Prove that the function f(x)=2x2+3x is strictly increasing on \(\left[ -\cfrac { 1 }{ 2 } ,\cfrac { 1 }{ 2 } \right] \)

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

  25. IF u(x, y) = x2 + 3xy + y2, x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

  26. If w=log(x2+y2),x=cosθ,y=sinθ, find \(\cfrac { dw }{ d\theta } \)

  27. Without using any kind of computational aid use linear approximation to find the value of e0.1

  28. Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x=0, x=4.

  29. Find the volume of the solid obtained by revolving the area of the triangle whose sides are x = 4, y = 0 and 3x - 4y = 0 about x - axis

  30. If \(\int _{ 0 }^{ \infty }{ \cfrac { { x }^{ 2 }dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right) \left( { x }^{ 2 }+{ c }^{ 2 } \right) } } =\cfrac { \pi }{ 2(a+b)(b+c)(c+a) } \) then find \(\int _{ 0 }^{ \infty }{ \cfrac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } } \)

  31. Find the area bounded by y=x2+2, x-axis, x=1 and x=2.

  32. Find the area of the region bounded by the curve y=sin x and the ordinate x=0.\(x=\cfrac { \pi }{ 3 } \)

  33. Find the area bounded by y=cosx,y=x+1,y=0.

  34. Determine the order and degree of \(\cfrac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

  35. Find the order and degree of \(\left( \cfrac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \cfrac { dy }{ dx } \right) =0\)

  36. Form the Differential Equation representing the family of curves y=Acos(x+B) where A and B are parameters.

  37. Solve : \(\cfrac { dy }{ dx } =\cfrac { { e }^{ x }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

  38. How many types of random variables are there? What are they?

  39. Define Probability Mass Function.

  40. Define Variance of a random variable X?

  41. What is the Probability Mass function of a binomial random variable?

  42. The probability distribution of a random variable X is given under :

    Find (i) k
    (ii) E(X)

  43. The p.d.f of a continuous random variable X is
    \(f(x)=\begin{cases} k,0\le x\le 4 \\ 0,otherwise \end{cases}\) Find k.

  44. Consider the binary operation ∗ defined on the set A = {a,b,c, d} by the following table:

    * a b c d
    a a c b d
    b c d a a
    d d b a c

    Is it commutative and associative?

  45. In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.

  46. In a non empty set S on which a binary operation * is defined and for an element a∈S.a✳️a✳️a=e,where e is the identity element. Find the inverses of a and a✳️a.

  47. If a☰b(mod n) and C☰d(mod n),check whether a+c☰(b+d) (mod n).

  48. Form the truth table of (~q)^p.

  49. Are ~pV(pVq) and pV(~pvq) tatology statements. Justify your answer.

  50. Give the truth table of ~p➝~q

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