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Important 2 Mark Book Back Questions (New Syllabus) 2020

12th Standard EM

    Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 76

    Part A

    38 x 2 = 76
  1. If A is a non-singular matrix of odd order, prove that |adj A| is positive

  2. Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

  3. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} -2 & 2 & -1 \\ 0 & 5 & 1 \\ 0 & 0 & 0 \end{matrix} \right] \)

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

  5. Write in polar form of the following complex numbers
    \(2+i2\sqrt { 3 } \)

  6. Simplify the following
    \(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

  7. Find the square roots of −6+8i

  8. Represent the complex numbe \(1+i\sqrt { 3 } \) in polar form.

  9. Find the monic polynomial equation of minimum degree with real coefficients having 2-\(\sqrt{3}\)i as a root.

  10. Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x9-4x8+4x7-3x6+2x5+x3+7x2+7x+2=0

  11. Find all the values of x such that
    -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x=0 

  12. Find the principal value of
    sec-1\((\frac{2}{\sqrt3})\)

  13. Find the period and amplitude of
    y=-sin\((\frac{1}{3}x)\)

  14. Find the principal value of
    cosec-1\((-\sqrt{2})\)

  15. Determine whether x+y−1=0 is the equation of a diameter of the circle x2+y2−6x+4y+c = 0 for all possible values of c .

  16. 3x2+2y2=14

  17. If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

  18. Find the vector and Cartesian form of the equations of a plane which is at a distance of 12 units from the origin and perpendicular to \(6\hat { i } +2\hat { j } -3\hat { k } \)

  19. Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\)=12 on the coordinate axes.

  20. Find the angle between the following lines.
    2x = 3y =  −z and 6x = − y = −4z.

  21. Find the point on the curve y = x2 − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

  22. Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals
    f(x) = |3x + 1|, x ∈ |-1, 3|

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

  24. A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9-8 cm. Find approximations for the following:
    change in the volume

  25. Find df for f(x) = x2 + x 3 and evaluate it for
    x = 3 and dx = 0.02

  26. In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
    \(U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right) \)

  27. Evaluate the following integrals using properties of integration:
    \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ({ x }^{ 5 }+xcos\quad x+{ tan }^{ 3 }x+1)dx } \)

  28. Evaluate the following
    \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)

  29. Find the area enclosed between the parabola y2=4ax and the line x=a, x=9a.

  30. A differential equation, determine its order, degree (if exists)
    \({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)

  31. Find the differential equation of the family of all non-vertical lines in a plane.

  32. Determine the order and degree (if exists) of the following differential equations: 
    \(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

  33. Solve the differential equation:
    \(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

  34. A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
    (i) the probability mass function
    (ii) the cumulative distribution function
    (iii) P(4 ≤ X < 10)
    (iv) P(X ≥ 6)

  35. A random variable X has the following probability mass function.

    x 1 2 3 4 5
    f(x) k2 2k2 3k2 2k 3k
  36. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function \(f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Find the expected life of this electronic equipment.

  37. Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining.’

  38. Which one of the following sentences is a proposition?
    (i) 4 + 7 =12
    (ii) What are you doing?
    (iii) 3n ≤ 8,1 n ∈ N
    (iv) Peacock is our national bird
    (v) How tall this mountain is!

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