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Sample 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. Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

  2. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)

  3. Find the rank of each of the following matrices:
    \(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

  4. If z1=3-2i and z2=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

  5. Obtain the Cartesian equation for the locus of z=x+iy in
    |z-4|=16

  6. Simplify the following
    \({ i }^{ 59 }+\cfrac { 1 }{ { i }^{ 59 } } \)

  7. Find the modulus of the following complex number \(\cfrac { 2-i }{ 1+i } +\cfrac { 1-2i }{ 1-i } \)

  8. Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3 \right| \)

  9. Find a polynomial equation of minimum degree with rational coefficients, having 2+\(\sqrt{3}\)i as a root.

  10. Construct a cubic equation with roots 1,1, and −2

  11. Find the period and amplitude of
    y=sin 7x

  12. If cot-1\(\frac{1}{7}=\theta\), find the value of cos\(\theta\).
     

  13. Show that \(cot(sin^{ -1 }x)=\frac { \sqrt { 1-x^{ 2 } } }{ x } -1\le x\le \)and x \(\neq \) 0

  14. Find the value of
    \({ cos }^{ -1 }\left( cos\frac { \pi }{ 7 } cos\frac { \pi }{ 17 } -sin\frac { \pi }{ 17 } sin\frac { \pi }{ 17 } \right) .\)

  15. Simplify
    \({ tan }^{ -1 }\left( tan\left( \frac { 3\pi }{ 4 } \right) \right) \)

  16. Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

  17. x2+y2+x−y=0

  18. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

  19. Find the vector and Cartesian equations of the plane passing through the point with position vector \(4\hat { i } +2\hat { j } -3\hat { k } \) and normal to vector \(2\hat { i } -\hat { j } +\hat { k } \)

  20. Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z =5.

  21. If the volume of a cube of side length x is v = x3. Find the rate of change of the volume with respect to x when x = 5 units.

  22. 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]

  23. Evaluate the following limit, if necessary use l’Hôpital Rule
    \(\underset { x\rightarrow 0 }{ lim } \frac { 1-cosx }{ { x }^{ 2 } } \)

  24. Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

  25. Find differential dy for each of the following function
    y = ex2-5x+7 cos (x2 - 1)

  26. In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
    f(x, y) = x2y + 6x3 + 7

  27. Evaluate the following definite integrals:
    \(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)

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

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

  30. A differential equation, determine its order, degree (if exists)
    \(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

  31. Express the physical statement in the form of the differential equation.
    The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.

  32. Find value of m so that the function y = emx is a solution of the given differential equation.
    y''− 5y' + 6y = 0

  33. Find the differential equation of the family of parabolas y2 ax = 4, where a is an arbitrary constant.

  34. Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred

  35. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
    (i) exactly 10 will have a useful life of at least 600 hours;
    (ii) at least 11 will have a useful life of at I least 600 hours;  
    (iii) at least 2 will not have a useful life of at : least 600 hours.

  36. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
    a*b = a + 3ab − 5b2;∀a,b∈Z

  37. Write each of the following sentences in symbolic form using statement variables p and q.
    (i) 19 is not a prime number and all the angles of a triangle are equal.
    (ii) 19 is a prime number or all the angles of a triangle are not equal
    (iii) 19 is a prime number and all the angles of a triangle are equal
    (iv) 19 is not a prime number

  38. Write the converse, inverse, and contrapositive of each of the following implication.
    (i) If x and y are numbers such that x = y, then x2 = y2
    (ii) If a quadrilateral is a square then it is a rectangle.

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