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Model 2 Mark Book Back Questions (New Syllabus) 2020

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 70

    Part A

    35 x 2 = 70
  1. Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal

  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. Solve the following system of homogenous equations.
    2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

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

  5. If the area of the triangle formed by the vertices z,iz and z+iz is 50 square units, find the value of |z|

  6. Represent the complex number −1−i

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

  8. Obtain the Cartesian form of the locus of z=x+iy in
    \(\overline { z } =2^{ -1 }\)

  9. If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

  10. Solve: \(2\sqrt { \frac { x }{ a } } +3\sqrt { \frac { a }{ x } } =\frac { b }{ a } +\frac { 6a }{ b } \)

  11. Find all values of x such that
    6\(\pi\le x \le 6\pi\) and cos x -0 

  12. Simplify
    \({ cos }^{ -1 }\left( cos\left( \frac { 13\pi }{ 3 } \right) \right) \)

  13. Find the period and amplitude of
    y=4sin(−2x)

  14. Find the principal value of
    \({ Sin }^{ -1 }\left( sin\left( -\frac { \pi }{ 3 } \right) \right) \)

  15. If y=4x+c is a tangent to the circle x2+y2=9 , find c .

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

  17. Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \),\(\hat { i } +\hat { 2j } -\hat { 3k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)

  18. Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

  19. Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z=15.

  20. Find the angle of intersection of the curve y = sin x with the positive x -axis.

  21. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
    \(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

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

  23. 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

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

  25. Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\)where [⋅] is the greatest integer function

  26. Evaluate the following:
    \(\int _{ 0 }^{ \infty }{ { x }^{ 5 }{ e }^{ -3x }dx } \)

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

  28. 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.

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

  30. Show that y=mx+\(\frac{7}{m}\),m≠0 is a solution of the differential equation xy'+7\(\frac{1}{y'}\)-y=0.

  31. 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)

  32. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
    \(f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 3 } & 0<x<30 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Obtain and interpret the expected value of the random variable X .

  33. 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.’

  34. Determine the truth value of each of the following statements
    (i) If 6 + 2 = 5 , then the milk is white.
    (ii) China is in Europe or \(\sqrt3\) is an integer
    (iii) It is not true that 5 + 5 = 9 or Earth is a planet
    (iv) 11 is a prime number and all the sides of a rectangle are equal

  35. Construct the truth table for the following statements.
    ( p V q) ∧ ¬q

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