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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