12th Standard English medium Maths Reduced Syllabus Two Mark Important Questions - 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. Find the inverse (if it exists) of the following:
$\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right]$

2. Find the rank of the following matrices by minor method:
$\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right]$

3. Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

4. Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

5. If z1=1-3i,z2=4i, and z3 = 5 , show that (z1+z2)+z3=z1+(z2+z3)

6. Which one of the points i,−2+i , and 3 is farthest from the origin?

7. Find the principal argument Arg z , when z = $\cfrac { -2 }{ 1+i\sqrt { 3 } }$

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

9. Find the modulus of the following complex numbers
(1-i)10

10. Find the modulus of the following complex numbers
2i(3−4i)(4−3i).

11. Find the modulus and principal argument of the following complex numbers.
$\sqrt { 3 }$-i

12. If (cosθ + i sinθ)2 = x + iy, then show that x2+y2 =1

13. Show that, if p,q,r are rational, the roots of the equation x2−2px+p2−q2+2qr−r2=0 are rational.

14. Discuss the nature of the roots of the following polynomials:
x5-19x4+2x3+5x2+11

15. Find the value of
$2{ cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right)$

16. If tan-1 x+tan-1y+tan-1z=$\pi$, show that x+y+z+=xyz

17. Find the value of
tan(tan-1(-0.2021)).

18. Simplify
${ sec }^{ -1 }\left( sec\left( \frac { 5\pi }{ 3 } \right) \right)$

19. Evaluate $sin\left( { cos }^{ -1 }\left( \cfrac { 1 }{ 2 } \right) \right)$

20. 11x2−25y2−44x+50y−256 = 0

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

22. Show that the lines $\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 }$ and $\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 }$ are parallel.

23. Find the angle between the straight line $\vec { r } =(2\hat { i } +\hat { j } +\hat { k } )+t(\hat { i } -\hat { j } +\hat { k } )$ and the plane 2x-y+z=5

24. 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

25. 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]

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

27. Verify Lagrange’s Mean Value theorem for $f(x)=\sqrt { x-2 }$ in the interva [2,6]

28. Prove that the function f(x)=e-x is strictky increasing on [0,1]

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

30. If U(x, y, z) = $\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y$, find $\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y }$ and $\frac { \partial U }{ \partial z }$

31. If u=x2+3xy2+y2, then prove that $\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } =\cfrac { { \partial }^{ 2 }u }{ \partial y\partial x }$

32. If f(x,y)=10x-x2+xy find
(i) fx(2,6)
(ii)fy(2,6)

33. If $u={ e }^{ \frac { x }{ y } }sin\left( \cfrac { x }{ y } \right) +{ e }^{ \frac { y }{ x } }cos\left( \cfrac { y }{ x } \right)$ ,then prove that $x\cfrac { \vartheta u }{ \vartheta x } +y\cfrac { \vartheta u }{ \vartheta y } =0$

34. Evaluate the following definite integrals:
$\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } }$

35. Evaluate the following
$\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx }$

36. Find the volume of the solid generated when the region enclosed by $y=\sqrt { x } ,y=2$  and x = 0 is revolved about y-axis.

37. Evaluate $\int _{ 1 }^{ 2 }{ \cfrac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx }$

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

39. Show that y = e−x + mx + n is a solution of the differential equation ex $\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right)$ -1 = 0

40. Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0

41. solve:xdy+ydx=xydx

42. The cumulative distribution function of a discrete random variable is given by

Find
(i) the probability mass function
(ii) P(X < 3) and
(iii) P(X $\ge$2).

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

44. Define Variance of a random variable X?

45. Find the variance of the binomial distribution with parameters 8 and $\cfrac { 1 }{ 4 }$

46. Let X be a continuous random variable with p.d.f
$f(x)=\begin{cases} 2x,0\le x\le 1 \\ 0,otherwise \end{cases}$.Find E(X)

47. Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give
verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

48. Construct the truth table for the following statements.
(¬p ⟶ r) ∧ ( p ↔️ q)

49. Is cross product commutative on the set of vectors? Justify your answer.

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