#### 12th Standard English Medium Maths Reduced Syllabus Two Mark Important Questions with Answer key - 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 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 by minor method:
$\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right]$

3. Find the rank of the following matrices by minor method:
$\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right]$

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

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

6. Represent the complex number −1−i

7. Find the square roots of −6+8i

8. Show that the following equations represent a circle, and, find its centre and radius
$\left| 2z+2-4i \right| =2$

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

10. Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

11. Show that the polynomial 9x9+2x5-x4-7x2+2 has at least six imaginary roots.

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

13. Show that cot−1$\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ={ sec }^{ -1 }x,|x|>1$

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

15. Find all values of x such that
-5$\pi\le x \le 5\pi$ and cos x =1

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

17. 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 .

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

19. 2x2−y2=7

20. Find the length of the tangent from (2, -3) to the circle x2 + y2 - 8x - 9y + 12 = 0.

21. Find the volume of the parallelepiped whose coterminus edges are given by the vectors $\hat { 2i } -\hat { 3j } +\hat { 4k }$$\hat { i } -\hat { 2j } +\hat { 4k }$ and $\hat {3 i } -\hat { j } +\hat { 2k }$

22. 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 $2\hat { i } +4\hat { j } -2\hat { k }$

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

24. A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) =100×(1− 0.1t)2, 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

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

26. Find two positive numbers whose sum is 12 and their product is maximum.

27.

Compute the limit

$\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})$.

28. Find the asymptotes of the curve $f(x)=\frac { { 2x }^{ 2 }-8 }{ { x }^{ 2 }-16 }$

29. Find the equation of the tangent to the curve y2=4x+5 and which is parallel to y=2x+7

30. Evaluate the following limits, if necessary using L’Hopitalrule
(i) $\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x }$
(ii) $\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 }$
(iii) $\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } }$
(iv) $\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } }$

31. If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

32. Find differential dy for each of the following function
y = (3 + sin(2x)) 2/3

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

34. Evaluate $\int _{ 0 }^{ 1 }{ \left( \cfrac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) }$

35. Find the slope of the tangent to the curve $y=\int _{ 0 }^{ x }{ \cfrac { dt }{ 1+{ t }^{ 3 } } stx=1 }$

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

37. Express the physical statement in the form of the differential equation.
For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.

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

39. Determine the order and degree of $\cfrac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k$

40. Find the order and degree of $\left( \cfrac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \cfrac { dy }{ dx } \right) =0$

41. An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

42. 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)

43. 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 .

44. A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability distribution of the number of tosses.

45. The probability distribution of a random variable X is given under :

Find (i) k
(ii) E(X)

46. Is it possible that the mean of a binomial distribution is 15 and its standard deviation is 5?

47. Let A =$\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$be any two boolean matrices of the same type. Find AvB and A^B.

48. Determine whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)

49. 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

50. Give the truth value of (~pvq)v(~q)