" /> -->

#### Important 2 Mark Book Back Questions (New Syllabus) 2020

12th Standard EM

Reg.No. :
•
•
•
•
•
•

Maths

Time : 01:00:00 Hrs
Total Marks : 76

Part A

38 x 2 = 76
1. If A is a non-singular matrix of odd order, prove that |adj A| is positive

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. Find the rank of the following matrices which are in row-echelon form :
$\left[ \begin{matrix} -2 & 2 & -1 \\ 0 & 5 & 1 \\ 0 & 0 & 0 \end{matrix} \right]$

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

5. Write in polar form of the following complex numbers
$2+i2\sqrt { 3 }$

6. Simplify the following
$\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } }$.

7. Find the square roots of −6+8i

8. Represent the complex numbe $1+i\sqrt { 3 }$ in polar form.

9. Find the monic polynomial equation of minimum degree with real coefficients having 2-$\sqrt{3}$i as a root.

10. Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x9-4x8+4x7-3x6+2x5+x3+7x2+7x+2=0

11. Find all the values of x such that
-10$\pi$$\le x\le$10$\pi$ and sin x=0

12. Find the principal value of
sec-1$(\frac{2}{\sqrt3})$

13. Find the period and amplitude of
y=-sin$(\frac{1}{3}x)$

14. Find the principal value of
cosec-1$(-\sqrt{2})$

15. 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 .

16. 3x2+2y2=14

17. If $\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k }$ are coplanar, find the value of m.

18. Find the vector and Cartesian form of the equations of a plane which is at a distance of 12 units from the origin and perpendicular to $6\hat { i } +2\hat { j } -3\hat { k }$

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

20. Find the angle between the following lines.
2x = 3y =  −z and 6x = − y = −4z.

21. Find the point on the curve y = x2 − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

22. Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals
f(x) = |3x + 1|, x ∈ |-1, 3|

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

24. 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

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

26. In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
$U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right)$

27. Evaluate the following integrals using properties of integration:
$\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ({ x }^{ 5 }+xcos\quad x+{ tan }^{ 3 }x+1)dx }$

28. Evaluate the following
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx }$

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

30. A differential equation, determine its order, degree (if exists)
${ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0$

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

32. Determine the order and degree (if exists) of the following differential equations:
$3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }$

33. Solve the differential equation:
$\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } }$

34. 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)

35. A random variable X has the following probability mass function.

 x 1 2 3 4 5 f(x) k2 2k2 3k2 2k 3k
36. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function $f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}$
Find the expected life of this electronic equipment.

37. 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.’

38. Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3n ≤ 8,1 n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!