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#### Full Portion Two Marks Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
25 x 2 = 50
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. Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

4. Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

5. Find z−1, if z=(2+3i)(1− i).

6. Simplify the following
${ i }^{ 59 }+\cfrac { 1 }{ { i }^{ 59 } }$

7. If z=$\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }$ , then show that Im (z) =0

8. Determine the number of positive and negative roots of the equation x9-5x4-14x2=0.

9. If sin ∝, cos ∝ are the roots of the equation ax2 + bx + c-0 (c ≠ 0), then prove that (n + c)2 - b2 + c2

10. Find the value of sec−1$\left( -\frac { 2\sqrt { 3 } }{ 3 } \right)$

11. Find the principal value of
${ Sin }^{ -1 }\left( sin\left( \frac { 5\pi }{ 6 } \right) \right)$

12. Ecalute $sin\left( { cos }^{ -1 }\left( \cfrac { 3 }{ 5 } \right) \right)$

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

14. For the ellipse x2 + 3y2 = a2, find the length of major and minor axis.

15. 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 $2\hat { i } +4\hat { j } -2\hat { k }$

16. Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line ${ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right)$

()

p=-1

17. A manufacturer wants to design an open box having a square base and a surface area of I 108 sq. em. Determine the dimensions of the box for the maximum volume.

18. Find the maximum and minimum values of f(x) = |x+3| ∀ $x\in R$.

19. Evaluate: $\int ^{log 2}_{-log 2} e ^{-|x|}$ dx.

20. Evaluate $\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx }$

21. Show that y = a cos bx is a solution of the differential equation $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0$.

22. Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values ofthe random variable X and number of points in its inverse images.

23. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
$a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q$

24. Let $A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right)$be any three boolean matrices of the same type.
Find (A∨B)∧C

25. In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.