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#### Term 1 Model Question Paper

12th Standard EM

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Maths

Time : 02:00:00 Hrs
Total Marks : 60

Part- A

10 x 1 = 10
1. If |adj(adj A)| = |A|9, then the order of the square matrix A is

(a)

3

(b)

4

(c)

2

(d)

5

2. The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

(a)

0

(b)

1

(c)

2

(d)

infinitely many

3. The value of $\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) }$ is

(a)

1+ i

(b)

i

(c)

1

(d)

0

4. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

(a)

-1

(b)

$\frac { 5 }{ 4 }$

(c)

$\frac { 4 }{ 5 }$

(d)

5

5. If sin-1 x+sin-1 y+sin-1 z=$\frac{3\pi}{2}$, the value of x2017+y2018+z2019$-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } }$is

(a)

0

(b)

1

(c)

2

(d)

3

6. ${ tan }^{ -1 }\left( \cfrac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \cfrac { 2 }{ 11 } \right)$ =

(a)

0

(b)

$\cfrac { 1 }{ 2 }$

(c)

-1

(d)

none

7. The radius of the circle3x2+by2+4bx−6by+b2 =0 is

(a)

1

(b)

3

(c)

$\sqrt {10}$

(d)

$\sqrt {11}$

8. If x+y=k is a normal to the parabola y2 =12x, then the value of k is

(a)

3

(b)

-1

(c)

1

(d)

9

9. If $\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c }$ where $\vec { a } ,\vec { b } ,\vec { c }$ are any three vectors such that $\vec { a } ,\vec { b }$ $\neq$ 0 and  $\vec { a } .\vec { b }$ $\neq$ 0 then $\vec { a }$ and $\vec { c }$ are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle $\frac{\pi}{3}$

(d)

inclined at an angle  $\frac{\pi}{6}$

10. If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is $\frac{1}{5}$ then the value of λ is

(a)

$2\sqrt { 3 }$

(b)

$3\sqrt { 2 }$

(c)

0

(d)

1

11. Part - B

5 x 1 = 5

13. (1)

14. Im(z)

15. (2)

$\frac { z-\bar { z } }{ 2 }$

16. $\left| -\sqrt { 3 } +i \right|$

17. (3)

Reciprocal equation of type I

18. p(x)=xn.p$\left( \frac { 1 }{ x } \right)$

19. (4)

$\cfrac { \pi }{ 3 }$

20. ${ sin }^{ -1 }\left( sin\cfrac { 2\pi }{ 3 } \right)$

21. (5)

2

Part - C

6 x 2 = 12
22. Prove that $\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right]$ is orthogonal

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

24. Construct a cubic equation with roots 1,2, and 3

25. Solve tan-1 $\left( \frac { 1-x }{ 1+x } \right) =\frac { 1 }{ 2 } { tan }^{ -1 }$ x for x>0

26. Find the general equation of the circle whose diameter is the line segment joining the points (−4,−2)and (1,1).

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

28. Part - D

6 x 3 = 18
29. If A = $\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right]$ and B = $\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right]$, verify that (AB)-1 = B-1A-1

30. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

31. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when $\theta =\cfrac { \pi }{ 3 }$.

32. Solve the equation x4-9x2+20=0.

33. Solve sin-1 x> cos-1x

34. Find the centre and radius of the circle3x2+(a+1)y2+6x−9y+a+4=0.

35. Part- E

3 x 5 = 15
36. If A = $\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right]$, find x and y such that A2 + xA + yI2 = O2. Hence, find A-1.

37. Find all cube roots of $\sqrt { 3 } +i$

38. ABCD is a quadrilateral with $\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha }$ and $\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta }$ and $\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta }$. If. the area of the quadrilateral is λ times the area of the parallelogram with $\overset { \rightarrow }{ AB }$ and $\overset { \rightarrow }{ AD }$ as adjacent sides, then prove that $\lambda =\frac { 5 }{ 2 }$

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