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#### Quarterly Model Question Paper

12th Standard EM

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Maths

Don't Write anything in the Question paper
Time : 02:45:00 Hrs
Total Marks : 90

Part - A

20 x 1 = 20
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. If A = $\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right]$ and AB = I , then B =

(a)

$\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A$

(b)

$\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }$

(c)

$\left( \cos ^{ 2 }{ \theta } \right) I$

(d)

(Sin2$\frac { \theta }{ 2 }$)A

3. If xayb = em, xcyd = en, Δ1 = $\left| \begin{matrix} m & b \\ n & d \end{matrix} \right|$, Δ2 = $\left| \begin{matrix} a & m \\ c & n \end{matrix} \right|$, Δ3 = $\left| \begin{matrix} a & b \\ c & d \end{matrix} \right|$, then the values of x and y are respectively,

(a)

e21), e31)

(b)

log (Δ13), log (Δ23)

(c)

log (Δ21), log(Δ31)

(d)

e(Δ13),e(Δ23)

4. If AT is the transpose of a square matrix A, then

(a)

|A| ≠ |AT|

(b)

|A| = |AT|

(c)

|A| + |AT| =0

(d)

|A| = |AT| only

5. If $\rho$(A) = $\rho$([A/B]) = number of unknowns, then the system is

(a)

consistent and has infinitely many solutions

(b)

consistent

(c)

inconsistent

(d)

consistent and has unique solution

6. If z = $\frac { 1 }{ (2+3i)^{ 2 } }$ then |z| =

(a)

$\frac { 1 }{ 13 }$

(b)

$\frac { 1 }{ 5}$

(c)

$\frac { 1 }{ 12 }$

(d)

none of these

7. If z1, z2, z3 are the vertices of a parallelogram, then the fourth vertex z4 opposite to z2 is _____

(a)

z1 + z2 - z2

(b)

z1 + z2 - z3

(c)

z1 + z2 - z3

(d)

z1 - z2 - z3

8. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

(a)

mn

(b)

m+n

(c)

mn

(d)

nm

9. 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

10. sin-1(2cos2x-1)+cos-1(1-2sin2x)=

(a)

$\frac{\pi}{2}$

(b)

$\frac{\pi}{3}$

(c)

$\frac{\pi}{4}$

(d)

$\frac{\pi}{6}$

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

(a)

0

(b)

$\cfrac { 1 }{ 2 }$

(c)

-1

(d)

none

12. The value of ${ sin }^{ -1 }\left( cos\cfrac { 33\pi }{ 5 } \right)$ is________

(a)

$\cfrac { 3\pi }{ 5 }$

(b)

$\cfrac { -\pi }{ 10 }$

(c)

$\cfrac { \pi }{ 10 }$

(d)

$\cfrac { 7\pi }{ 5 }$

13. If the coordinates at one end of a diameter of the circle x2+y2−8x−4y+c = 0 are (11,2) ,
the coordinates of the other end are

(a)

(-5,2)

(b)

(2,-5)

(c)

(5,-2)

(d)

(-2,5)

14. When the eccentricity of a ellipse becomes zero, then it becomes a

(a)

straight line

(b)

circle

(c)

point

(d)

parabola

15. The angle between the tangents drawn from (1, 4) to the parabola y2 = 4x is __________

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { \pi }{ 3 }$

(c)

$\frac { \pi }{ 5 }$

(d)

$\frac { \pi }{ 5 }$

16. If e1,e2 are eccentricities of the ellipse $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 and the hyperbola $\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } }$ = 1 then

(a)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$ = 1

(b)

${ e }_{ 1 }^{ 2 }$  + ${ e }_{ 2 }^{ 2 }$ = 1

(c)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$ = 2

(d)

${ e }_{ 1 }^{ 2 }$ - ${ e }_{ 2 }^{ 2 }$=2

17. If $\vec { a }$ and $\vec { b }$ are unit vectors such that $[\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 }$, then the angle between $\vec { a }$ and $\vec { b }$ is

(a)

$\frac { \pi }{ 6 }$

(b)

$\frac { \pi }{ 4 }$

(c)

$\frac { \pi }{ 3 }$

(d)

$\frac { \pi }{ 2 }$

18. If the planes $\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3$ and $\vec { r } =(4+\hat { j } -\mu \hat { k } )=5$ are parallel, then the value of λ and μ are

(a)

$\frac { 1 }{ 2 } ,-2$

(b)

$-\frac { 1 }{ 2 } ,2$

(c)

$-\frac { 1 }{ 2 } ,-2$

(d)

$\frac { 1 }{ 2 } ,2$

19. If $\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right)$, then

(a)

$\left| \overset { \rightarrow }{ d } \right|$

(b)

$\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c }$

(c)

$\overset { \rightarrow }{ d } =\overset { \rightarrow }{ 0 }$

(d)

a, b, c are coplanar

20. If $\left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b }$then the angle between the vector $\overset { \rightarrow }{ a }$ and $\overset { \rightarrow }{ b }$ is _____________

(a)

$\frac { \pi }{ 4 }$

(b)

$\frac { \pi }{ 3 }$

(c)

$\frac { \pi }{ 6 }$

(d)

$\frac { \pi }{ 2 }$

21. Part -B

7 x 2 = 14
22. If A is symmetric, prove that then adj Ais also symmetric.

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

24. Find the modulus of the complex number i25.

25. If cot-1$\frac{1}{7}=\theta$, find the value of cos$\theta$.

26. If ${ cot }^{ -1 }\left( \cfrac { 1 }{ 7 } \right) =\theta$ find the value of cos $\theta$

27. 3x2+3y2−4x+3y+10 = 0

28. 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

29. Part -C

7 x 3 = 21
30. Verify (AB)-1 = B-1A-1 with A = $\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right]$, B = $\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right]$.

31. Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.

32. Solve: ${ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 }$

33. Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
x2 = 24y

34. Find the condition for the line lx + my + n = 0 is Rtangent to the circle x2 + y2 = a2

35. Show that the lines $\vec { r } =(6\hat { i } +\hat { j } +2\hat { k } )+s(\hat { i } +2\hat { j } -3\hat { k } )$ and $\vec { r } =(3\hat { i } +2\hat { j } +2\hat { k } )+t(2\hat { i } +4\hat { j } -5\hat { k } )$ are skew lines and hence find the shortest distance between them.

36. Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

()

Cartesian equation

37. Part - D

7 x 5 = 35
38. If A = $\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & 4 \\ 2 & -4 & 3 \end{matrix} \right]$, verify thatA(adj A)=(adj A)A = |A| I3.

39. If A = $\left[ \begin{matrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{matrix} \right]$ and B = $\left[ \begin{matrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} \right]$, find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2.

40. Find the value of k for which the equations kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution

41. Show that $\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }$=-1

42. If 2+i and 3-$\sqrt{2}$ are roots of the equation x6-13x5+62x4-126x3+65x2+127x-140=0, find all roots.

43. If ${ tan }^{ -1 }\left( \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a$ than prove that x2=sin2a

44. If $\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k }$ are two given vector, then find a vector B satisfying the equations $\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B }$$\overset { \rightarrow }{ C }$ and $\overset { \rightarrow }{ A }$.$\overset { \rightarrow }{ B }$=3