Quarterly Model Question Paper

12th Standard EM

    Reg.No. :
  •  
  •  
  •  
  •  
  •  
  •  

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

*****************************************

TN 12th Standard EM Maths free Online practice tests

Reviews & Comments about 12th Maths Quarterly Model Question Paper

Write your Comment