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12th Standard Maths English Medium Free Online Test Volume 1 One Mark Questions with Answer Key 2020

12th Standard

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Maths

Time : 00:10:00 Hrs
Total Marks : 10
    Answer all the questions
    25 x 1 = 25
  1. Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

    (a)

    ±2

    (b)

    ±4

    (c)

    ±8

    (d)

    ±12

  2. If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is

    (a)

    sinx cosx

    (b)

    1

    (c)

    2

    (d)

    none

  3. If A is a non-singular matrix then IA-1|= ______

    (a)

    \(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)

    (b)

    \(\frac { 1 }{ |A^{ 2 }| } \)

    (c)

    \(\left| \frac { 1 }{ A } \right| \)

    (d)

    \(\frac { 1 }{ |A| } \)

  4. The area of the triangle formed by the complex numbers z,iz, and z+iz in the Argand’s diagram is

    (a)

    \(\cfrac { 1 }{ 2 } \left| z \right| ^{ 2 }\)

    (b)

    |z|2

    (c)

    \(\cfrac { 3 }{ 2 } \left| z \right| ^{ 2 }\)

    (d)

    2|z|2

  5. If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \) =3k, then k is equal to 

    (a)

    1

    (b)

    -1

    (c)

    \(\sqrt { 3i } \)

    (d)

    \(-\sqrt { 3i } \)

  6. If z=\(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) =

    (a)

    0

    (b)

    \(\frac{1}{2}\)

    (c)

    cot\(\frac { \theta }{ 2 } \)

    (d)

    \(\frac{1}{2}\)cot\(\frac { \theta }{ 2 } \)

  7. The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

    (a)

    \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

    (b)

    \(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

    (c)

    \(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

    (d)

    \(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

  8. The polynomial x3+2x+3 has

    (a)

    one negative and two real roots

    (b)

    one positive and two imaginary roots

    (c)

    three real roots

    (d)

    no solution

  9. Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

    (a)

    n

    (b)

    n -1

    (c)

    n+1

    (d)

    (n-r)

  10. If (2+√3)x2-2x+1+(2-√3)x2-2x-1=\(\frac { 2 }{ 2-\sqrt { 3 } } \) then x=

    (a)

    0,2

    (b)

    0,1

    (c)

    0,3

    (d)

    0, √3

  11. If cot-1 2 and cot-1 3 are two angles of a triangle, then the third angle is

    (a)

    \(\frac{\pi}{4}\)

    (b)

    \(\frac{3\pi}{4}\)

    (c)

    \(\frac{\pi}{6}\)

    (d)

    \(\frac{\pi}{3}\)

  12. If sin-1 \(\frac{x}{5}+ cosec^{-1}\frac{5}{4}=\frac{\pi}{2}\), then the value of x is

    (a)

    4

    (b)

    5

    (c)

    2

    (d)

    3

  13. ·If \(\alpha ={ tan }^{ -1 }\left( \cfrac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \cfrac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \)

    (a)

    \(\cfrac { \pi }{ 6 } \)

    (b)

    \(\cfrac { \pi }{ 3 } \)

    (c)

    \(\cfrac { \pi }{ 2 } \)

    (d)

    \(\cfrac { -\pi }{ 3 } \)

  14. The value of sin(2(tan-1 0.75) is___________

    (a)

    0.75

    (b)

    1.5

    (c)

    0.96

    (d)

    sin-1(1.5)

  15. The centre of the circle inscribed in a square formed by the lines x2−8x−12=0 and y2−14y+45 = 0 is

    (a)

    (4,7)

    (b)

    (7,4)

    (c)

    (9,4)

    (d)

    (4,9)

  16. The ellipse E1\(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point(0,4) circumscribes the rectangle R . The eccentricity of the ellipse is

    (a)

    \(\frac { \sqrt { 2 } }{ 2 } \)

    (b)

    \(\frac { \sqrt { 3 } }{ 2 } \)

    (c)

    \(\frac { 1 }{ 2 } \)

    (d)

    \(\frac { 3 }{ 4 } \)

  17. Area of the greatest rectangle inscribed in the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1.\) is

    (a)

    2ab

    (b)

    ab

    (c)

    \( \sqrt{ ab}\)

    (d)

    \(\frac { a }{ b } \)

  18. The equation of tangent at (1, 2) to the circle x+ y2 = 5 is

    (a)

    x+y=3

    (b)

    x + 2y = 3

    (c)

    x- y= 5

    (d)

    x - 2y = 5

  19. The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is

    (a)

    2

    (b)

    4

    (c)

    6

    (d)

    5

  20. The locus of the foot of perpendicular from the focus on any tangent to y2 = 4ax is

    (a)

    x2 + y2 = a2 - b2

    (b)

    x2 + y2 = a2

    (c)

    x2 + y2 = a2 - b2

    (d)

    x = 0

  21. 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)

    \(\frac { \pi }{ 2 } \)

  22. 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

  23. The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is

    (a)

    30o

    (b)

    60o

    (c)

    45o

    (d)

    90o

  24. The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is

    (a)

    \(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

    (b)

    \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \)

    (c)

    \(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }-{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

    (d)

    \({ \left| \overset { \rightarrow }{ a } \right| }^{ 2 }{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 }\)

  25. If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are three non - coplanar vectors, then \(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } +\frac { \overset { \rightarrow }{ b } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } } \)=_____________

    (a)

    0

    (b)

    1

    (c)

    -1

    (d)

    \(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } .\overset { \rightarrow }{ c } } \)

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