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

12th Standard

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Maths

Time : 00:25:00 Hrs
Total Marks : 25

    Answer all the questions

    25 x 1 = 25
  1. If A is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = 

    (a)

    A

    (b)

    B

    (c)

    I

    (d)

    BT

  2. If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then

    (a)

    a2 + b2 + c2 = 1

    (b)

    abc ≠ 1

    (c)

    a + b + c =0

    (d)

    a2 + b2 + c2 + 2abc =1

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

    (a)

    1+ i

    (b)

    i

    (c)

    1

    (d)

    0

  5. The principal value of the amplitude of (1+i) is

    (a)

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

    (b)

    \(\frac { \pi }{ 12 } \)

    (c)

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

    (d)

    \(\pi \)

  6. The amplitude of \(\frac{1}{i}\) is equal to

    (a)

    0

    (b)

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

    (c)

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

    (d)

    \(\pi \)

  7. 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

  8. 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

  9. sin−1(cos x)\(=\frac{\pi}{2}-x \) is valid for

    (a)

    \(-\pi \le x\le 0\)

    (b)

    \(0\pi \le x\le 0\)

    (c)

    \(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

    (d)

    \(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

  10. ·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)

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

  11. \(cot\left( \cfrac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

    (a)

    7

    (b)

    6

    (c)

    5

    (d)

    none

  12. The circle x2+y2=4x+8y+5intersects the line3x−4y=m at two distinct points if

    (a)

    15< m < 65

    (b)

    35< m <85

    (c)

    −85<m < −35

    (d)

    −35<m <15

  13. The length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is

    (a)

    \(\frac { 98 }{ 6 } \)

    (b)

    \(\frac { 72 }{ 7 } \)

    (c)

    \(\frac { 72 }{ 14 } \)

    (d)

    \(\frac { 98 }{ 12 } \)

  14. 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)

    \(\frac { \pi }{ 5 } \)

  15. If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \) , then

    (a)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\) = 1

    (b)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= -1

    (c)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 0

    (d)

    \([\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]\)= 2

  16. Let \(\overset { \rightarrow }{ a } \),\(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of  \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)=

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  17. If \(\overset { \rightarrow }{ a } \),\(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are any three vectors, then \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \) if and only if 

    (a)

    \(\overset { \rightarrow }{ b } \)\(\overset { \rightarrow }{ c } \) are collinear

    (b)

    \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) are collinear

    (c)

    \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are collinear

    (d)

    none

  18. The straight lines \(\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 } \) and \(\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 } \) are

    (a)

    parallel

    (b)

    perpendicular

    (c)

    inclined at 45o

    (d)

    none

  19. Let \(\overset { \rightarrow }{ u } ,\overset { \rightarrow }{ v } ,\overset { \rightarrow }{ w } \) be vectors such that \(\overset { \rightarrow }{ u } +\overset { \rightarrow }{ v } +\overset { \rightarrow }{ w } =\overset { \rightarrow }{ 0 } \). If \(\left| \overset { \rightarrow }{ u } \right| \)= 3\(\left| \overset { \rightarrow }{ v } \right| \)= 4\(\left| \overset { \rightarrow }{ w } \right| \)=5  then \(\overset { \rightarrow }{ u } .\overset { \rightarrow }{ v } +\overset { \rightarrow }{ v } .\overset { \rightarrow }{ w } +\overset { \rightarrow }{ w } .\overset { \rightarrow }{ u } \) is ______________

    (a)

    25

    (b)

    -25

    (c)

    5

    (d)

    \(\sqrt { 5 } \)

  20. If the slope of the curve 2y2=ax2+b at (1,-1) is - 1, then the values of a, b is

    (a)

    2, 0

    (b)

    0, 2

    (c)

    0, 0

    (d)

    2, 2

  21. The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

    (a)

    \(\frac{1}{31}\)

    (b)

    \(\frac15\)

    (c)

    5

    (d)

    31

  22. The value of \(\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx\)

    (a)

    \(\frac { { \pi }^{ 2 } }{ 4 } -1\)

    (b)

    \(\frac { { \pi }^{ 2 } }{ 4 } +2\)

    (c)

    \(\frac { { \pi }^{ 2 } }{ 4 } +1\)

    (d)

    \(\frac { { \pi }^{ 2 } }{ 4 } -2\)

  23. The solution of the differential equation \(\frac { dy }{ dx } =2xy\)is

    (a)

    y = Cex2

    (b)

    y= 2x2 +C

    (c)

    y = Ce−x2 +C

    (d)

    y = x2 +C

  24. A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
    \(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0<x<l \\\ 0 & l \leq x<2 l \end{array}\right.\)
    The mean and variance of the shorter of the two pieces are respectively

    (a)

    \(\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 3 } \)

    (b)

    \(\\ \cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 6 } \)

    (c)

    \(l,\cfrac { { l }^{ 2 } }{ 12 } \)

    (d)

    \(\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 12 } \)

  25. Subtraction is not a binary operation in

    (a)

    R

    (b)

    Z

    (c)

    N

    (d)

    Q

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