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#### Quarterly Exam Model One Mark Questions

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
50 x 1 = 50
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 = $\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right]$, B = adj A and C = 3A, then $\frac { \left| adjB \right| }{ \left| C \right| }$

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 9 }$

(c)

$\frac { 1 }{ 4 }$

(d)

1

3. If A$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right]$, then A =

(a)

$\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right]$

4. If A = $\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right]$, then 9I - A =

(a)

A-1

(b)

$\frac { { A }^{ -1 } }{ 2 }$

(c)

3A-1

(d)

2A-1

5. If A = $\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right]$ and AT = A−1 , then the value of x is

(a)

$\frac { -4 }{ 5 }$

(b)

$\frac { -3 }{ 5 }$

(c)

$\frac { 3 }{ 5 }$

(d)

$\frac { 4 }{ 5 }$

6. If adj A = $\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right]$ and adj B = $\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right]$ then adj (AB) is

(a)

$\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right]$

7. 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

8. 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

9. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

(a)

0

(b)

2n

(c)

2n

(d)

n2

10. Which of the following is not an elementary transformation?

(a)

Ri ↔️ Rj

(b)

Ri ⟶ 2Ri + Rj

(c)

Cj ⟶ Cj + Ci

(d)

Ri ⟶ Ri + Cj

11. In the system of equations with 3 unknowns, if Δ = 0, and one of Δx, Δy of Δz is non zero then the system is ______

(a)

Consistent

(b)

inconsistent

(c)

consistent with one parameter family of solutions

(d)

consistent with two parameter family of solutions

12. If A = [2 0 1] then the rank of AAT is ______

(a)

1

(b)

2

(c)

3

(d)

0

13. The conjugate of a complex number is $\cfrac { 1 }{ i-2 }$/Then the complex number is

(a)

$\cfrac { 1 }{ i+2 }$

(b)

$\cfrac { -1 }{ i+2 }$

(c)

$\cfrac { -1 }{ i-2 }$

(d)

$\cfrac { 1 }{ i-2 }$

14. If $\left| z-\cfrac { 3 }{ z } \right| =2$ then the least value |z| is

(a)

1

(b)

2

(c)

3

(d)

5

15. If |z1|=1,|z2|=2|z3|=3 and |9z1z2+4z1z3+z2z3|=12, then the value of |z1+z2+z3| is

(a)

1

(b)

2

(c)

3

(d)

4

16. If $\omega \neq 1$ is a cubic root of unity and $\left( 1+\omega \right) ^{ 7 }=A+B\omega$ ,then (A,B) equals

(a)

(1,0)

(b)

(−1,1)

(c)

(0,1)

(d)

(1,1)

17. The principal argument of the complex number $\cfrac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) }$ is

(a)

$\cfrac { 2\pi }{ 3 }$

(b)

$\cfrac { \pi }{ 6 }$

(c)

$\cfrac { 5\pi }{ 6 }$

(d)

$\cfrac { \pi }{ 2 }$

18. If $\omega =cis\cfrac { 2\pi }{ 3 }$, then the number of distinct roots of $\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right|$

(a)

1

(b)

2

(c)

3

(d)

4

19. If a=cosθ + i sinθ, then $\frac { 1+a }{ 1-a }$ =

(a)

cot $\frac { \theta }{ 2 }$

(b)

cot θ

(c)

i cot $\frac { \theta }{ 2 }$

(d)

i tan$\frac { \theta }{ 2 }$

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

(a)

$\frac { \pi }{ 4 }$

(b)

$\frac { \pi }{ 12 }$

(c)

$\frac { 3\pi }{ 4 }$

(d)

$\pi$

21. If ω is the cube root of unity, then the value of (1-ω) (1-ω2) (1-ω4) (1-ω8) is

(a)

9

(b)

-9

(c)

16

(d)

32

22. 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)

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

23. If x=cosθ + i sinθ, then xn+$\frac { 1 }{ { x }^{ n } }$ is ______

(a)

2 cos nθ

(b)

2 i sin nθ

(c)

2n cosθ

(d)

2n i sinθ

24. 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

25. 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

26. The number of positive zeros of the polynomial $\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }$(-1)rxr is

(a)

0

(b)

n

(c)

< n

(d)

r

27. The quadratic equation whose roots are ∝ and β is

(a)

(x - ∝)(x -β) =0

(b)

(x - ∝)(x + β) =0

(c)

∝+β=$\frac{b}{a}$

(d)

∝.β=$\frac{-c}{a}$

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

(a)

n

(b)

n -1

(c)

n+1

(d)

(n-r)

29. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

30. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

31. If x=$\frac{1}{5}$, the valur of cos (cos-1x+2sin-1x) is

(a)

$-\sqrt { \frac { 24 }{ 25 } }$

(b)

$\sqrt { \frac { 24 }{ 25 } }$

(c)

$\frac{1}{5}$

(d)

-$\frac{1}{5}$

32. ${ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right)$is equal to

(a)

$\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(b)

$\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(c)

$\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right)$

(d)

${ tan}^{ -1 }\left( \frac { 1}{ 2 } \right)$

33. If |x|$\le$1, then 2tan-1 x-sin-1 $\frac{2x}{1+x^2}$ is equal to

(a)

tan-1x

(b)

sin-1x

(c)

0

(d)

$\pi$

34. 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

35. If ${ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha$ then x2 =

(a)

$sin2\alpha$

(b)

$sin\alpha$

(c)

$cos2\alpha$

(d)

$cos\alpha$

36. If ${ sin }^{ -1 }x-cos^{ -1 }x=\cfrac { \pi }{ 6 }$ then

(a)

$\cfrac { 1 }{ 2 }$

(b)

$\cfrac { \sqrt { 3 } }{ 2 }$

(c)

$\cfrac { -1 }{ 2 }$

(d)

none of these

37. If $4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi$ then x is

(a)

$\cfrac { 3 }{ 2 }$

(b)

$\cfrac { 1 }{ \sqrt { 2 } }$

(c)

$\cfrac { \sqrt { 3 } }{ 2 }$

(d)

$\cfrac { 2 }{ \sqrt { 3 } }$

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

(a)

0.75

(b)

1.5

(c)

0.96

(d)

sin-1(1.5)

39. ${ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right)$

(a)

$\cfrac { 9\pi }{ 8 }$

(b)

$\cfrac { 9\pi }{ 8 }$

(c)

$\cfrac { \pi }{ 8 }$

(d)

$\cfrac { -\pi }{ 8 }$

40. The radius of the circle passing through the point(6,2) two of whose diameter arex+y=6
and x+2y=4 is

(a)

10

(b)

${2} \sqrt {5}$

(c)

6

(d)

4

41. 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

42. Let C be the circle with centre at(1,1) and radius =1. If T is the circle centered at(0, y)
passing through the origin and touching the circleC externally, then the radius of T is equal to

(a)

$\frac { \sqrt { 3 } }{ 2 }$

(b)

$\frac { \sqrt { 3 } }{ \sqrt { 2 } }$

(c)

$\frac { 1 }{ 2 }$

(d)

$\frac { 1 }{ 4 }$

43. An ellipse hasOB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

(a)

$\frac { 1 }{ \sqrt { 2 } }$

(b)

$\frac { 1 }{ 2 }$

(c)

$\frac { 1 }{ 4 }$

(d)

$\frac { 1 }{ \sqrt { 3 } }$

44. The values of m for which the line y=mx+ $2\sqrt { 5 }$ touches the hyperbola 16x2−9y2=144 are the roots of x2−(a+b)x−4=0, then the value of (a+b) is

(a)

2

(b)

4

(c)

0

(d)

-2

45. 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)

46. In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is

(a)

$\frac { 4 }{ 5 }$

(b)

$\frac { 1 }{ \sqrt { 52 } }$

(c)

$\frac { 3 }{ 5 }$

(d)

$\frac { 1 }{ 2 }$

47. The area of the circle (x - 2)2 + (y - k)2 = 25 is

(a)

25ㅠ

(b)

5ㅠ

(c)

10ㅠ

(d)

25

48. 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

49. 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

50. The locus of the point of intersection of perpendicular tangents of the parabola y2 = 4ax is

(a)

latus rectum

(b)

directrix

(c)

tangent at the vertex

(d)

axis of the parabola