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#### All Chapter 1 Marks

12th Standard EM

Reg.No. :
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Maths

Time : 00:30:00 Hrs
Total Marks : 46

46 x 1 = 46
1. If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

(a)

$\frac { 2\pi }{ 3 }$

(b)

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

(c)

$\frac { 5\pi }{ 6 }$

(d)

$\frac { \pi }{ 4 }$

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

(a)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right]$

(b)

$\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right]$

(c)

$\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right]$

(d)

$\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right]$

3. 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

4. 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

5. in+in+1+in+2+in+3 is

(a)

0

(b)

1

(c)

-1

(d)

i

6. z1, z2 and z3 are complex number such that z1+z2+z3=0 and |z1|=|z2|=|z3|=1 then z12+z22+z33 is

(a)

3

(b)

2

(c)

1

(d)

0

7. The value of (1+i) (1+i2) (1+i3) (1+i4) is

(a)

2

(b)

0

(c)

1

(d)

i

8. 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

9. If α,β and γ are the roots of x3+px2+qx+r, then $\Sigma \frac { 1 }{ \alpha }$ is

(a)

-$\frac { q }{ r }$

(b)

$\frac { p }{ r }$

(c)

$\frac { q }{ r }$

(d)

-$\frac { q }{ p }$

10. 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

11. 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=$\frac{-c}{a}$

12. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

(a)

0

(b)

1

(c)

4

(d)

3

13. 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)

$\pi$

14. 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

15. 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

16. If ${ cos }^{ -1 }x>x>{ sin }^{ -1 }x$ then

(a)

$\cfrac { 1 }{ \sqrt { 2 } } <x\le 1$

(b)

$0\le x<\cfrac { 1 }{ \sqrt { 2 } }$

(c)

$-1\le x<\cfrac { 1 }{ \sqrt { 2 } }$

(d)

x>0

17. 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)

$\frac { 3 }{ 4 }$

18. 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)

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

19. If a parabolic reflector is 20 em in diameter and 5 em deep, then its focus is

(a)

(0,5)

(b)

(5,0)

(c)

(10,0)

(d)

(0, 10)

20. The point of contact of y2 = 4ax and the tangent y = mx + c is

(a)

$\left( \frac { 2a }{ { m }^{ 2 } } ,\frac { a }{ m } \right)$

(b)

$\left( \frac { a }{ { m }^{ 2 } } ,\frac { 2a }{ m } \right)$

(c)

$\left( \frac { a }{ m} ,\frac { 2a }{ { m }^{ 2 } } \right)$

(d)

$\left( \frac {-a }{ { m }^{ 2 }} ,\frac {- 2a }{ m } \right)$

21. If $\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c }$ where $\vec { a } ,\vec { b } ,\vec { c }$ are any three vectors such that $\vec { a } ,\vec { b }$ $\neq$ 0 and  $\vec { a } .\vec { b }$ $\neq$ 0 then $\vec { a }$ and $\vec { c }$ are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle $\frac{\pi}{3}$

(d)

inclined at an angle  $\frac{\pi}{6}$

22. If the distance of the point (1,1,1) from the origin is half of its distance from the plane x + y + z + k =0, then the values of k are

(a)

$\pm 3$

(b)

$\pm 6$

(c)

-3, 9

(d)

3, 9

23. If $\overset { \rightarrow }{ p } \times \overset { \rightarrow }{ q } =2\overset { \wedge }{ i } +3\overset { \wedge }{ j }$$\overset { \rightarrow }{ r } \times \overset { \rightarrow }{ s } =3\overset { \wedge }{ i } +2\overset { \wedge }{ k }$ then $\overset { \rightarrow }{ p } .\left( \overset { \rightarrow }{ q } \left( \overset { \rightarrow }{ r } \times \overset { \rightarrow }{ s } \right) \right)$ is

(a)

9

(b)

6

(c)

2

(d)

5

24. The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are

(a)

mutually perpendicular

(b)

parallel

(c)

inclined at 45o

(d)

inclined at 30

25. The slope of the line normal to the curve f(x) = 2cos 4x at $x=\cfrac { \pi }{ 12 }$

(a)

$-4\sqrt { 3 }$

(b)

-4

(c)

$\cfrac { \sqrt { 3 } }{ 12 }$

(d)

$4\sqrt { 3 }$

26. The maximum slope of the tangent to the curve y = t:r sin x, x ∈ [0, 2π] is at

(a)

$x=\cfrac { \pi }{ 4 }$

(b)

$x=\cfrac { \pi }{ 2 }$

(c)

$x=\pi$

(d)

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

27. The law of linear motion of a particle is given $\frac{1}{3}$ t3-16t, the acceleration at the time when the velocity vanishes is

(a)

4

(b)

6

(c)

2

(d)

8

28. The value of $\underset { x\rightarrow \infty }{ lim } { e }^{ -x }$ is

(a)

0

(b)

(c)

e

(d)

$\frac{1}{e}$

29. A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is

(a)

0.2%

(b)

0.4%

(c)

0.04%

(d)

0.08%

30. If w (x, y) = xy, x > 0, then $\frac { \partial w }{ \partial x }$ is equal to

(a)

xy log x

(b)

y log x

(c)

yxy-1

(d)

x log y

31. If f (x, y) = x3 + y3 - 3xythen $\frac { { \partial }f }{ \partial { x } }$ at x = 2,

(a)

-15

(b)

15

(c)

-9

(d)

16

32. If u = $(\frac{y}{x})$ then x $x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y }$ = .....................

(a)

0

(b)

1

(c)

2u

(d)

u

33. The value of $\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx }$ is

(a)

4

(b)

3

(c)

2

(d)

0

34. The value of  $\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx }$ is

(a)

$\frac{3\pi}{10}$

(b)

$\frac{3\pi}{8}$

(c)

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

(d)

$\frac{3\pi}{2}$

35. If $\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) }$ then

(a)

f(2a -x) = - f(x)

(b)

f(2a - x) = f(x)

(c)

f(x) is odd

(d)

f(x) is even

36. The area of the ellipse $\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1$

(a)

(b)

36π

(c)

2

(d)

36π2

37. The solution of the differential equation $\frac { dy }{ dx } =\frac { y }{ x } +\frac { \phi \left( \frac { y }{ x } \right) }{ \phi '\left( \frac { y }{ x } \right) }$is

(a)

$x\phi \left( \frac { y }{ x } \right) =k$

(b)

$\phi \left( \frac { y }{ x } \right) =kx$

(c)

$y\phi \left( \frac { y }{ x } \right) =k$

(d)

$\phi \left( \frac { y }{ x } \right) =ky$

38. The population P in any year t is such that the rate of increase in the population is proportional to the population. Then

(a)

P=Cekt

(b)

P=Ce-kt

(c)

P=Ckt

(d)

P=C

39. The differential equation corresponding to xy=c2 where c is an arbitrary constant is ________.

(a)

xy"+x=0

(b)

y"=0

(c)

xy'+y=0

(d)

xy"-x=0

40. The I.F. of (1+y2)dx=(tan-1-t-x)dy is ________.

(a)

etan-1 y

(b)

etan-1 x

(c)

tan-1 y

(d)

tan-1x

41. A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
$f(x)=\begin{cases} \begin{matrix} \frac { 2 }{ { x }^{ 3 } } & 0<x>l \end{matrix} \\ \begin{matrix} 0 & 1\le x<2l \end{matrix} \end{cases}$

(a)

$\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 3 }$

(b)

$\\ \cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 6 }$

(c)

$1,\cfrac { { l }^{ 2 } }{ 12 }$

(d)

$\cfrac { 1 }{ 2 } ,\cfrac { { l }^{ 2 } }{ 12 }$

42. Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E[X] is

(a)

0.11

(b)

1.1

(c)

1.1

(d)

1

43. A binary operation on a set S is a function from

(a)

S ⟶ S

(b)

(SxS) ⟶ S

(c)

S⟶ (SxS)

(d)

(SxS) ⟶ (SxS)

44. In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

(a)

1

(b)

2

(c)

3

(d)

4

45. Which of the following is a tautology?

(a)

p ν q

(b)

p ៱ q

(c)

q v ~ q

(d)

q ៱ ~ q

46. A binary operation * is defined on the set of positive rational numbers Q+ by a*b = $\frac { ab }{ 4 }$. Then 3 * $\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right)$ is

(a)

$\frac { 3 }{ 160 }$

(b)

$\frac { 5 }{ 160 }$

(c)

$\frac { 3 }{ 10 }$

(d)

$\frac { 3 }{ 40 }$