Question Set C

Tamilnadu 12th Standard Maths Complex Numbers Important Questions Study Material - QB365

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Questions:

Multiple Choice Question

50 x 1 = 50

Determine the truth value of each of the following statements:
(a) 4 + 2 = 5 and 6 + 3 = 9
(b) 3 + 2 = 5 and 6 + 1 = 7
(c) 4 + 5 = 9 and 1 + 2 = 4
(d) 3 + 2 = 5 and 4 + 7 = 11

(a)

(a)	(b)	(c)	(d)
F	T	F	T

(b)

(a)	(b)	(c)	(d)
T	F	T	F

(c)

(a)	(b)	(c)	(d)
T	T	F	F

(d)

(a)	(b)	(c)	(d)
F	F	T	T

The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is

(a)

ᄀ(p ∧ q) ∧ [p V (p ∧ ᄀr)]

(b)

(p ∧ q) ∧ [p ∧ (p V ᄀr)]

(c)

ᄀ(p ∧ q) ∧ [p ∧ (p ∧ r)]

(d)

ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

Which one of the following statements has the truth value T?

(a)

sin x is an even function

(b)

Every square matrix is non-singular

(c)

The product of complex number and its conjugate is purely imaginary

(d)

\(\sqrt 5\) is an irrational number

In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?

(a)

y = \(\frac{2}{3}\)

(b)

y = \(\frac{-2}{3}\)

(c)

y = \(\frac{-3}{2}\)

(d)

y = 4

The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is

(a)

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

(b)

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

(c)

(d)

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

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)

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

If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

(a)

z - x

(b)

y - z

(c)

x - z

(d)

y - x

If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

(a)

xy + yz + zx

(b)

x(y + z)

(c)

y(z + x)

(d)

If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is

(a)

(b)

(c)

(d)

10.

The area between y² = 4x and its latus rectum is

(a)

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

(b)

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

(c)

\(\frac{8}{3}\)

(d)

\(\frac{5}{3}\)

11.

If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

(a)

xye^xy

(b)

(1 +xy)e^xy

(c)

(1 +y)e^xy

(d)

(1 + x)e^xy

12.

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%

13.

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

14.

If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

(a)

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

(b)

\(\frac { 2 }{ 5 } \)

(c)

\(\frac { 1 }{ 5 } \)

(d)

\(\frac { 1 }{ 3 } \)

15.

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)

(d)

16.

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

(b)

(c)

(d)

17.

The number of arbitrary constants in the particular solution of a differential equation of third order is

(a)

(b)

(c)

(d)

18.

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

(a)

2^x+ 2^y= C

(b)

2^x- 2^y= C

(c)

\(\frac { 1 }{ { 2 }^{ x } } -\frac { 1 }{ { 2 }^{ y } } =C\)

(d)

x + y = C

19.

The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

(a)

\(\frac { x }{ { e }^{ \lambda } } \)

(b)

\(\frac { { e }^{ \lambda} }{ x } \)

(c)

\({ \lambda e }^{ x }\)

(d)

e^x

20.

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

(a)

1, 2

(b)

2, 2

(c)

1, 1

(d)

2, 1

21.

The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively

(a)

2, 3

(b)

3, 3

(c)

2, 6

(d)

2, 4

22.

The point of inflection of the curve y = (x - 1)³ is

(a)

(0, 0)

(b)

(0, 1)

(c)

(1, 0)

(d)

(1, 1)

23.

The maximum value of the function \(x^{2} e^{-2 x}, x>0\) is

(a)

\(\frac { 1 }{ e } \)

(b)

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

(c)

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

(d)

\(\frac { 4 }{ { e }^{ 4 } } \)

24.

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)

(a)

(b)

(c)

(d)

∞

25.

Angle between y² = x and x²= y at the origin is

(a)

\({ tan }^{ -1 }\cfrac { 3 }{ 4 } \)

(b)

\({ tan }^{ -1 }\left( \cfrac { 4 }{ 3 } \right) \)

(c)

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

(d)

\(\cfrac { \pi }{ 4 } \)

26.

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)

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

27.

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

(a)

Only (i)

(b)

(ii) and (iii)

(c)

(iii) and (iv)

(d)

(i), (ii) and (iv)

28.

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

(a)

(b)

(c)

(d)

29.

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)

30.

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

(a)

\(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)

(b)

\(\frac{7}{2}\)

(c)

\(\frac { \sqrt { 7 } }{ 2 } \)

(d)

\(\frac { 7 }{ 2\sqrt { 2 } } \)

31.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is

(a)

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

(b)

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

(c)

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

(d)

\( { \pi }\)

32.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

(a)

\(\vec { a } \)

(b)

\(\vec { b} \)

(c)

\(\vec { c } \)

(d)

\(\vec { 0 } \)

33.

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

34.

If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is

(a)

2x + 1 = 0

(b)

x = −1

(c)

2x −1 = 0

(d)

x = 1

35.

Tangents are drawn to the hyperbola \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

(a)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{-1}{\sqrt{2}}\right)\)

(b)

\(\left(\frac{-9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(c)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(d)

\((3 \sqrt{3},-2 \sqrt{2})\)

36.

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

(a)

(b)

-1

(c)

(d)

37.

The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is

(a)

(b)

(c)

\( \sqrt {10}\)

(d)

\( \sqrt {11}\)

38.

The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).

(a)

\(\frac { 6 }{ 5 } \)

(b)

\(\frac { 5 }{ 3 } \)

(c)

\(\frac { 10 }{ 3 } \)

(d)

\(\frac { 3 }{ 5 } \)

39.

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

(a)

(b)

(c)

(d)

40.

If \(\cot ^{-1}(\sqrt{\sin \alpha})+\tan ^{-1}(\sqrt{\sin \alpha})=u\), then cos2u is equal to

(a)

tan²\(\alpha\)

(b)

(c)

-1

(d)

tan2\(\alpha\)

41.

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

(a)

\(-\sqrt { \frac { 24 }{ 25 } } \)

(b)

\(\sqrt { \frac { 24 }{ 25 } } \)

(c)

\(\frac{1}{5}\)

(d)

\(-\frac{1}{5}\)

42.

The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

(a)

[1, 2]

(b)

[-1, 1]

(c)

[0, 1]

(d)

[-1, 0]

43.

If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

(a)

(b)

(c)

(d)

44.

The value of \(\left( \cfrac { 1+\sqrt { 3 } i}{ 1-\sqrt { 3}i } \right) ^{ 10 }\) is

(a)

\(cis\cfrac { 2\pi }{ 3 } \)

(b)

\(cis\cfrac { 4\pi }{ 3 } \)

(c)

\(-cis\cfrac { 2\pi }{ 3 }\)

(d)

\(-cis\cfrac { 4\pi }{ 3 }\)

45.

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

(a)

(b)

(c)

< n

(d)

46.

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

(a)

a ≥ 0

(b)

a > 0

(c)

a < 0

(d)

a ≤ 0

47.

The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

(a)

(b)

(c)

(d)

∞

48.

A polynomial equation in x of degree n always has

(a)

n distinct roots

(b)

n real roots

(c)

n complex roots

(d)

at most one root

49.

The solution of the equation |z| - z = 1 + 2i is

(a)

\(\frac { 3 }{ 2 } -2i\)

(b)

\(-\frac { 3 }{ 2 } +2i\)

(c)

\(2-\frac { 3 }{ 2 } i\)

(d)

\(2+\frac { 3 }{ 2 } i\)

50.

If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

(a)

(b)

(c)

(d)

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Answers Key:

Multiple Choice Question

50 x 1 = 50

(a)

(a) (b) (c) (d)

F T F T

(d)
ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

(d)
\(\sqrt 5\) is an irrational number

(b)
y = \(\frac{-2}{3}\)

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

(b)
\(\frac{3\pi}{8}\)

(b)
y - z

(d)
0

(d)
9

10.

11.

(b)
(1 +xy)e^xy

12.

(b)
0.4%

13.

(d)
0.96

14.

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

15.

(b)
1.1

16.

(d)
2

17.

(d)
0

18.

(b)
2^x- 2^y= C

19.

(d)
e^x

20.

\({ I }_{ n }=\int _{ 0 }^{ \pi /2 }{ { cos }^{ 7 }xdx=\frac { n-1 }{ n } { I }_{ n-2 },n\ge 2 } \)
\(\therefore { I }_{ 7 }=\int _{ 0 }^{ \pi /2 }{ { cos }^{ 7 }xdx=\frac { 6 }{ 7 } \times \frac { 4 }{ 5 } \times \frac { 2 }{ 3 } \times 1 } =\frac { 16 }{ 35 } \)

21.

(a)
2, 3

22.

23.

24.

(a)
0

25.

26.

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

27.

(d)
(i), (ii) and (iv)

28.

(a)
1

29.

(b)
\(\frac { 1 }{ 9 } \)

30.

(a)
\(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)

31.

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

32.

(d)
\(\vec { 0 } \)

33.

34.

(b)
x = −1

35.

36.

(d)
9

37.

38.

39.

(d)
3

40.

41.

(d)
\(-\frac{1}{5}\)

42.

(a)
[1, 2]

43.

Comparing the two given lines with
\(\vec { r } =\vec { a } +t\vec { b } ,\vec { r } =\vec { c } +s\vec { d } \)
we have, \(\vec { a } =-\hat { -1 } -3\hat { j } -5\hat { k } ,\vec { b } =3\hat { i } +5\hat { j } +7\hat { k } ,\vec { c } =2\hat { i } +4\hat { j } +6\hat { k } \) and \(\vec { d } =\hat { i } +4\hat { j } +7\hat { k } \)
We know that the two given lines are coplar, if \((\vec { c } -\vec { a } ).(\vec { b } \times \vec { d } )\)=0
Here, \(\vec { b } \times \vec { d } \left| \begin{matrix} \hat { i } & \hat { j } & \hat { k } \\ 3 & 5 & 7 \\ 1 & 4 & 7 \end{matrix} \right| =7\hat { i } -14\hat { j } +7\hat { k } \) and \(\vec { c } -\vec { a } =3\hat { i } +7\hat { j } +11\hat { k } \)
Then, \((\vec { c } -\vec { a } ).(\vec { b } \times \vec { d } )=(3\hat { i } +7\hat { j } +11\hat { k } )(7\hat { i } -14\hat { j } +7\hat { k } )\)
Therefore the two given lines are coplanar.Then we find the non parametric form of vector equation of the plane containing the two given coplanar lines. We know that the plane containing the two given coplanar lines is
\((\vec { r } -\vec { a } ).(\vec { b } \times \vec { d } )\)=0
which implies that \((\vec { r } -(-\hat { i } -3\hat { j } -5\hat { k } )).(7\hat { i } -14\hat { j } +7\hat { k } )\)=0. Thus, the required non-parametric vector equation of the plane containing the two given coplanar lines is \(\vec { r } .(\hat { i } -2\hat { j } +\hat { k } )\)=0.