Question Set B

TN 12th Standard Maths Inverse Trigonometric Functions Previous Year Question Papers Study Material - QB365

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Questions:

Multiple Choice Question

60 x 1 = 60

The proposition p ∧ (¬p ∨ q) is

(a)

a tautology

(b)

a contradiction

(c)

logically equivalent to p ∧ q

(d)

logically equivalent to p ∨ q

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

(a)

(b)

(c)

(d)

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

The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

(a)

Q⁺

(b)

(c)

(d)

Subtraction is not a binary operation in

(a)

(b)

(c)

(d)

The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is

(a)

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

(b)

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

(c)

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

(d)

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

The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is

(a)

\(\frac{7}{27}\)

(b)

\(\frac{5}{27}\)

(c)

\(\frac{4}{27}\)

(d)

\(\frac{2}{27}\)

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 \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) 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.

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)

(b)

(c)

(d)

12.

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13.

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

(a)

12 x_o+dx

(b)

12x_o dx

(c)

6x_o dx

(d)

6x_o+ dx

14.

If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

(a)

x^y log x

(b)

y log x

(c)

yx^y-1

(d)

x log y

15.

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)

(d)

16.

Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.

(a)

I and II

(b)

II only

(c)

III only

(d)

II and III

17.

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)

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

18.

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)

19.

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

(a)

(b)

(c)

(d)

20.

Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is

(a)

\(\cfrac { 19 }{ 6 } \)

(b)

\(-\cfrac { 19 }{ 6 } \)

(c)

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

(d)

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

21.

If sin x is the integrating factor of the linear differential equation \(\frac { dy }{ dx } +Py=Q,\) then P is

(a)

log sin x

(b)

cos x

(c)

tan x

(d)

cot x

22.

The general solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) is

(a)

e^x+ e^y= C

(b)

e^x+ e^-y= C

(c)

e^-x+ e^y= C

(d)

e^-x+ e^-y= C

23.

The solution of \(\frac{d y}{d x}+p(x) y=0\) is

(a)

\(y={ ce }^{ \int { pdx } }\)

(b)

\(y={ ce }^{ -\int { pdx } }\)

(c)

\(x={ ce }^{ -\int { pdy } }\)

(d)

\(x={ce }^{ \int { pdy } }\)

24.

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

25.

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is

(a)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(b)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }+y=0\)

(c)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }=0\)

(d)

\(\frac { { d }^{ 2 }x }{ { dy }^{ 2 } }=0\)

26.

The maximum product of two positive numbers, when their sum of the squares is 200, is

(a)

100

(b)

\(25\sqrt { 7 } \)

(c)

(d)

\(24\sqrt { 14 } \)

27.

The minimum value of the function |3 - x| + 9 is

(a)

(b)

(c)

(d)

28.

The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is

(a)

(b)

\(\sqrt { 2 } \)

(c)

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

(d)

29.

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

(a)

-8

(b)

-4

(c)

-2

(d)

30.

The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is

(a)

t = 0

(b)

\(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

(c)

t =1

(d)

t = 3

31.

The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc} 1 & 2 & 7 & 3 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & \lambda-7 & \mu+5 \end{array}\right]\). The system has infinitely many solutions if

(a)

\(\lambda=7, \mu \neq-5\)

(b)

\(\lambda=-7, \mu=5\)

(c)

\(\lambda \neq 7, \mu \neq-5\)

(d)

\(\lambda=7, \mu=-5\)

32.

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)

33.

If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a)

adj A = |A|A^-1

(b)

adj(AB) = (adj A)(adj B)

(c)

det A^-1 = (det A)^-1

(d)

(ABC)^-1 = C^-1B^-1A^-1

34.

If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is

(a)

(b)

-2

(c)

-3

(d)

-1

35.

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)

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

36.

If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\), then the value of λ is

(a)

\(2\sqrt { 3 } \)

(b)

\(3\sqrt { 2 } \)

(c)

(d)

37.

Distance from the origin to the plane 3x − 6y + 2z + 7 = 0 is

(a)

(b)

(c)

(d)

38.

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 \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle \(\frac{\pi}{3}\)

(d)

inclined at an angle \(\frac{\pi}{6}\)

39.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to

(a)

(b)

(c)

(d)

40.

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)

\(\vec { 0 } \)

41.

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)

(-3, 2)

(c)

(5, -2)

(d)

(-2, 5)

42.

An ellipse has OB 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)

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

43.

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

(a)

(b)

-1

(c)

(d)

44.

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

(a)

4(a²+b²)

(b)

2(a²+b²)

(c)

a²+b²

(d)

\(\frac { 1 }{ 2 } \)(a²+b²)

45.

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)

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

46.

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

(a)

tan^-1x

(b)

sin^-1x

(c)

(d)

\(\pi\)

47.

\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation

(a)

x²−x−6 = 0

(b)

x²−x−12 = 0

(c)

x²+x−12 = 0

(d)

x²+x−6 = 0

48.

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) 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)

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

49.

If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is

(a)

\(-\frac{\pi}{10}\)

(b)

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

(c)

\(\frac{\pi}{10}\)

(d)

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

50.

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to

(a)

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

(b)

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

(c)

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

(d)

\(\pi\)

51.

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

(a)

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

(b)

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

(c)

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

(d)

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

52.

If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is

(a)

real axis

(b)

imaginary axis

(c)

ellipse

(d)

circle

53.

The polynomial x³+ 2x + 3 has

(a)

one negative and two imaginary zeros

(b)

one positive and two imaginary zeros

(c)

three real zeros

(d)

no zeros

54.

The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies

(a)

|k| ≤ 6

(b)

k = 0

(c)

|k| > 6

(d)

|k| ≥ 6

55.

If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is

(a)

\(-\frac { q }{ r } \)

(b)

\(-\frac { p }{ r } \)

(c)

\(\frac { q }{ r } \)

(d)

\(-\frac { q }{ p } \)

56.

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

57.

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is

(a)

(b)

m+n

(c)

mⁿ

(d)

n^m

58.

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

(a)

(b)

(c)

(d)

59.

If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to

(a)

(b)

(c)

(d)

60.

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

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Answers Key:

Multiple Choice Question

60 x 1 = 60

(b)
8

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

(b)
Z

(d)
\(\frac{2}{27}\)

(b)
y - z

(d)
9

10.

11.

12.

13.

(b)
12x_o dx

14.

15.

(b)
\(\frac15\)

16.

(a)
I and II

17.

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

18.

(b)
1.1

19.

(d)
2

20.

(d)
\(-\cfrac { 3 }{ 2 } \)

21.

(d)
cot x

22.

23.

(b)
\(y={ ce }^{ -\int { pdx } }\)

24.

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

25.