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)

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

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)

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

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

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z 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

iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

(a)

0

(b)

1

(c)

-1

(d)

i

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

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

(a)

2

(b)

0

(c)

1

(d)

i

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

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)

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

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

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 = \(\frac{-c}{a}\)

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

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)

0

(d)

\(\pi\)

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

(a)

4

(b)

5

(c)

2

(d)

3

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

(a)

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

(b)

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

(c)

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

(d)

none of these

If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________

(a)

\(\cfrac { 1 }{ \sqrt { 2 } }

(b)

\(0\le x<\frac { 1 }{ \sqrt { 2 } } \)

(c)

\(-1\le x<\frac { 1 }{ \sqrt { 2 } } \)

(d)

x>0

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) 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)

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

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)

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

If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________

(a)

(0, 5)

(b)

(5, 0)

(c)

(10, 0)

(d)

(0, 10)

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)

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

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  \(\frac{\pi}{6}\)

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

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

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

(a)

mutually perpendicular

(b)

parallel

(c)

inclined at 45^o

(d)

inclined at 30

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

(a)

\(-4\sqrt { 3 } \)

(b)

-4

(c)

\(\cfrac { \sqrt { 3 } }{ 12 } \)

(d)

\(4\sqrt { 3 } \)

The maximum slope of the tangent to the curve y = e^x sin x, x ∈ [0, 2π] is at

(a)

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

(b)

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

(c)

\(x=\pi \)

(d)

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

The law of linear motion of a particle is given s = \(\frac{1}{3}\) t³-16t, the acceleration at the time when the velocity vanishes is __________

(a)

4

(b)

6

(c)

2

(d)

8

The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________

(a)

0

(b)

∞

(c)

e

(d)

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

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%

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

If f (x, y) = x³ + y³ - 3xy² then \(\frac { { \partial }f }{ \partial { x } } \) at x = 2,_____________

(a)

-15

(b)

15

(c)

-9

(d)

16

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

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

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 \(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) } \) then __________

(a)

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

(b)

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

(c)

f(x) is odd

(d)

f(x) is even

The area of the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) __________

(a)

6π

(b)

36π

(c)

6π²

(d)

36π²

The solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}\) 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)

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

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=Ce^kt

(b)

P=Ce^-kt

(c)

P=Ckt

(d)

P=C

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

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

(a)

e^tan-1 y

(b)

e^tan-1 x

(c)

tan^-1 y

(d)

tan^-1x

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

(a)

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

(b)

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

(c)

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

(d)

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

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)

11

(d)

1

A binary operation on a set S is a function from

(a)

S ⟶ S

(b)

(SxS) ⟶ S

(c)

S⟶ (SxS)

(d)

(SxS) ⟶ (SxS)

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

Which of the following is a tautology?

(a)

p ν q

(b)

p ∧ q

(c)

q v ~ q

(d)

q ∧ ~ q

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)

\(\frac { 3 }{ 40 } \)