The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________

(a)

k ≠ 0

(b)

-1 < k < 1

(c)

-2 < k < 2

(d)

k = 0

If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--

(a)

consistent and has infinitely many solutions

(b)

consistent

(c)

inconsistent

(d)

consistent and has unique solution

If \(\rho\) (A) = r then which of the following is correct?

(a)

all the minors of order n which do not vanish

(b)

'A' has at least one minor of order r which does not vanish and all higher order minors vanish

(c)

'A' has at least one (r + 1) order minor which vanish

(d)

all (r + 1) and higher order minors should not vanish

If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________

(a)

consistent and has infinitely many solutions

(b)

consistent and has a unique solution

(c)

consistent

(d)

inconsistent

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

(a)

|z| = 1, arg(z) =\(\frac { \pi }{ 4 } \)

(b)

|z| = 1, arg(z) = \(\frac { \pi }{ 6 } \)

(c)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg(z) = \(\frac { 5\pi }{ 24 } \)

(d)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg (z) = tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

(a)

16

(b)

8

(c)

4

(d)

2

The value of (1+i)⁴ + (1-i)⁴ is __________

(a)

8

(b)

4

(c)

-8

(d)

-4

If zⁿ = \(cos\frac { n\pi }{ 3 } +isin\frac { n\pi }{ 3 } \), then z₁, z₂ ..... z₆ is _________

(a)

1

(b)

-1

(c)

i

(d)

-i

The points represented by 3 - 3i, 4 - 2i, 3 - i and 2 - 2i form _____ in the argand plane.

(a)

collinear points

(b)

Vertices of a parallelogram

(c)

Vertices of a rectangle

(d)

Vertices of a square

If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

(a)

-2i sin(α - β)

(b)

2i sin(α - β)

(c)

2 cos(α - β)

(d)

-2 cos(α - β)

If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

(a)

2 cos nθ

(b)

2 i sin nθ

(c)

2ⁿ cosθ

(d)

2ⁿ i sinθ

If the equation ax²+ bx+c = 0(a > 0) has two roots ∝ and β such that ∝ <- 2 and β > 2, then __________

(a)

b²-4ac = 0

(b)

b² - 4ac <0

(c)

b² - 4ac >0

(d)

b² - 4ac ≥ 0

If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________ 

(a)

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

(b)

\(\cfrac { 5\pi }{ 3 } \)

(c)

\(\cfrac { 10\pi }{ 3 } \)

(d)

0

\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

(a)

\(\frac { 6 }{ 15 } \)

(b)

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

(c)

\(\frac { 4 }{ 5 } \)

(d)

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

If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________

(a)

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

(b)

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

(c)

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

(d)

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

\(cot\left( \frac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

(a)

7

(b)

6

(c)

5

(d)

none

If x > 1, then \(2{ tan }^{ -1 }x+{ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \) ________

(a)

4 tan^-1x

(b)

0

(c)

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

(d)

\(\pi \)

The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is ___________

(a)

x = -1

(b)

x = 1

(c)

x = \(\frac{-3}{2}\)

(d)

x = \(\frac{3}{2}\)

If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is __________

(a)

6x² + 10y² = 5

(b)

6x² + 10y² = 15

(c)

x² + 3y² = 10

(d)

none

The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

(a)

parabola

(b)

ellipse

(c)

hyperbola

(d)

rectangular hyperbola

The director circle of the ellipse \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 5 } =1\) is ____________

(a)

x² + y² = 4

(b)

x² +y² = 9

(c)

x² +y² = 45

(d)

x² +y² = 14

The length of the diameter of a circle with centre (1, 2) and passing through (5, 5) is ____________

(a)

5

(b)

\(\sqrt{45}\)

(c)

10

(d)

\(\sqrt{50}\)

If (1, -3) is the centre of the circle x² + y² + ax + by + 9 = 0 its radius is _________

(a)

\(\sqrt{10}\)

(b)

1

(c)

5

(d)

\(\sqrt{19}\)

If e₁, e₂ are eccentricities of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 then 

(a)

\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 1

(b)

\({ e }_{ 1 }^{ 2 }\)  + \({ e }_{ 2 }^{ 2 }\) = 1

(c)

\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 2

(d)

\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 2

If B, B¹ are the ends of minor axis, F₁, F₂ are foci of the ellipse \(\frac { { x }^{ 2 } }{ 8 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 then area of F₁BF₂B¹ is __________

(a)

16

(b)

8

(c)

16\(\sqrt2\)

(d)

32\(\sqrt2\)

The length of major and minor axes of 4x² + 3y² = 12 are ____________

(a)

4, 2\(\sqrt3\)

(b)

2, \(\sqrt3\)

(c)

2\(\sqrt3\), 4

(d)

\(\sqrt3\), 2

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

(a)

1

(b)

2

(c)

3

(d)

\(\infty\)

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +\overset { \wedge }{ 4k } \) is ____________

(a)

4

(b)

2\(\sqrt { 3 } \)

(c)

4\(\sqrt { 3 } \)

(d)

5\(\sqrt { 3 } \)

If \(\overset { \rightarrow }{ a } =\left| \overset { \rightarrow }{ a } \right| \overset { \rightarrow }{ e } \) then \(\overset { \rightarrow }{ e } .\overset { \rightarrow }{ e } \) is _____________

(a)

0

(b)

e

(c)

1

(d)

\(\overset { \rightarrow }{ 0 } \)

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

If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is _____________

(a)

土 \(\frac { 1 }{ 3 } \)

(b)

土 \(\frac { 1 }{ 4 } \)

(c)

土 \(\frac { 1 }{ 9 } \)

(d)

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

If the vectors \(a\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \), \(\overset { \wedge }{ i } +b\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +c\overset { \wedge }{ k } \) (a ≠ b ≠ c ≠ 1) are coplaner, then \(\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } =\) _____________

(a)

0

(b)

1

(c)

2

(d)

\(\frac { abc }{ (1-a)(1-b)(1-c) } \)

If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are three non - coplanar vectors, then \(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } +\frac { \overset { \rightarrow }{ b } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } } \) = _____________

(a)

0

(b)

1

(c)

-1

(d)

\(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } .\overset { \rightarrow }{ c } } \)

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

(a)

-1, 2

(b)

1, \(\frac { 1 }{ 2 } \)

(c)

1, 2

(d)

none

In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

(a)

<b

(b)

≤b

(c)

=b

(d)

≠b

If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________

(a)

\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

(b)

0

(c)

u

(d)

2u

lf u = (x-y)⁴+(y-z)⁴ +(z-x)⁴ then \(\sum { \frac { \partial u }{ \partial x } } \) = _____________

(a)

4

(b)

1

(c)

0

(d)

-4

The approximate value of (627)^\(\frac14\) is ................

(a)

5.002

(b)

5.003

(c)

5.005

(d)

5.004

The value of \(\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \sqrt { \frac { 1-cos2x }{ 2x } } } \) dx is __________

(a)

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

(b)

2

(c)

0

(d)

1

The area enclosed by the curve y² = 4x, the x-axis and its latus rectum is ________ sq.units.

(a)

\(\frac23\)

(b)

\(\frac43\)

(c)

\(\frac83\)

(d)

\(\frac{16}{3}\)

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

(a)

6π

(b)

36π

(c)

6π²

(d)

36π²

The volume generated by the curve y² = 16x from x = 2 to x = 3 rotating about x - axis ......... cu. units

(a)

72π

(b)

\(\frac { 256\times 19 }{ 3 } \)ㅠ

(c)

40ㅠ

(d)

80ㅠ

 

The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________

(a)

tan x+tan y = c

(b)

sec x + sec y = c

(c)

tan x tan y = c

(d)

sec x- sec y = c

The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________

(a)

\(y={ e }^{ \frac { x }{ 2 } }\left[ A\ cos\frac { x }{ 2 } +B\ sin\frac { x }{ 2 } \right] \)

(b)

\(y={ e }^{ \frac { x }{ 2 } }\left[ A\ cos\frac { x }{ 2 } -B\ sin\frac { x }{ 2 } \right] \)

(c)

\(y=Acos\frac { x }{ 2 } +Bsin\frac { x }{ 2 } \)

(d)

\(y={ Ae }^{ \frac { x }{ 2 } }+B{ e }^{ \frac { -x }{ 2 } }\)

If \(f(x)=\frac { 1 }{ 2 } \) ,\(E\left( { x }^{ 2 } \right) =\frac { 1 }{ 4 } \) then var(x) is _____________

(a)

0

(b)

\(\frac { 1 }{ 4 } \)

(c)

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

(d)

1

Var (2x ± 5) is =________

(a)

5

(b)

var (2x) ± 5

(c)

4 var (X)

(d)

0

The sum of the mean and variance of a binomial distribution for 6 total is 2.16. Then the probability of success p =__________

(a)

0.4

(b)

0.6

(c)

0.8

(d)

0.2

If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________

(a)

20

(b)

40

(c)

400

(d)

445

In (N, *), x * y = max(x, y), x, y \(\in \) N then 7 * (-7)

(a)

7

(b)

-7

(c)

0

(d)

-49