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

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 the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________

(a)

-3, -\(\frac{1}{2}\)

(b)

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

(c)

1

(d)

2

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 z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________

(a)

0

(b)

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

(c)

cot\(\frac { \theta }{ 2 } \)

(d)

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

The amplitude of \(\frac{1}{i}\) is equal to _______

(a)

0

(b)

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

(c)

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

(d)

\(\pi \)

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

\(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

(a)

\(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

(b)

\(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

(c)

\(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

(d)

\(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is __________

(a)

\(\frac { 1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } \)

(b)

\(\frac { 1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(-\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

(d)

\(\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________

(a)

-p+√q

(b)

p-iq

(c)

p-√q

(d)

-p-√q

Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________

(a)

real and negative

(b)

real and positive

(c)

rational numbers

(d)

none

lf the root of the equation x³ + bx²+ cx - 1 = 0 form an lncreasing G.P, then ___________

(a)

one of the roots is 2

(b)

one of the roots is 1

(c)

one of the roots is -1

(d)

one of the roots is -2

If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

(a)

0, 2

(b)

0, 1

(c)

0, 3

(d)

0, √3

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 p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________

(a)

\(4\alpha =3\beta \quad \)

(b)

\(3\alpha =4\beta \)

(c)

\(\alpha -\beta =\frac { 7\pi }{ 12 } \)

(d)

none

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

(a)

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

(b)

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

(c)

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

(d)

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

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

(a)

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

(b)

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

(c)

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

(d)

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

If tan^-1(cot \(\theta\)) = 2\(\theta\), then\(\theta\) = _____________

(a)

\(\pm 3\)

(b)

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

(c)

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

(d)

none

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

(a)

\(\frac { 19 }{ 8 } \)

(b)

\(\frac { 8 }{ 19 } \)

(c)

\(\frac { 19 }{ 12 } \)

(d)

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

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)

\(\pi \)

Equation of tangent at (-4, -4) on x² = -4y is _____________

(a)

2x - y + 4 = 0

(b)

2x + y - 4 = 0

(c)

2x - y - 12 = 0

(d)

2x + y + 4 = 0

The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________

(a)

x² - y² = 32

(b)

y² - x² = 32

(c)

x² - y² = 16

(d)

y² - x² = 16

When the eccentricity of a ellipse becomes zero, then it becomes a __________

(a)

straight line

(b)

circle

(c)

point

(d)

parabola