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

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

(a)

unique solution

(b)

one parameter family of solution

(c)

two parameter family of solutions

(d)

in consistent

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 modular of \(\frac { (-1+i)(1-i) }{ 1+i\sqrt { 3 } } \) is ______

(a)

\(\sqrt{2}\)

(b)

2

(c)

1

(d)

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

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

The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

(a)

2

(b)

3

(c)

1

(d)

none

If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________ 

(a)

5

(b)

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

(c)

\(\frac { 5 }{ 14 } \)

(d)

\(\frac { 14 }{ 5 } \)

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 length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________

(a)

\(\frac { 98 }{ 6 } \)

(b)

\(\frac { 72 }{ 7 } \)

(c)

\(\frac { 72 }{ 14 } \)

(d)

\(\frac { 98 }{ 12 } \)

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)

\(\sqrt{19}\)

The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

(a)

\(\sqrt6\)

(b)

3

(c)

\(\sqrt3\)

(d)

6

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector ___________

(a)

\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)

(b)

\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

(c)

2\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)

(d)

2\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

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

If the rate of increase of s = x³-5x²+ 5x + 8 is twice the rate of increase of x, then one value of x is __________

(a)

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

(b)

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

(c)

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

(d)

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

The curve y = e^x is ________

(a)

convex

(b)

concave

(c)

convex upwards

(d)

concave upwards

If u = x^y + y^x then u_x + u_y at x = y = 1 is _____________

(a)

0

(b)

2

(c)

1

(d)

∞

If u = y^x then \(\frac { \partial u }{ \partial y } \) = ............

(a)

xy^x-1

(b)

yx^y-1

(c)

0

(d)

1

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

\(\int _{ a }^{ b }{ f(x) } dx=\) ..............

(a)

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

(b)

\(\int _{ a }^{ b }{ f(a-x) } dx\)

(c)

\(\int _{ b }^{ a }{ f(b-x) } dx\)

(d)

\(\int _{ a }^{ b }{ f(a+b-x) } dx\)

The I.F of y log y \(\frac{dx}{dy}+x-log\ y=0\) is __________

(a)

log(log y)

(b)

log y

(c)

\(\frac{1}{log\ y}\)

(d)

\(\frac{1}{log(log\ y)}\)

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

The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________

(a)

\(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \)

(b)

\(\left( \begin{matrix} \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \\ \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \end{matrix} \right) \)

(c)

\(\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right) \)

(d)

\(\left( \begin{matrix} \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \\ \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \end{matrix} \right) \)

If p is true and q is unknown, then _________

(a)

~ p is true

(b)

p v (~p) is false

(c)

p ∧ (~p) is true

(d)

p v q is true