Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.

Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

Find Re (z) and im (z) if z = 5i¹¹ + 7i³

If z₁ and z₂ are 1-i, -2+4i then find Im\(\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right) \).

Find the argument of -2

Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)

If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,

If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?

For the ellipse x² + 3y² = a², find the length of major and minor axis.

Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.

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Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7

Expand the polynomial f(x)=x²-3x+2 in power of (x-2)

Use differentials to find \(\sqrt{25.2}\)

If of f(x, y) = x² + y³ + 2xy² find f_xx, f_yy, f_xy and f_yx.

If \(w={ e }^{ { x }^{ 2 }+{ y }^{ 2 } }\) ,x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

Using differentials, find the approximate value of \(sin\left( \frac { 22 }{ 14 } \right) \) 

Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log(tanx)dx } =0\)

Find the area of the region bounded by the curve \(\sqrt { x } +\sqrt { y } =\sqrt { a } \) (x,y>0) and the co-ordinate axes.

Find the order and degree of \(y+\frac { dy }{ dx } =\frac { 1 }{ 4 } \int { ydx } \)