If the vertices of a· triangle are A(0, 4, 1), B(2, 3, - 1) and C(4, 5, 0), then the orthocentre of $∆$ABC, is

• (4, 5, 0)

• (2, 3, - 1)

• (- 2, 3, - 1)

• (2, 0, 2)

If ${∆}_{\mathrm{r}} = \left|\begin{array}{ccc}2\mathrm{r} - 1& {}^{\mathrm{m}}\mathrm{C}_{\mathrm{r}}& 1\\ {\mathrm{m}}^2 - 1& 2^{\mathrm{m}}& \mathrm{m} + 1\\ {\sin }^2\left({\mathrm{m}}^2\right)& {\sin }^2\left(\mathrm{m}\right)& {\sin }^2\left(\mathrm{m} + 1\right)\end{array}\right|$, then the value of $\sum _{\mathrm{r} = 0}^{\mathrm{m}}{∆}_{\mathrm{r}}$

• 1

• 0

• 2

• None of these

Two lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - \mathrm{k}}{2} = \mathrm{z}$ intersect at a point, if k is equal to

• $\frac{2}{9}$

• $\frac{1}{2}$

• $\frac{9}{2}$

• $\frac{1}{6}$

The statement $\left(\mathrm{p} \Rightarrow \mathrm{q}\right) \iff \left(~ \mathrm{p} \wedge \mathrm{q}\right)$ is

• tautology

• contradiction

• Neither (a) nor (b)

• None of these

5.

If x + iy = $\frac{3}{2 + \cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)}$,  then x² + y² is equal to

• 3x - 4

• 4x - 3

• 4x + 3

• None of these

The negation of $\left(~ \mathrm{p} \wedge \mathrm{q}\right) \vee \left(\mathrm{p} \wedge ~ \mathrm{q}\right)$ is

• $\left(\mathrm{p} \vee ~ \mathrm{q}\right) \vee \left(~ \mathrm{p} \vee \mathrm{q}\right)$

• $\left(~ \mathrm{p} \vee \mathrm{q}\right) \vee \left(~ \mathrm{p} \vee \mathrm{q}\right)$

• $\left(\mathrm{p} \wedge ~ \mathrm{q}\right) \wedge \left(~ \mathrm{p} \vee \mathrm{q}\right)$

• $\left(\mathrm{p} \wedge ~ \mathrm{q}\right) \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

The normals at three points· P,Q and R of the parabola y² = 4ax meet at (h, k). The centroid of the $∆$PQR lies on

• x = 0

• y = 0

• x = - a

• y = a

The minimum area of the triangle formed by any tangent to the ellipse ( x²/a² ) + ( y²/b² ) = 1 with the coordinate axes is

• a² + b²

• ( a + b )²/2

• ab

• ( a - b )²/2

If the line lx + my - n = 0 will be a normal to the hyperbola, then $\frac{{\mathrm{a}}^2}{{\mathrm{l}}^2} - \frac{{\mathrm{b}}^2}{{\mathrm{m}}^2} = \frac{{\displaystyle {\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2}}{\mathrm{k}}$, where k is equal to

• n

• n²

• n³

• None of these

If $\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right), \mathrm{b} = \cos \left(\mathrm{\beta }\right) + \mathrm{isin}\left(\mathrm{\beta }\right), \mathrm{c} = \cos \left(\mathrm{\gamma }\right) + \mathrm{isin}\left(\mathrm{\gamma }\right)$ and $\frac{\mathrm{b}}{\mathrm{c}} + \frac{\mathrm{c}}{\mathrm{a}} + \frac{\mathrm{a}}{\mathrm{b}} = 1$, then $\cos \left(\mathrm{\beta } - \mathrm{\gamma }\right) + \cos \left(\mathrm{\gamma } - \mathrm{\alpha }\right) + \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)$ is equal to

• $\frac{3}{2}$

• - $\frac{3}{2}$

• 0

• 1