21). If a point \( \Large P \left(4,\ 3\right) \) is shifted by a distance \( \Large \sqrt{2} \) unit parallel to the line y = x, then co-ordinates of P in new position are:
| A). \( \Large \left(5,\ 4\right) \) |
| B). \( \Large \left(5+\sqrt{2},\ 4+\sqrt{2}\right) \) |
| C). \( \Large \left(5-\sqrt{2},\ 4-\sqrt{2}\right) \) |
| D). none of these |
|
22). The orthocentre of a triangle formed by lines \( \Large 2x+y=2,\ x-2y=1\ and\ x+y=1 \) is:
| A). \( \Large \left(\frac{1}{3},\ \frac{2}{3}\right) \) |
| B). \( \Large \left(0,\ 1\right) \) |
| C). \( \Large \left(\frac{2}{3},\ \frac{1}{3} \right) \) |
| D). \( \Large \left(1,\ 0\right) \) |
|
23). The incentre of triangle formed by lines x = 0, y = 0 and 3x+4y = 12 is at:
| A). \( \Large \left(\frac{1}{2},\ \frac{1}{2}\right) \) |
| B). \( \Large \left(1,\ 1\right) \) |
| C). \( \Large \left(1,\ \frac{1}{2} \right) \) |
| D). \( \Large \left(\frac{1}{2},\ 1\right) \) |
|
24). If the vertices of a triangle have integral co ordinates, the triangle cannot be:
| A). an equilateral triangle |
| B). a right angled triangle |
| C). an isosceles triangle |
| D). none of these |
|
25). If \( \Large t_{1}+t_{2}+t_{3}=-t_{1}t_{2}t_{3} \) then orthocentre of the triangle formed by the points \( \Large \left[ at_{1}t_{2},\ a \left(t_{1}+t_{2}\right) \right],\ \left[ at_{2}t_{3},\ a \left(t_{2}+t_{3}\right) \right]\ and\ \left[ at_{3}t_{1},\ a \left(t_{3}-t_{1}\right) \right] \) lies on;
| A). \( \Large \left(a,\ 0\right) \) |
| B). \( \Large \left(-a,\ 0\right) \) |
| C). \( \Large \left(0,\ a\right) \) |
| D). \( \Large \left(0,\ -a\right) \) |
|
26). P\( \Large \left(2,\ 1\right) \) is image of the point Q\( \Large \left(4,\ 3\right) \) about the line:
| A). x - y = 1 |
| B). 2x - 3y = 0 |
| C). x + y = 5 |
| D). none of these |
|
27). If the points \( \Large \left(a_{1},b_{1}\right), \left(a_{2},b_{2}\right) \) and \( \Large \left( a_{3},b_{3} \right) \) are collinear then lines \( \Large a_{i}x+b_{i}y+1=0 \) for i = 1, 2, 3 are:
| A). concurrent |
| B). indentical |
| C). parallel |
| D). none of these |
|
28). A point P\( \Large \left(2,\ 4\right) \) translates to the point Q along the parallel to the positive direction of x-axis by 2 unit. If O be the origin, then \( \Large \angle OPQ \) is:
| A). \( \Large \sin^{-1}\sqrt{\frac{399}{400}} \) |
| B). \( \Large \cos^{-1} \left(\frac{1}{20}\right) \) |
| C). \( \Large -\sin^{-1} \left(\sqrt{\frac{399}{400}}\right) \) |
| D). none of these |
|
29). The line joining \( \Large A \left(b \cos \alpha ,\ b \sin \alpha \right)\ and\ B \left(a \cos \beta ,\ a \sin \beta \right) \) is produced to the point \( \Large M \left(x,\ y\right) \) so that \( \Large AM:MB=b:a\ then\ x \cos \left(\frac{ \alpha + \beta }{2}\right)\ +\ y \sin \left(\frac{ \alpha + \beta }{2}\right) \) is:
| A). -1 |
| B). 0 |
| C). 1 |
| D). \( \Large a^{2} + b^{2} \) |
|
30). If \( \Large P_{1},\ P_{2} \) denote the length of perpendiculars from the origin on the lines \( \Large x \sec \alpha + y \cosh \alpha = 2 \alpha \ and\ x \cos \alpha + y \sin \alpha = \alpha \cos 2 \alpha \) respectively, then \( \Large \left(\frac{P_{1}}{P_{2}}+\frac{P_{2}}{P_{1}}\right)^{2} \) is equal to:
| A). \( \Large 4 \sin^{2} 4 \alpha \) |
| B). \( \Large 4 \cos^{2} 4 \alpha \) |
| C). \( \Large 4 cosec^{2} 4 \alpha \) |
| D). \( \Large 4 \sec^{2} 4 \alpha \) |
|
