1). If O be the origin and if \( \Large P_{1} \left(x_{1},\ y_{1}\right)\ and\ P_{2} \left(x_{2},\ y_{2}\right) \) be two points, then \( \Large \left(OP_{1} \parallel \ OP_{2} \right) \cos \left( \angle P_{1}\ OP_{2}\right) \) is equal to:
A). \( \Large x_{1}y_{2}+x_{2}y_{1} \) |
B). \( \Large \left(x^{2}_{1}+x^{2}_{2}+y^{2}_{2}\right) \) |
C). \( \Large \left(x_{1}-x_{2}\right)^{2}+ \left(y_{1}-y_{2}\right)^{2} \) |
D). \( \Large x_{1}x_{2}+y_{1}y_{2} \) |
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2). The median BE and AD of triangle with vertices \( \Large A \left(0,\ b\right), B \left(0,\ 0\right)\ and\ C \left(a,\ 0\right) \) are perpendicular to each other if;
A). \( \Large a=\frac{b}{2} \) |
B). \( \Large b=\frac{a}{2} \) |
C). ab = 1 |
D). \( \Large a\ =\ \pm \sqrt{2}b \) |
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3). ABC is a triangle with vertices A = (-1, 4), B = (6, -2) and C = (-2, 4). D, E and F are points which divide each AB, BC and CA respectively in the ratio 3 : 1 internally. Then centroid of the triangle DEF is:
A). \( \Large \left(3,\ 6\right) \) |
B). \( \Large \left(1,\ 2\right) \) |
C). \( \Large \left(4,\ 8\right) \) |
D). \( \Large \left(-3,\ 6\right) \) |
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4). If \( \Large A \left(x_{1},\ y_{1}\right), B \left(x_{2},\ y_{2}\right)\ and\ C \left(x_{3},\ y_{3}\right) \) are the vertices of a triangle, then the excentre with respect to B is:
A). \( \Large \frac{ \left(ax_{1}-bx_{2}+cx_{3}\right) }{a-b+c},\ \frac{ay_{1}-by_{2}+cy_{3}}{a-b+c} \) |
B). \( \Large \frac{ \left(ax_{1}+bx_{2}-cx_{3}\right) }{a+b-c},\ \frac{ay_{1}+by_{2}-cy_{3}}{a-b-c} \) |
C). \( \Large \frac{ \left(ax_{1}-bx_{2}-cx_{3}\right) }{a-b-c},\ \frac{ay_{1}-by_{2}-cy_{3}}{a-b-c} \) |
D). none of these |
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5). The four distinct points \( \Large \left(0,\ 0 \right),\ \left(2,\ 0\right),\ \left(0,\ -2\right)\ and\ \left(k,\ -2\right) \) are conocyclic, if k is equal to:
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6). The co-ordinate axis rotated through an angle \( \Large 135 ^{\circ} \). If the co-ordinates of a points P in the new system are known to be \( \Large \left(4,\ -3\right) \), then the co-ordinates of P in the original systems are:
A). \( \Large \left(\frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \) |
B). \( \Large \left(\frac{1}{\sqrt{2}},\ -\frac{7}{\sqrt{2}}\right) \) |
C). \( \Large \left( - \frac{1}{\sqrt{2}},\ -\frac{7}{\sqrt{2}}\right) \) |
D). \( \Large \left( - \frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \) |
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7). Area of quadrilateral whose vertices are \( \Large \left(2,\ 3\right),\ \left(3,\ 4\right),\ \left(4,\ 5\right)\ and\ \left(5,\ 6\right) \)
A). 0 |
B). 4 |
C). 6 |
D). none of these |
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8). Co-ordinates of the foot of the perpendicular drawn from \( \Large \left(0,\ 0\right) \) to the line joining \( \Large \left(a \cos \alpha ,\ a \sin \alpha\right)\ and\ \left(a \cos \beta ,\ a \sin \beta \right) \) are:
A). \( \Large \left(\frac{a}{2},\ \frac{b}{2}\right) \) |
B). \( \Large \left(\frac{a}{2} \left(\cos \alpha + \cos \beta \right), \frac{a}{2} \left(\sin \alpha + \sin \beta \right) \right) \) |
C). \( \Large \left(\cos\frac{ \alpha + \beta }{2},\ \sin\frac{ \alpha + \beta }{2}\right) \) |
D). \( \Large \left(0,\ \frac{b}{2}\right) \) |
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9). An equilateral triangle has each side equal to a the co-ordinates of its vertices are \( \Large \left(x_{1},\ y_{1}\right),\ \left(x_{2},\ y_{2}\right)\ and\ \left(x_{3},\ y_{3}\right) \), then the square of determinant \( \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} \) equals
A). \( \Large 3a^{4} \) |
B). \( \Large \frac{3}{4}a^{4} \) |
C). \( \Large 4a^{4} \) |
D). none of these |
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10). The incentre of triangle with vertices \( \Large \left(1,\ \sqrt{3}\right),\ \left(0,\ 0\right)\ and\ \left(2,\ 0\right) \) is:
A). \( \Large \left(1,\ \frac{\sqrt{3}}{2}\right) \) |
B). \( \Large \left(\frac{2}{3},\ \frac{1}{\sqrt{3}}\right) \) |
C). \( \Large \left(\frac{2}{3},\ \frac{\sqrt{3}}{2}\right) \) |
D). \( \Large \left(1,\ \frac{1}{\sqrt{3}} \right) \) |
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