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>> Elementary Mathematics >> Rectangular and Cartesian products
1). If the points (1, 1), (-1, 1), \( \left( -\sqrt{3}, \sqrt{3} \right) \) are the vertices of a triangle, then this triangle is:
A). right angled
B). isoscels
C). Equilateral
D). none of these
2). The length of altitude through A of the triangle ABC where A = (-3, 0), B = (4, 1), C = (5, 2) is:
A). \( \Large \frac{2}{\sqrt{10}} \)
B). \( \Large \frac{4}{\sqrt{10}} \)
C). \( \Large \frac{11}{\sqrt{10}} \)
D). \( \Large \frac{22}{\sqrt{10}} \)
3). If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2) then the co-ordinates of its centroid are:
A). \( \Large \left(\frac{7}{3},\ \frac{5}{3}\right) \)
B). \( \Large \left(\frac{5}{3},\ \frac{7}{3}\right) \)
C). (7, 5)
D). none of these
4). If
\( \begin{vmatrix} 
x_{1} & y_{1} & 1 \\ 
x_{2} & y_{2} & 1 \\ 
x_{3} & y_{3} & 1  
\end{vmatrix} = 0 \) ,  then the points \( \Large \left(x_{1},\ y_{1}\right),\ \left(x_{2},\ y_{2}\right)\ and\ \left(x_{3},\ y_{3}\right) \) are:

A). Vertices of an equilateral triangle
B). Vertices of a right angled triangle
C). Vertices of an isosceles triangle
D). none of these
5). If two vertices of an equilateral triangle are \( \Large \left(0,\ 0\right)\ and\ \left(3,\ 3\sqrt{3}\right) \) then the third vertex is:
A). \( \Large \left(3,\ -3\right) \)
B). \( \Large \left(-3,\ 3\right) \)
C). \( \Large \left(-3,\ 3\sqrt{3}\right) \)
D). none of these


6). If origin is shifted to \( \Large \left(7,\ -4\right) \) then point \( \Large \left(4,\ 5\right) \) shifted to
A). \( \Large \left(-3,\ 9\right) \)
B). \( \Large \left(3,\ 9\right) \)
C). \( \Large \left(11,\ 1\right) \)
D). none of these
7). The feet of the perpendicular drawn from P to the sides of a triangle ABC are collinear, then P is:
A). circumcentre of triangle ABC
B). lies on the circumcircle of triangle ABC
C). excentre of triangle ABC
D). none of these
8). The points \( \Large \left(k,\ 2-2k\right) \), \( \Large \left(-k+1,\ 2k\right) \), \( \Large \left(-4-k,\ 6-2k\right) \) are collinear then k is equal to:
A). 2, 3
B). 1, 0
C). \( \Large \frac{1}{2},\ 1 \)
D). 1, 2
9). Let AB is divided internally and externally at P and Q in the same ratio. Then AP, AB, AQ are in
A). AP
B). GP
C). HP
D). none of these
10). 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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