Rectangular and Cartesian products Questions and answers

  1. Elementary Mathematics
    1. Quadratic Equations
    2. Simplification
    3. Area and perimeter
    4. Volume and surface area
    5. Geometry
    6. Trigonometry
    7. Polynomials
    8. Height and Distance
    9. Simple and Decimal fraction
    10. Indices and Surd
    11. Logarithms
    12. Trigonometric ratio
    13. Straight lines
    14. Triangle
    15. Circles
    16. Quadrilateral and parallelogram
    17. Loci and concurrency
    18. Statistics
    19. Rectangular and Cartesian products
    20. Rational expression
    21. Set theory
    22. Factorisation
    23. LCM and HCF
    24. Clocks
    25. Real Analysis
11). 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). \( \Large a\ =\ \pm \sqrt{2}b \)
12). 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). \( \Large \left(-3,\ 6\right) \)
13). 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
14). 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:
A). -2
B). 2
C). 1
D). 0
15). 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). \( \Large \left( - \frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \)


16). 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
17). 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). \( \Large \left(0,\ \frac{b}{2}\right) \)
18). 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
19). 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). \( \Large \left(1,\ \frac{1}{\sqrt{3}} \right) \)
20). If \( \Large P \left(a_{1},\ b_{1}\right)\ and\ Q \left(a_{2},\ b_{2}\right) \) are two points, the \( \Large OP.OQ \cos \left( \angle POQ\right) \) is (O is origin ):
A). \( \Large a_{1}a_{2}+b_{1}b_{2} \)
B). \( \Large a^{2}_{1}+a^{2}_{2}+b^{2}_{1}+b^{2}_{2} \)
C). \( \Large a^{2}_{1}-a^{2}_{2}+b^{2}_{1}-b^{2}_{2} \)
D). none of these
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