31). The number of integeral points (integral point means both the co-ordinates should be integer) exactly in the interior of the triangle with vertices (0, 0)(0, 21) and (21, 0) is
| A). 133 |
| B). 190 |
| C). 233 |
| D). 105 |
|
32). The arc (in sq unit) of the quadrilateral formed by two pairs of lines \( \Large l^{2}x^{2}-m^{2}y^{2}-n \left(lx+my\right)=0\ and\ l^{2}m^{2}-m^{2}y^{2}+n \left(lx-my\right)=0 \) is
| A). \( \Large \frac{n^{2}}{2|lm|} \) |
| B). \( \Large \frac{n^{2}}{|lm|} \) |
| C). \( \Large \frac{n}{2|lm|} \) |
| D). \( \Large \frac{n^{2}}{4|lm|} \) |
|
33). The point of lines represented by\( \Large 3ax^{2} + 5xy + \left(a^{2}-2\right)y^{2}=0 \) and perpendicular to each other for
| A). two value of a |
| B). for all value of a |
| C). for one values of a |
| D). for no value of a |
|
34). If the pair of lines \( \Large ax^{2}+2hxy+by^{2}+2gx+2fy+c=0 \) intersect on the y-axis then:
| A). \( \Large 2fgh=bg^{2}+ch^{2} \) |
| B). \( \Large bg^{2} \ne ch^{2}\) |
| C). \( \Large abc = 2fgh \) |
| D). none of these |
|
35). Equation of straight line belonging to families of straight lines \( \Large \left(x+2y\right)+h \left(3x+2y+1\right)=0\ and\ \left(x-2y\right)+u \left(x-y+1\right)=0 \)
| A). \( \Large 6x+5y=2 \) |
| B). \( \Large 5x-6y+4=0 \) |
| C). \( \Large 5x+6y=4 \) |
| D). none of these |
|
36). In the adjacent figure equation of refracted ray is
| A). \( \Large y=\sqrt{3}+1 \) |
| B). \( \Large y+\sqrt{3}x-3=0 \) |
| C). \( \Large \sqrt{3}x+y-\sqrt{3}=0 \) |
| D). none of these |
|
37). An equilateral triangle ABC is in first quadrant such that A lies on x-axis, B lies on y-axis and BC is parallel to x-axis, then equation of straight line through C parallel to AB is ('a' is length of the side)
| A). \( \Large y-\sqrt{3}x=\frac{3a\sqrt{3}}{2} \) |
| B). \( \Large \sqrt{3}y+x=\frac{3a\sqrt{3}}{2} \) |
| C). \( \Large y+\sqrt{3}x=\frac{3a\sqrt{3}}{2} \) |
| D). none of these |
|
38). A point moves in such a way that the square of its distances from point \( \Large \left(3,\ -2\right) \) is equal to numerically its distance from the lines \( \Large 5x-12y=13 \). The equation of the locus of the point is:
| A). \( \Large x^{2}+y^{2}-11x-16y+26=0 \) |
| B). \( \Large x^{2}+y^{2}-11x+16y=0 \) |
| C). \( \Large 13 \left(x^{2}+y^{2}\right)-83x+64y+182=0 \) |
| D). \( \Large x^{2}+y^{2}-83x+64y+182=0 \) |
|
39). The centroid of the triangle whose three sides are given by the combined equation \( \Large \left(x^{2}+7xy+2y^{2}\right) \left(y-1\right)=0 \) is:
| A). \( \Large \left(\frac{2}{3}, 0 \right) \) |
| B). \( \Large \left(\frac{7}{3},\ \frac{2}{3}\right) \) |
| C). \( \Large \left(-\frac{7}{3},\ \frac{2}{3}\right) \) |
| D). none of these |
|
40). If the point \( \Large \left(a,\ a\right) \) falls between the lines \( \Large |x+y|=2 \) then:\( \Large |x+y|=2 \)
| A). \( \Large |a| = 2 \) |
| B). \( \Large |a| 1 \) |
| C). \( \Large |a| < 1 \) |
| D). \( \Large |a| < \frac{1}{2}\) |
|
