A point moves in such a way that the difference of its distance from two points (8, 0) and (- 8, 0) always remains 4. Then, the locus of the point is

• a circle

• a parabola

• an ellipse

• a hyperbola

The number of integer values of m, for which the x - coordinate of the point of intersection of the lines 3x + 4y= 9 and y =mx + 1 is also an integer, is

• 0

• 2

• 4

• 1

If a straight line passes through the point ($\mathrm{\alpha }, \mathrm{\beta }$) and the portion of the line intercepted between the axes is divided equally at that point, then $\frac{\mathrm{x}}{\mathrm{\alpha }} + \frac{\mathrm{y}}{\mathrm{\beta }}$ is

• 0

• 1

• 2

• 4

The coefficient of x¹⁰ in the expansion of 1 + (1 + x) + ... + (1 + x)²⁰

• ${}_{19}\mathrm{C}_9$

• ${}_{20}\mathrm{C}_{10}$

• ${}_{21}\mathrm{C}_{11}$

• ${}_{22}\mathrm{C}_{12}$

Let A and B be two events with $\mathrm{P}\left({\mathrm{A}}^{\mathrm{C}}\right) = 0.3, \mathrm{P}\left(\mathrm{B}\right) = 0.4 \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap {\mathrm{B}}^{\mathrm{C}}\right)$. Then, $\mathrm{P}\left(\mathrm{B} | \mathrm{A} \cup {\mathrm{B}}^{\mathrm{C}}\right)$ is equal to

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

Let p, q and r be the altitudes of a triangle with area S and perimeter 2 t. Then, the value of $\frac{1}{\mathrm{p}} + \frac{1}{\mathrm{q}} + \frac{1}{\mathrm{r}}$ is

• $\frac{\mathrm{S}}{\mathrm{t}}$

• $\frac{\mathrm{t}}{\mathrm{S}}$

• $\frac{\mathrm{S}}{2\mathrm{t}}$

• $\frac{2\mathrm{S}}{\mathrm{t}}$

Let C₁ and C₂ denote the centres of the circles x² + y² = 4 and (x - 2)² + y² = 1 respectively and let P and Q be their points of intersection. Then, the areas of $∆$C₂PQ and CPQ are in the ratio

• 3 : 1

• 5 : 1

• 7 : 1

• 9 : 1

A straight line through the point of intersection of the lines x + 2y = 4 and 2x + y = 4 meets the coordinate axes at A and B. The locus of the mid-point of AB is

• 3(x + y) = 2xy

• 2(x + y) = 3xy

• 2(x + y) = xy

• x + y = 3xy

49.

Let P and Q be the points on the parabola y² = 4x so that the line segment PQ subtends right angle at the vertex. If PQ intersects the axis of the parabola at R, then the distance of the vertex from R is

• 1

• 2

• 4

• 6

The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0. Then, the equation of the circumcircle of the triangle is

• x² + y² - 2x - 2y - 2 = 0

• x² + y² - 2x - 2y - 14 = 0

• x² + y² - 2x - 2y + 2 = 0

• x² + y² - 2x - 2y + 14 = 0