List of practice Questions

Let ABC be a triangle formed by the lines $ 7x - 6y + 3 = 0 $, $ x + 2y - 31 = 0 $, and $ 9x - 2y - 19 = 0 $. Let the point $ (h, k) $ be the image of the centroid of $ \triangle ABC $ in the line $ 3x + 6y - 53 = 0 $. Then $ h^2 + k^2 + hk $ is equal to:

Let the line $ x + y = 1 $ meet the circle $ x^2 + y^2 = 4 $ at the points A and B. If the line perpendicular to AB and passing through the midpoint of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to:

Let $ |z_1 - 8 - 2i| \leq 1 $ and $ |z_2 - 2 + 6i| \leq 2 $, where $ z_1, z_2 \in \mathbb{C} $. Then the minimum value of $ |z_1 - z_2| $ is:

Let the line $x + y = 1$ meet the axes of x and y at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and $AN : NB = \lambda : 1$, then the sum of all possible values of $\lambda$ is:

If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] whose midpoint is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha + \beta$ is equal to:

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on $3x + 2y + 2 = 0$. Then the length of the chord of the circle C, whose midpoint is (1, 2), is:

Let $ A(6,8) $, $ B(10\cos\alpha, -10\sin\alpha) $, and $ C(-10\sin\alpha, 10\cos\alpha) $ be the vertices of a triangle. If $ L(a,9) $ and $ G(h,k) $ be its orthocenter and centroid respectively, then $ 5a - 3h + 6k + 100\sin2\alpha $ is equal to _____ .

Let the distance between two parallel lines be 5 units and a point $ P $ lies between the lines at a unit distance from one of them. An equilateral triangle $ POR $ is formed such that $ Q $ lies on one of the parallel lines, while $ R $ lies on the other. Then $ (QR)^2 $ is equal to _______.

Let $ a $ be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $ \alpha $ with the positive $ x $-axis and the equations of its diagonals are $\left( \sqrt{3} + 1 \right) x + \left( \sqrt{3} - 1 \right) y = 0$ and $\left( \sqrt{3} - 1 \right) x - \left( \sqrt{3} + 1 \right) y + 8\sqrt{3} = 0.$ Then $ a^2 $ is equal to

A line passing through the point $ P(a, 0) $ makes an acute angle $ \alpha $ with the positive $ x $-axis. Let this line be rotated about the point $ P $ through an angle $ \frac{\alpha}{2} $ in the clock-wise direction. If in the new position, the slope of the line is $ 2 - \sqrt{3} $ and its distance from the origin is $ \frac{1}{\sqrt{2}} $, then the value of $ 3a^2 \tan^2 \alpha - 2\sqrt{3} $ is

A line passing through the point P$(\sqrt{5}, \sqrt{5})$ intersects the ellipse $ \frac{x^2}{36} + \frac{y^2}{25} = 1 $ at A and B such that (PA).(PB) is maximum. Then 5(PA$^2$ + PB$^2$) is equal to :

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $ L_1 : 2x + y + 6 = 0 $ and $ L_2 : 4x + 2y - p = 0 $, $ p>0 $, at the points A and B, respectively. If $ AB = \frac{9}{\sqrt{2}} $ and the foot of the perpendicular from the point A on the line $ L_2 $ is M, then $ \frac{AM}{BM} $ is equal to

The radius of the smallest circle which touches the parabolas $ y = x^2 + 2 $ and $ x = y^2 + 2 $ is

Let the product of the focal distances of the point P(4, $ 2\sqrt{3} $) on the hyperbola H: $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then $ p^2 + q^2 $ is equal to .....

If the four distinct points $ (4, 6) $, $ (-1, 5) $, $ (0, 0) $ and $ (k, 3k) $ lie on a circle of radius $ r $, then $ 10k + r^2 $ is equal to

Consider the lines $ x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5 $. If P is the point through which all these lines pass and the distance of L from the point $ Q(3, 6) $ is $ d $, then the distance of L from the point $ (3, 6) $ is $ d $, then the value of $ d^2 $ is

Let the equation $ x(x+2) * (12-k) = 2 $ have equal roots. The distance of the point $ \left(k, \frac{k}{2}\right) $ from the line $ 3x + 4y + 5 = 0 $ is

Let C be the circle of minimum area enclosing the ellipse E: $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ with eccentricity $ \frac{1}{2} $ and foci $ (\pm 2, 0) $. Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 29 is parallel to the major axis and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is:

The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:

If the equation of the hyperbola with foci $ (4, 2) $ and $ (8, 2) $ is $ 3x^2 - y^2 - \alpha x + \beta y + \gamma = 0 $, then $ \alpha + \beta + \gamma $ is equal to _______

Let the area of the triangle formed by a straight line $ L: x + by + c = 0 $ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $ L $ makes an angle of $ 45^\circ $ with the positive x-axis, then the value of $ b^2 + c^2 $ is:

If the line $ \arg(z) = \frac{\pi}{3} $ intersects the curve $ |z - 2\sqrt{3}i| = 2 $, $ z \in \mathbb{C} $, at two distinct points $A$ and $B$, then $AB$ equals:

Let the distance between the foci of an ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] be $4$ and the distance between its directrices be $10$. Then the length of its latus rectum is:

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $2x^2+9y^2=36$ and the hyperbola $4x^2-9y^2=36$, respectively. Then the distance between the point of intersection of the lines $5x-7y=3$ and $3x+y=7$, and the point $ (9e_1^2,\;9e_2^2) $ is:

Let $A(3,4)$, $B(5,-2)$ and $P(\alpha,\beta)$, $\alpha\beta \neq 0$, be three points such that $PA = PB$ and the area of $\triangle PAB$ is $10$. Then the distance of the point $Q(2\alpha-5\beta,\; \alpha-\beta^2)$ from the line having intercepts $3$ and $1$ on the $x$- and $y$-axis respectively, is: