List of top Mathematics Questions

If the area of the region enclosed by the curve $ ay = x^2 $ and the line $ x + y = 2a $ is $ k a^3 $, then $ k $ is:

If the equation of the circle which cuts each of the circles \[ x^2 + y^2 = 4, \] \[ x^2 + y^2 - 6x - 8y + 10 = 0, \] \[ x^2 + y^2 + 2x - 4y - 2 = 0 \] at the extremities of a diameter of these circles is \[ x^2 + y^2 + 2gx + 2fy + c = 0, \] then the value of $ g + f + c $ is:

Consider the curves $ y = f(x) $ and $ x = g(y) $, and let $ P(x,y) $ be a common point of these curves. If at $ P $, on the curve $ y = f(x) $, \[ \frac{dy}{dx} = Q(x), \] and at the same point $ P $ on the curve $ x = g(y) $, \[ \frac{dx}{dy} = -Q(x), \] then:

If $ L $ is the line of intersection of two planes \[ x + 2y + 2z = 15 \quad \text{and} \quad x - y + z = 4 \] and the direction ratios of the line $ L $ are $ (a, b, c) $, then the value of \[ \frac{a^2 + b^2 + c^2}{b^2} \] is:

The equations \[ 7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0 \] represent the equal sides of an isosceles triangle. If the third side passes through $ (-1,1) $, then a possible equation for the third side is: \

If the locus of the centroid of the triangle with vertices $ A(a,0) $, $ B(\cos t, a\sin t) $ and $ C(\sin t, -b\cos t) $ ($ t $ is a parameter) is given by \[ 9x^2 + 9y^2 - 6x = 49, \] then the area of the triangle formed by the line \[ \frac{x}{a} + \frac{y}{b} = 1 \] with the coordinate axes is: \

If the extremities of the latus recta having positive ordinate of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b) $$ lie on the parabola $$ x^2 + 2ay - 4 = 0, $$ then the points $(a, b)$ lie on the curve:

The foot of the perpendicular drawn from $ A(1,2,2) $ onto the plane \[ x + 2y + 2z - 5 = 0 \] is $ B(a, \beta, \gamma) $. If $ \pi(x,y,z) = x + 2y + 2z + 5 = 0 $ is a plane then $-\pi(A):\pi(B) $ is:

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is:

If the tangent drawn at a point $P(t)$ on the hyperbola $$ x^2 - y^2 = c^2 $$ cuts the X-axis at $T$ and the normal drawn at the same point $P$ cuts the Y-axis at $N$, then the equation of the locus of the midpoint of $TN$ is:

By shifting the origin to the point $ (h,5) $ by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to $ Y = X^2 $, then $ \left( \frac{d - c}{h} \right) $ is: \

If the harmonic conjugate of $ P(2,3,4) $ with respect to the line segment joining the points \[ A(3,-2,2) \quad \text{and} \quad B(6,-17,-4) \] is $ Q(\alpha, \beta, \gamma) $, then the value of $ \alpha + \beta + \gamma $ is:

The equation of the circle passing through the origin and cutting the circles $x^2 + y^2 + 6x - 15 = 0$ and $x^2 + y^2 - 8y - 10 = 0$ orthogonally is:

Given that $ a $ and $ b $ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes. If its latus rectum is of length 4 units and the distance between its foci is $ 4\sqrt{2} $, then the value of $ a^2 + b^2 $ is:

Given the two parabolas: \[ S = y^2 - 4ax = 0, \quad S' = y^2 + ax = 0 \] where $ P(t) $ is a point on the parabola $ S' = 0 $. If $ A $ and $ B $ are the feet of the perpendiculars from $ P $ to the coordinate axes and $ AB $ is a tangent to the parabola $ S = 0 $ at the point $ Q(t_1) $, then the value of $ t_1 $ is:

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is:

If the line \[ 2x + by + 5 = 0 \] forms an equilateral triangle with \[ ax^2 - 96bxy + ky^2 = 0, \] then $ a + 3k $ is: \

S = (-1,1) is the focus, $ 2x - 3y + 1 = 0 $ is the directrix corresponding to S and $ \frac{1}{2} $ is the eccentricity of an ellipse. If $ (a,b) $ is the centre of the ellipse, then $ 3a + 2b $ is:

The combined equation of a possible pair of adjacent sides of a square with area 16 square units whose centre is the point of intersection of the lines \[ x + 2y - 3 = 0 \quad \text{and} \quad 2x - y - 1 = 0 \] is: \

Given two planes with equations $ \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 5 $ and $ \mathbf{r} \cdot (2\mathbf{i} + \mathbf{j} - \mathbf{k}) = 3 $. A plane $ \pi $ passing through the line of intersection of these two planes also passes through the point (0,1,2). If the equation of $ \pi $ is $ \mathbf{r} \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = m $, determine the value of $ \frac{bc}{a^2} $:

A plane $ \pi_1 $ passing through the point $ 3\hat{i} - 7\hat{j} + 5\hat{k} $ is perpendicular to the vector $ \hat{i} + 2\hat{j} - 2\hat{k} $ and another plane $ \pi_2 $ passing through the point $ 2\hat{i} + 7\hat{j} - 8\hat{k} $ is perpendicular to the vector $ 3\hat{i} + 2\hat{j} + 6\hat{k} $. If $ p_1 $ and $ p_2 $ are the perpendicular distances from the origin to the planes $ \pi_1 $ and $ \pi_2 $ respectively, then $ p_1 - p_2 $ is:

In triangle ABC, if $ A = 45^\circ $, $ C = 75^\circ $, and $ R = \sqrt{2} $, then the value of $ r $ is:

If $ A(1, 2, -3) $, $ B(2, 3, -1) $, and $ C(3, 1, 1) $ are the vertices of triangle $ \Delta ABC $, then find $ \left| \frac{\cos A}{\cos B} \right|.$

If $ a, b, c $ are the intercepts made on X, Y, Z-axes respectively by the plane passing through the points $ (1, 0, -2) $, $ (3, -1, 2) $, and $ (0, -3, 4) $, then $ 3a + 4b + 7c = $?

$ \vec{a}, \vec{b}, \vec{c} $ are three vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 2\sqrt{2}$, $|\vec{c}| = 5$, and $ \vec{c} $ is perpendicular to the plane of $ \vec{a} $ and $ \vec{b} $.

If the angle between the vectors $ \vec{a} $ and $ \vec{b} $ is $ \frac{\pi}{4} $, then

\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]