List of top Mathematics Questions

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
If \(a = 1 + \sum_{r=1}^{6} (-3)^{r-1} \binom{12}{2r-1}\), then the distance of the point \((12, \sqrt{3})\) from the line \(\alpha x - \sqrt{3}y + 1 = 0\) is:

Let \( C_{t-1} = 28, C_t = 56 \) and \( C_{t+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \text{ and } C(3r - n_1, r^2 - n - 1) \) be the vertices of a triangle ABC, where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \) is the locus of the centroid of triangle ABC, then \( \alpha \) equals:

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 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 \(\vec a = 2\hat i + 3\hat j + 5\hat k\), \(\vec b = \hat i - \hat j + 3\hat k\) and \(\vec c\) be a vector such that \[ \vec a \cdot \vec c = 104 \quad \text{and} \quad \vec a \times \vec c = \vec c \times \vec b. \] Then \(\vec b \cdot \vec c\) is equal to ______

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 the points \(A(a,-1,2)\), \(B(1,b,-4)\), \(C(-1,1,c)\) and \(D(1,-2,8)\) be the vertices of a parallelogram \(ABCD\). Then its area is equal to:
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:

If the circles \[ x^2+y^2-2x-8y+17=r \quad \text{and} \quad x^2+y^2-26x-18y+234=0 \] intersect at exactly one point, then the sum of all possible values of \(r\) is _______

Let \((\alpha,\beta,\gamma)\) be the foot of the perpendicular from the point \((25,2,41)\) on the line \[ \frac{x-4}{3}=\frac{y+1}{7}=\frac{z-2}{3}. \] Then \(\alpha+\beta+\gamma\) is equal to:
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:
Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.
Let \( A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} -5 \\ -2 \end{bmatrix}, C = [9 \ \ 7], \) which of the following is defined?

For the matrix [A] given below, the transpose is __________. 

\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]

Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
 

If \( E \) and \( F \) are two independent events such that \( P(E) = \frac{2}{3} \), \( P(F) = \frac{3}{7} \), then \( P(E \,|\, F) \) is equal to:
If \( f(x) = |x| + |x - 1| \), then which of the following is correct?
Find the domain of the function $f(x) = \sin^{-1}(-x^2)$.
A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm$^3$/s from the leak. If the radius of the container is 15 cm, find the rate at which the height of water is decreasing inside the container, when the height of water is 2 meters.
If \( A \) denotes the set of continuous functions and \( B \) denotes the set of differentiable functions, then which of the following depicts the correct relation between set \( A \) and \( B \)?
If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.

There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter. An entry gate is to be constructed at a point P on the boundary of the park such that distance of P from A is 35 m more than the distance of P from B. Find distance of point P from A and B respectively.