List of top Mathematics Questions

Let \( y = y(x) \) be the solution of the differential equation: \[ (x^2 + 4)^2 \, dy + \left( 2x^3 y + 8xy - 2 \right) dx = 0. \] If \( y(0) = 0 \), then \( y(2) \) is equal to:

Let \( P \) be the point of intersection of the lines \[ \frac{x - 2}{1} = \frac{y - 4}{5} = \frac{z - 2}{1} \quad \text{and} \quad \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 3}{2}. \] Then, the shortest distance of \( P \) from the line \( 4x = 2y = z \) is:

If the value of the integral \[ \int_{-1}^{1} \frac{\cos \alpha x}{1 + 3^x} \, dx = \frac{2}{\pi}, \] then a value of \( \alpha \) is:

Consider a hyperbola \( H \) having its centre at the origin and foci on the \( x \)-axis. Let \( C_1 \) be the circle touching the hyperbola \( H \) and having its centre at the origin. Let \( C_2 \) be the circle touching the hyperbola \( H \) at its vertex and having its centre at one of its foci. If the areas (in square units) of \( C_1 \) and \( C_2 \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of the latus rectum of \( H \) is:

If the coefficients of \( x^4 \), \( x^5 \), and \( x^6 \) in the expansion of \( (1 + x)^n \) are in arithmetic progression, then the maximum value of \( n \) is:

A variable line \( L \) passes through the point \( (3, 5) \) and intersects the positive coordinate axes at the points \( A \) and \( B \). The minimum area of the triangle \( OAB \), where \( O \) is the origin, is:

The coefficient of \( x^{70} \) in \( x^2 (1+x)^{98} + x^3 (1+x)^{97} + x^4 (1+x)^{96} + \ldots + x^{54} (1+x)^{46} \) is \( ^{99}C_p - ^{46}C_q \).
Then a possible value to \( p + q \) is:

Let \( \lambda, \mu \in \mathbb{R} \). If the system of equations
\( 3x + 5y + \lambda z = 3 \) 
\( 7x + 11y - 9z = 2 \) 
\( 97x + 155y - 189z = \mu \) 
has infinitely many solutions, then \( \mu + 2\lambda \) is equal to:

Let \( f(x) = ax^3 + bx^2 + ex + 41 \) be such that \( f(1) = 40 \), \( f'(1) = 2 \) and \( f''(1) = 4 \). Then \( a^2 + b^2 + c^2 \) is equal to:

The shortest distance between the line:
\[ \frac{x-3}{4} = \frac{y+7}{-11} = \frac{z-1}{5} \] and \[ \frac{x-5}{3} = \frac{y-9}{-6} = \frac{z+2}{1} \] is:

If the domain of the function \(f(x) = \sin^{-1}\left( \frac{x - 1}{2x + 3} \right)\) is \(\mathbb{R} - (\alpha, \beta)\), then \(12 \alpha \beta\) is equal to:

Let \[ \left| \cos \theta \cos (60^\circ - \theta) \cos (60^\circ - \theta) \right| \leq \frac{1}{8}, \quad \theta \in [0, 2\pi] \] Then, the sum of all \( \theta \in [0, 2\pi] \), where \( \cos 3\theta \) attains its maximum value, is:

Let \(\overrightarrow{OA} = 2\vec{a}\), \(\overrightarrow{OB} = 6\vec{a} + 5\vec{b}\), and \(\overrightarrow{OC} = 3\vec{b}\), where \(O\) is the origin. If the area of the parallelogram with adjacent sides \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) is 15 sq. units, then the area (in sq. units) of the quadrilateral \(OABC\) is equal to:

Let \( \int \frac{2 - \tan x}{3 + \tan x} \, dx = \frac{1}{2} \left( \alpha x + \log_e \lvert \beta \sin x + \gamma \cos x \rvert \right) + C \), where \( C \) is the constant of integration.

The solution curve, of the differential equation \( 2y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx} \), passing through the point \( (0, 1) \), is a conic, whose vertex lies on the line:

A ray of light coming from the point \( P(1, 2) \) gets reflected from the point \( Q \) on the x-axis and then passes through the point \( R(4, 3) \). If the point \( S(h, k) \) is such that \( PQRS \) is a parallelogram, then \( hk^2 \) is equal to:

The parabola \( y^2 = 4x \) divides the area of the circle \( x^2 + y^2 = 5 \) in two parts. The area of the smaller part is equal to:

If the mirror image of the point \(P(3,4,9)\) in the Line 
\(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}\) is \((α,β,γ)\) then find \(14(a+ẞ+y)\) is:

Let \( a_1, a_2, a_3, \dots \) be in an arithmetic progression of positive terms.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).  
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.

Let \( f \) be a differentiable function in the interval \( (0, \infty) \) such that \( f(1) = 1 \) and \(\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1\) for each \( x > 0 \).
Then \( 2f(2) + 3f(3) \) is equal to _________.

The number of ways of getting a sum 16 on throwing a dice four times is __________.

The area of the region enclosed by the parabola \((y - 2)^2 = x - 1\), the line \(x - 2y + 4 = 0\) and the positive coordinate axes is ______.

\[\text{Let } A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} \text{ and } P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}.\]The sum of the prime factors of \( |P^{-1}AP - 2I| \) is equal to.

Let \( A = \{1, 2, 3, \ldots, 100\} \). Let \( R \) be a relation on \( A \) defined by \( (x, y) \in R \) if and only if \( 2x = 3y \). Let \( R_1 \) be a symmetric relation on \( A \) such that \( R \subset R_1 \) and the number of elements in \( R_1 \) is \( n \). Then, the minimum value of \( n \) is \(\_\_\_\_\_\).

Let \( y = y(x) \) be the solution of the differential equation \(\sec^2 x \, dx + \left( e^{2y} \tan^2 x + \tan x \right) dy = 0,\)\( 0 < x < \frac{\pi}{2} \), \( y \left( \frac{\pi}{4} \right) = 0 \). If \( y \left( \frac{\pi}{6} \right) = \alpha \), then \( e^{8\alpha} \) is equal to\(\_\_\_\_\_\).