List of top Mathematics Questions

If \( \cos^{-1}(2x) + \cos^{-1}(3x) = \frac{\pi}{3} \) and \( 4x^2 = \frac{a}{b} \), then \( a + b = \):
If \[ \cos x \frac{dy}{dx} - y \sin x = 6x, \quad (0<x<\frac{\pi}{2}) \quad {and} \quad y(\frac{\pi}{3}) = 0, \quad {then} \quad y(\frac{\pi}{6}) = \]
If all chords of the curve \( 2x^2 - y^2 + 3x + 2y = 0 \), which subtend a right angle at the origin, always pass through the point \( (a, \beta) \), then \( (a, \beta) = \):
If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then \( (\alpha, \beta) \) is:
If \( m, n \) are respectively the least positive and greatest negative integer values of such that \( (\frac{1-i}{1+i})^k = -i\), then \( m - n = \):
Equation of the normal to the curve \( y = x^2 + x \) at the point \( (1,2) \) is:

Let a and b be two non-collinear vectors of unit modulus. If u = a − (a · b)b and v = a × b, then ∥v∥ = ?

If the line segment joining the points \( (1,0) \) and \( (0,1) \) subtends an angle of \( 45^\circ \) at a variable point \( P \), then the equation of the locus of \( P \) is: 

If \( L, M, N \) are the midpoints of the sides PQ, QR, and RP of triangle \( \Delta PQR \), then \( \overline{QM} + \overline{LN} + \overline{ML} + \overline{RN} - \overline{MN} - \overline{QL} = \):

Evaluate \( \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx \):
The equation \( 2x^2 - 3xy - 2y^2 = 0 \) represents two lines \( L_1 \) and \( L_2 \). The equation \( 2x^2 - 3xy - 2y^2 - x + 7y - 3 = 0 \) represents another two lines \( L_3 \) and \( L_4 \). Let \( A \) be the point of intersection of lines \( L_1 \) and \( L_3 \), and \( B \) be the point of intersection of lines \( L_2 \) and \( L_4 \). The area of the triangle formed by the lines \( AB \), \( L_3 \), and \( L_4 \) is: .
If \( 1 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 7 + 5 \cdot 7 \cdot 9 + \dots \) (n terms) = \( n(n + 1)f(n) - 3n \), then \( f(1) = \):
If the roots of the equation \( x^3 - 13x^2 + Kx - 27 = 0 \) are in geometric progression, then \( K = \):
Evaluate the integral: \[ I = \int_{-\pi}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \,dx. \]
If \[ \frac{x^2+3}{x^4+2x^2+9} = \frac{Ax+B}{x^2+ax+b} + \frac{Cx+D}{x^2+cx+d} \] then \( aA + bB + cC + dD = \)
In \( \triangle ABC \), if \( A, B, C \) are in arithmetic progression, \( \Delta = \frac{\sqrt{3}}{2} \) and \( r_1 r_2 = r_3 r \), then \( R \) is:
If a function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = x^3 - x \), then \( f \) is:
If \[ \int \log \left( 6\sin^2x + 17\sin x + 12 \right)^{\cos x} \, dx = f(x) + c \] then, \[ f\left(\frac{\pi}{2}\right) = \]
If \( a \) is a common root of \( x^2 - 5x + \lambda = 0 \) and \( x^2 - 8x - 2\lambda = 0 \) (\( \lambda \neq 0 \)) and \( \beta, \gamma \) are the other roots of them, then \( a + \beta + \gamma + \lambda = \):

Let \( F \) and \( F' \) be the foci of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( b<2 \)), and let \( B \) be one end of the minor axis. If the area of the triangle \( FBF' \) is \( \sqrt{3} \) sq. units, then the eccentricity of the ellipse is: 

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:
If a five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4, and 5 without repetition, then the total number of ways this can be done is:
If \( y = \frac{ax + \beta}{\gamma x + \delta} \), then \( 2y_1 y_3 = \) ?

.The function \( f(x) \) is given by: \[ f(x) = \begin{cases} \frac{2}{5 - x}, & x<3 \\ 5 - x, & x \geq 3 \end{cases} \] Which of the following is true

Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \]