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List of top Mathematics Questions

The length of x-intercept made by the pair of lines $ 2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0 $ is:

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 $ 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 $:

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 = $

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then $ f'(2) $ 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 $ 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 = $:

If the line $ 5x - 2y - 6 = 0 $ is a tangent to the hyperbola $ 5x^2 - ky^2 = 12 $, then the equation of the normal to this hyperbola at $ (\sqrt{6}, p) $ is:

If \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] then $ \frac{2}{3} (A + B + C + D) = $}

Let $[r]$ denote the largest integer not exceeding $r$, and the roots of the equation $ 3z^2 + 6z + 5 + \alpha(x^2 + 2x + 2) = 0 $ are complex numbers whenever $ \alpha > L $ and $ \alpha < M $. If $ (L - M) $ is minimum, then the greatest value of $[r]$ such that $ Ly^2 + My + r < 0 $ for all $ y \in \mathbb{R} $ is:

If $ \cos^{-1}(2x) + \cos^{-1}(3x) = \frac{\pi}{3} $ and $ 4x^2 = \frac{a}{b} $, then $ a + b = $:

The angle between the line with the direction ratios $ (2, 5, 1) $ and the plane $ 8x + 2y - z = 4 $ is given by

If the roots of the equation $ x^3 - 13x^2 + Kx - 27 = 0 $ are in geometric progression, then $ K = $:

If $ A = (1,2,3), B = (3,4,7) $ and $ C = (-3,-2,-5) $ are three points then the ratio in which the point C divides AB externally is

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 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:

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 the lines $ 3x+y-4=0 $, $ x - \alpha y + 10 = 0 $, $ \beta x + 2y + 4 = 0 $ and $ 3x + y + k = 0 $ represent the sides of a square, then find $ \alpha \beta (k+4)^2 $.

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 $ y = \frac{ax + \beta}{\gamma x + \delta} $, then $ 2y_1 y_3 = $ ?

If \[ f(x) = 5 \cos^3 x - 3 \sin^3 x \quad \text{and} \quad g(x) = 4 \sin^3 x + \cos^2 x, \] then the derivative of $ f(x) $ with respect to $ g(x) $ is:

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 = $:

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then $ f'(2) $ is: