List of top Mathematics Questions

If \(P(\alpha, \beta)\) is a point on the curve \(9x^2 + 4 y^2 = 144\) in the first quadrant and the minimum area of the triangle formed by the tangent of the curve at \(P\) with the coordinate axes is \(S\), then find \(S\).

Evaluate \[ \int \frac{x + 1}{(x - 2) \sqrt{1 - x}} \, dx = ? \]

If the tangent to the curve \(xy + ax + by = 0\) at \((1,1)\) makes an angle \(\tan^{-1} 2\) with X-axis, then find \(\frac{ab}{a+b}\).

If the displacement \(S\) of a particle travelling along a straight line in \(t\) seconds is given by \[ S = 2t^3 + 2t^2 - 2t - 3, \] then the time taken (in seconds) by the particle to change its direction is?

If \(f(x) = x^{\sec^{-1} x}\), then find \(f'(2)\).

If \(y = \sin^{-1} \left(\frac{2x}{1 + x^2}\right)\) and \(\left(\frac{d^2 y}{dx^2}\right)_{x=2} = k\), then find \(25k\).

If \(f(x) = \sec^{-1} \left(\frac{1}{2x^2 -1}\right)\) and \(g(x) = \tan^{-1} \left(\frac{\sqrt{1 + x^2} - 1}{x}\right)\), then the derivative of \(f(x)\) with respect to \(g(x)\) is?

If the image of the point \(A(1,1,1)\) with respect to the plane \(4x + 2y + 4z + 1 = 0\) is \(B(\alpha, \beta, \gamma)\), then find \(\alpha + \beta + \gamma\).

Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]

Evaluate \[ \lim_{x \to 0} \sqrt{\frac{x + 2 \sin x + 3 \tan x - \tan^3 x}{x^2 + 2 \sin x + \tan x + 3 - \sqrt{\sin^2 x - 2 \tan x - x + 3}}} = ? \]

The point in the \(xy\)-plane which is equidistant from the points \(A(2,0,3), B(0,3,2)\) and \(C(0,0,1)\) has the coordinates?

If the direction ratios of two lines \(L_1\) and \(L_2\) are \((1,-2,2)\) and \((-2,3,-6)\) respectively, then the direction ratios of the line which is perpendicular to both \(L_1\) and \(L_2\) are?

Let \(P(a \sec \theta, b \tan \theta)\) and \(Q(a \sec \phi, b \tan \phi)\) where \(\theta + \phi = \frac{\pi}{2}\) be two points on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). If \((h,k)\) is the point of intersection of the normals drawn at \(P\) and \(Q\), then find \(k\).

If the chord joining points \((1,2)\) and \((2,-1)\) on a circle subtends an angle \(\frac{\pi}{4}\) at any point on its circumference, then the equation of such a circle is?

If the angle between the asymptotes of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is \(2 \tan^{-1} \left(\frac{1}{3}\right)\) and \(a^2 - b^2 = 45\), then find \(ab\).

If the normal chord drawn at the point \(\left(\frac{15}{2\sqrt{2}}, \frac{15}{2\sqrt{2}}\right)\) to the parabola \(y^2 = 15x\) subtends an angle \(\theta\) at the vertex of the parabola, then \(\sin \frac{\theta}{3} + \cos \frac{2\theta}{3} - \sec \frac{4\theta}{3} =\) ?

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b)\) is normal to the circle \((x+1)^2 + (y+1)^2 = 1\), then \(a^2\) lies in the interval?

The equation of the circle which cuts all the three circles \[ 4(x-1)^2 + 4(y-1)^2 = 1,
4(x+1)^2 + 4(y-1)^2 = 1,
4(x+1)^2 + 4(y+1)^2 = 1, \] orthogonally is?

The equation of the circle touching the lines \(|x-2| + |y-3| = 4\) is?

Lines \(L_1\) and \(L_2\) have slopes 2 and \(-\frac{1}{2}\) respectively. If both \(L_1\) and \(L_2\) are concurrent with the lines \(x - y + 2 = 0\) and \(2x + y + 3 = 0\), then the sum of the absolute values of the intercepts made by the lines \(L_1\) and \(L_2\) on the coordinate axes is?

If \((\alpha, \beta)\) is the external centre of similitude of the circles \[ x^2 + y^2 = 3 \] and \[ x^2 + y^2 - 2x + 4y + 4 = 0, \] then find \(\frac{\beta}{\alpha}\).

The angle made by a line \(L\) with positive X-axis measured in the positive direction is \(\frac{\pi}{6}\) and the intercept made by \(L\) on Y-axis is negative. If \(L\) is at a distance 5 units from the origin, then the perpendicular distance from the point \(\left(1,-\sqrt{3}\right)\) to the line \(L\) is?

If \[ 2x^2 + 3xy - 2y^2 - 5x + 2fy - 3 = 0 \] represents a pair of straight lines, then one of the possible values of \(f\) is?

A circle passing through origin cuts the coordinate axes at \(A\) and \(B\). If the straight line \(AB\) passes through a fixed point \((x_1,y_1)\), then the locus of the centre of the circle is?

The lines \(L_1: y - x = 0\) and \(L_2: 2x + y = 0\) intersect the line \(L_3: y + 2 = 0\) at points \(P\) and \(Q\) respectively. The bisector of the angle between \(L_1\) and \(L_2\) divides the segment \(PQ\) internally at \(R\). Consider: Statement-I: \(PR : RQ = 2\sqrt{2} : \sqrt{5}\).
Statement-II: In any triangle, bisector of an angle divides that triangle into two similar triangles.
Which statement(s) is/are correct?