List of top Mathematics Questions

Bag \( A \) contains 9 white and 8 black balls and bag \( B \) contains 6 white and 4 black balls. A ball is randomly transferred from bag \( B \) to bag \( A \), then a ball is drawn from bag \( A \). If the probability that the drawn ball is white is \( \dfrac{p}{q} \) (where \( p \) and \( q \) are coprime), then find \( p + q \):

If two numbers \( a \) and \( b \) are selected from \( S = \{1, 2, 3, \dots, 100\} \), then the probability that \( |a - b| \geq 10 \) is:

If probability distribution is given by \[ P(x) = \begin{array}{c|c|c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(x) & k & 2k^2 & 6k^2 & 2k^2 + k & 4k & k & k \\ \end{array} \] Then, the value of \( P(3<x \leq 6) \) is:

Let $S$ has 5 elements and $P(S)$ is the power set of $S$. Let an ordered pair $(A,B)$ is selected at random from $P(S)\times P(S)$. If the probability that $A\cap B=\varnothing$ is $\dfrac{3^m}{2^n}$, then the value of $(m+n)$ is

If a line \(ax + y = 1\) does not intersect the hyperbola \(x^2 - 9y^2 = 9\) then a possible value of \(\alpha\) is :

Let $f(x)$ be a differentiable function satisfying the equations $\lim_{t \to x} \dfrac{t^2 f(x)-x^2 f(t)}{t-x} = 3$ and $f(1)=2$. Find the value of $2f(2)$.

Given \[ f(t)=\left|\frac{t+1}{t^2}\right|,\ (t<0) \] is strictly decreasing in the interval $(2\alpha,\alpha)$, then the maximum value of \[ g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha \] is:

If \( x^4 \, dy + (4x^3 y + 2 \sin x) \, dx = 0 \) and \( f\left( \frac{\pi}{2} \right) = 0 \), then find the value of \( \pi^4 f\left( \frac{\pi}{3} \right) \) (where \( y = f(x) \)):

The solution of the differential equation \[ x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx \] is (where \( c \) is the integration constant):

If \( y = y(x) \) and \[ (1 + x^2)\,dy + (1 - \tan^{-1}x)\,dx = 0 \] and \( y(0) = 1 \), then \( y(1) \) is equal to:

The minimum value of \( \cos^2\theta + 6\sin\theta\cos\theta + 3\sin^2\theta \) is:

Let \(O\) be the vertex of the parabola \(y^{2}=16x\). The locus of the centroid of \(\triangle OPA\), when point \(P\) lies on the parabola and point \(A\) lies on the \(x\)-axis such that \(\angle OPA = 90^\circ\), is:

The area enclosed by \[ x^2 + 4y^2 \le 4,\qquad y \le |x| - 1,\qquad y \ge 1 - |x| \] is equal to:

Ellipse \( E:\; \dfrac{x^2}{36} + \dfrac{y^2}{16} = 1 \). A hyperbola is confocal with the ellipse and the eccentricity of the hyperbola is equal to \(5\). If the principal axis of the hyperbola is the \(x\)-axis, then the length of the latus rectum of the hyperbola is:

Image of point P (1, 2, a) with respect to line mirror \(\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}\) is point Q (5, b, c), then value of \((a^2 + b^2 + c^2)\) is :

If \( PQ \) is a chord perpendicular to the transverse axis of \[ \frac{x^2}{4} - \frac{y^2}{b^2} = 1 \] of eccentricity \( \sqrt{3} \) such that \( \triangle OPQ \) is an equilateral triangle (where \( O \) is the origin), then the area of \( \triangle OPQ \) is:

If $P(h,k)$ is a variable point on $x^2 + y^2 = 4$ and $Q(2h+1,\,3k+3)$ always lies on an ellipse, if eccentricity of the ellipse is $e$, then $\dfrac{5}{e^2}$ is equal to

The locus of point of intersection of tangent drawn to the circle \( (x - 2)^2 + (y - 3)^2 = 16 \), which subtends an angle of \( 120^\circ \) is

Given triangle $OAB$ where $O$ is the origin, $A=(0,-\sqrt{3}a)$ and $B=(-\sqrt{2}b,0)$. Let the circumradius of $\triangle OAB$ be $4$ units. If the locus of the centroid of $\triangle OAB$ is a circle, then its radius is:

If \( O \) is the vertex of the parabola \( x^2 = 4ay \), \( Q \) is the point on the parabola. If \( C \) is the locus of the point which divides \( OQ \) in ratio 2:3, the equation of the chord of \( C \) which is bisected at point \( (1, 2) \) is

If \( L_1 \) and \( L_2 \) are two parallel lines and \( \triangle ABC \) is an equilateral triangle, then the area of triangle ABC is

If two circles \(x^2 + y^2 - 4x - 2y - 4 = 0\) \& \((x+1)^2 + (y+4)^2 = r^2\) intersect at two distinct points and range of r \(\in (\alpha, \beta)\), then the value of \(\alpha\beta\) is :

Let \( O \) be the vertex of the parabola \( y^2 = 16x \). The locus of centroid of \( \triangle OPA \) when \( P \) lies on the parabola and \( A \) lies on the x-axis and \( \angle OPA = 90^\circ \).

Let \( A(1, 6, 3) \) and point \( B \) and \( C \) lie on the line \[ \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] where \( B(4, 9, \alpha) \) and point \( C \) is 10 units from \( B \). Find the area of \( \triangle ABC \):

Let one end of a focal chord of the parabola \( y^{2} = 20x \) be \( (20, -20) \). If \( P(\alpha, \beta) \) divides the chord internally in the ratio \( 2 : 3 \), find the minimum value of \( \alpha + \beta \).