List of top Mathematics Questions

The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to
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 $2f(2)$.
If both the roots of the equation \[ x^2-2ax+a^2-1=0 \quad (a\in\mathbb{R}) \] lie in the interval \((-2,2)\), then the equation \[ x^2-(a^2+1)x-(a^2+2)=0 \] has:
If the domain of the function \[ f(x)=\frac{1}{\ln(10-x)}+\sin^{-1}\!\left(\frac{x+2}{2x+3}\right) \] is \[ (-\infty,-a]\ \cup\ (-1,b)\ \cup\ (b,c), \] then find the value of \( (b+c+3a) \).
Let \(Q(a,b,c)\) be the image of the point \(P(3,2,1)\) in the line \[ \frac{x-1}{1}=\frac{y-2}{2}=\frac{z-1}{1}. \] The distance of \(Q\) from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:
Orthocentre of equilateral \(\Delta ABC\) is at the origin. If side \(BC\) lies along \(x + 2\sqrt{2}y = 4\). If coordinates of vertex \(A\) are (a,b). Find the value of \(\lfloor |a + \sqrt{2}b| \rfloor\), where \([ \cdot ]\) denotes G.I.F. :
The no. of solution in \(x \in \left(-\frac{1}{2\sqrt{6}}, \frac{1}{2\sqrt{6}}\right)\) of equation \(\tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6}\) is :
The largest \( n \in \mathbb{N} \), for which \( 7^n \) divides \( 101! \), is :
If \[ f(x)=1-2x+\int_{0}^{x} e^{x-t} f(t)\,dt \] and \[ g(x)=\int_{0}^{x} (f(t)+2)^{11}(t+12)^{17}(t-4)^4\,dt, \] If local minima and local maxima of \(g(x)\) occur at \(x=p\) and \(x=q\) respectively, find \(|p|+q\).
Let \( z \) be the complex number satisfying \( |z - 5| \leq 3 \) and having maximum possible positive argument. Then the value of \[ \left| \frac{5z - 12}{5iz + 16} \right|^2 \] is equal to:
Let \[ A=\{z\in\mathbb{C}:|z-2|\le 4\} \quad\text{and}\quad B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}. \] Then the maximum value of \(|z_1-z_2|\), where \(z_1\in A\) and \(z_2\in B\), is:
Mean deviation about median for \( k, 2k, 3k, \ldots, 1000k \) is 500, then the value of \( k^2 \) is:
Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(2x) - 2\sin x}{x^3} \]
An ellipse has its centre at \((1,-2)\), one focus at \((3,-2)\) and one vertex at \((5,-2)\). Then the length of its latus rectum is:
If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \sqrt{5\beta}}{2} \] then the value of \( (\alpha + \beta) \) is:

If xdy - ydx = \(\sqrt{x^2 + y^2}\) dx. If y = y(x) \(\&\) y(1) = 0 then y(3) is :

Show that\( a^3 + b^3 + c^3 - 3abc = \dfrac{1}{2}(a+b+c)\left\{(a-b)^2 + (b-c)^2 + (c-a)^2\right\}\)

If \( \cos A = \dfrac{3}{5} \), calculate \( \sin A \) and \( \tan A \).
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(h\). At a point on the plane, the angle of elevation of the bottom of the flagstaff is \(\alpha\) and that of the top of the flagstaff is \(\beta\). Prove that the height of the tower is \(\dfrac{h\tan\alpha}{\tan\beta-\tan\alpha}\).
Find the area of the quadrilateral whose vertices are \( (1,1) \), \( (3,4) \), \( (5,-2) \) and \( (4,-7) \) taken in order.
Prove that \(x^n - y^n\) is divisible by \(x+y\) only when \(n\) is even.
Show that the square of an odd integer is of the form \( 8k + 1 \).
A metallic sphere of radius \(9\) cm is melted and recast to form a cylinder of radius \(3\) cm. Find the curved surface area of the cylinder.
If \(x, y, z\) are real numbers, \(x \neq 0\), and \(xy = xz\), prove that \(y = z\).
If three consecutive vertices of a parallelogram are \( A(1,-2) \), \( B(3,6) \) and \( C(5,10) \), find its fourth vertex.