List of top Mathematics Questions

The value of \[ \sin20^\circ\sin40^\circ\sin80^\circ \] is:

If the displacement of a particle at time $t$ ($0 < t < \pi$) is given by $s = 3 \sin 2t - 6 \cos t$, then the acceleration for the values of $t$ at which its velocity is zero is:

The maximum value of $y = x(\log x)^2$ is:

If $f(x) = \sqrt{3}\sin x - \cos x - 2ax + b$ decreases for all $x \in \mathbb{R}$, then:

The maximum area of a rectangle inscribed in a circle of radius $r$ is:

Evaluate $\int \frac{1+x^2}{\sqrt{1-x^2}} dx$:

If \[ \sin\theta+\cos\theta=\sqrt{2}\cos\alpha, \] where \(0<\theta<\frac{\pi}{2}\), then the value of \[ \sin2\theta \] is:

If \[ \tan\theta+\cot\theta=4, \] then the value of \[ \sec^2\theta+\csc^2\theta \] is:

The value of \[ \lim_{x\to0}\frac{\sin5x-5\sin x}{x^3} \] is:

Let \(\theta\) be the angle between the line \[ \frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2} \] and the plane \[ 2x-y+\sqrt{\lambda}\,z+4=0. \] If \[ \sin\theta=\frac13, \] then the value of \(\lambda\) is:

Let \[ \vec a=\hat i+2\hat j+2\hat k, \qquad \vec b=2\hat i+\hat j+2\hat k. \] If \(\theta\) is the angle between \(\vec a\) and \(\vec b\), then the value of \[ \cos\theta \] is:

Let \[ A= \begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}. \] Then the determinant of \(A^2\) is:

If the feet of the perpendiculars drawn from the point \[ (3,4,5) \] to the \(X\)-, \(Y\)- and \(Z\)-coordinate axes are \(A,B,C\) respectively and the angle between \(AB\) and \(AC\) is \[ \cos^{-1}\left(\frac{9}{a}\right), \] then the value of \(a\) is:

The product of the slopes of the non-horizontal normals drawn through the point \[ (6,0) \] to the parabola \[ y^2=8x \] is:

If the ends of the major axis \(A'\) and \(A\) of the ellipse \[ \frac{(x-2)^2}{a^2}+\frac{(y-3)^2}{b^2}=1 \] are respectively at distances \(9\) and \(3\) units from a directrix \(L\), then the foci of the ellipse are:

Let \[ O(0,0,0) \] and \[ A(2,1,-3) \] be vertices of a triangle \(OAB\). If \[ (-1,2,1) \] is the midpoint of side \(AB\), and the perimeter of the triangle is \[ \sqrt2\,(k+l\sqrt7+m\sqrt{13}), \] then the value of \[ k+l+m \] is:

Consider the hyperbola \[ S \equiv \frac{x^2}{25}-\frac{y^2}{16}-1=0. \] Let \(B,B'\) be the ends of the transverse axis of the conjugate hyperbola of \(S=0\). If \(C\) is the circle with \(B,B'\) as ends of a diameter, then the slope of a common tangent to \(C\) and the given hyperbola is:

The value of \[ \sin^2 15^\circ+\cos^2 15^\circ \] is

If the equation \[ 2x^2-5x+k=0 \] has equal roots, then the value of \(k\) is

If the pole of the line \[ 2x+3y-20=0 \] with respect to the circle \[ x^2+y^2-4x+6y-12=0 \] is \((\alpha,\beta)\), then the number of tangents that can be drawn from \((\alpha,\beta)\) to the given circle is:

If \[ \log_2(x-1)+\log_2(x-3)=3, \] then the value of \(x\) is

The centre of the circle which intersects the circles \[ x^2+y^2-8x+10y+5=0 \] and \[ x^2+y^2-2x+2y+1=0 \] orthogonally is:

If \[ y=mx+\frac{3}{m} \] is a tangent to the parabola \[ y^2=4ax \] at the point \(P(3,\beta)\), where \(\beta<0\), then the value of \[ 3m-\beta \] is:

If the equation \[ x^2-6x+y^2+8y+9=0 \] represents a circle, then its radius is:

The equation of the normal to the parabola \[ y^2=12x \] at the point \((3\lambda^2,6\lambda)\) is