The roots of the equation \(x^3-3 x-2=0\) are

If \(\cos A=\frac{-60}{61}\) and \(\tan B=-\frac{7}{24}\) and neither \(A\) nor \(B\) is the second quadrant, then the angle \(A+\frac{B}{2}\) lies in the quadrant

Let \(A\) and \(B\) represent \(z_1\) and \(z_2\) in the Argand plane and \(z_1, z_2\) be the roots of the equation \(Z^2+p Z+q=0\), where \(p, q\) are complex numbers. If \(O\) is the origin, \(O A=O B\) and \(\lfloor A O B=\alpha\), then \(p^2=\)

If \(\frac{x^2+x+1}{x^2+2 x+1}=A+\frac{B}{x+1}+\frac{C}{(x+1)^2}\), then \(A-B\) is equal to

If \(3+i\) and \(2-\sqrt{3}\) are the roots of the equation \(\mathrm{f}(\mathrm{x})=\mathrm{a}_0+\mathrm{a}_1 \mathrm{x}+\mathrm{a}_2 \mathrm{x}^2+\ldots .+\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} ; \mathrm{a}_0, \mathrm{a}_1 \ldots . . \mathrm{a}_{\mathrm{n}} \in \mathbb{Z}\), then the least value of \(\mathrm{n}\) and value of \(\mathrm{a}_0\) are respectively.

Which among the following equations has roots which are negatives of the roots of the equation \(x^3-x^2+x-4=0\) ?

If \(f(x)= \begin{cases}\frac{\cos a x-\cos b x}{x^2}, & x \neq 0 \\ \frac{1}{2}\left(a^2-b^2\right) & , x=0\end{cases}\)
where \(a, b\) are real and district constants, then

The magnetic needle of a vibration magnetometer makes 12 oscillations per minute in the horizontal component of earth's magnetic field. When an external short bar magnet is placed at some distance along the axis of the needle in the same line, it makes 15 oscillations per minute. If the poles of the bar magnet are interchanged, the number of oscillations it makes per minute is

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

A plane meets the \(X, Y, Z\)-axes in \(A, B, C\) respectively. If the centroid of the \(\triangle A B C\) is \((2,-3,5)\), then the perpendicular distance from origin to the given plane is

If \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is a real number and \(0 < \theta < 2 \pi\), then \(\theta\) is equal to

If \(n \geq 2\) is a natural number and \(0 \lt \theta \lt \frac{\pi}{2}\), then \(\int \frac{\left(\cos ^n \theta-\cos \theta\right)^{1 / n}}{\cos ^{n+1} \theta} \sin \theta d \theta=\)

If the lines \(2 x-3 y=5\) and \(3 x-4 y=7\) are two diameters of a circle of radius 7 , then the equation of the circle is