List of practice Questions

If \( \alpha \) and \( \beta \) are the roots of \( x^{2} - 2x + 4 = 0 \), then the value of \( \alpha^{6} + \beta^{6} \) is

If \( n \) is an integer which leaves remainder one when divided by three, then \( (1+\sqrt{3}i)^{n} + (1-\sqrt{3}i)^{n} \) equals

The period of \( \sin(4x + \cos^{4} x) \) is

\( \frac{\cos x}{\cos(x - 2y)} = \lambda \Rightarrow \tan(x - y)\tan y \) is equal to

\( \cos A \cos 2A \cos 4A \cdots \cos 2^{\,n-1}A \) equals

\( \cos^{-1}\left(\frac{-1}{2}\right) - 2\sin^{-1}\left(\frac{1}{2}\right) + 3\cos^{-1}\left(\frac{-1}{\sqrt{2}}\right) - 4\tan^{-1}(-1) \) equals

\( \sinh^{-1} 2 + \sinh^{-1} 3 = x \Rightarrow \cosh x \) is equal to

In a \( \Delta ABC \), \( \frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4b^{2}c^{2}} \) equals

The solution of the equation \( 4^{x} - 3 \cdot 2^{x+2} + 32 = 0 \) is:

The value of \( \sin^{-1}(\sin 10) \) is:

The center of the ellipse \( \frac{(x+1)^{2}}{9} + \frac{(y-2)^{2}}{4} = 1 \) is:

The slope of the tangent to the curve \( y = x^{2 - x} \) at the point where \( x = 2 \) is:

The value of \( \int e^{x(\sin x + \cos x)} \, dx \) is:

The value of \( \vec{i} \cdot (\vec{j} \times \vec{k}) + \vec{j} \cdot (\vec{i} \times \vec{k}) + \vec{k} \cdot (\vec{i} \times \vec{j}) \) is:

The point \( (3, -4) \) lies on both the circles \( x^{2} + y^{2} - 2x + 8y + 13 = 0 \) and \( x^{2} + y^{2} - 4x + 6y + 11 = 0 \). Then, the angle between the circles is

The equation of the circle which passes through the origin and cuts orthogonally each of the circles \( x^{2} + y^{2} - 6x + 8 = 0 \) and \( x^{2} + y^{2} - 2x - 2y = 7 \) is

The number of normals drawn to the parabola \( y^{2} = 4x \) from the point \( (1, 0) \) is

If the circle \( x^{2} + y^{2} = a^{2} \) intersects the hyperbola \( xy = c^{2} \) in four points \( (x_i, y_i) \), for \( i = 1, 2, 3, 4 \), then \( y_{1} + y_{2} + y_{3} + y_{4} \) equals

The mid point of the chord \( 4x - 3y = 5 \) of the hyperbola \( 2x^{2} - 3y^{2} = 12 \) is

The eccentricity of the conic \( \frac{5}{r} = 2 + 3\cos\theta + 4\sin\theta \) is

If a line in the space makes angle \( \alpha, \beta, \gamma \) with the coordinate axes, then \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^{2}\alpha + \sin^{2}\beta + \sin^{2}\gamma \) equals

\( \lim_{x \to \infty} \left(\frac{x+5}{x+2}\right)^{x+3} \) equals

If \( f: \mathbb{R} \to \mathbb{R} \) is defined by \[ f(x) = \begin{cases} \dfrac{2 \sin x - \sin 2x}{2x \cos x}, & \text{if } x \ne 0 \\ a, & \text{if } x = 0 \end{cases} \] then the value of \( a \) so that \( f \) is continuous at \( 0 \) is

\( x = \frac{1 - \sqrt{y}}{1 + \sqrt{y}} \implies \frac{dy}{dx} \) is equal to

\( x = \cos^{-1}\left(\frac{1}{\sqrt{1+t^{2}}}\right), \; y = \sin^{-1}\left(\frac{t}{\sqrt{1+t^{2}}}\right) \implies \frac{dy}{dx} \) is equal to