The minimum value of the function \( f(x) = x^4 - 4x - 5 \), where \( x \in \mathbb{R} \), is:
Let \( f(x) = \frac{x^2 + 40}{7x} \), \( x \neq 0 \), \( x \in [4,5] \). The value of \( c \) in \( [4,5] \) at which \( f'(c) = -\frac{1}{7} \) is equal to:
If \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \), where \( 0<a, b<\frac{\pi}{2} \), then \( a - b \) is:
Let \[ A = \begin{pmatrix} 3 & -2 & 1 \\ -1 & 3 & -1 \end{pmatrix} \] and \[ B = \begin{pmatrix} 1 \\ \alpha \\ -1 \end{pmatrix}. \] If \[ AB = \begin{pmatrix} -2 \\ 6 \end{pmatrix}, \] then the value of \( \alpha \) is equal to:
The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:
The value of the limit \(\lim_{t \to 0} \frac{(5-t)^2 - 25}{t}\) is equal to:
For \(1 \leq x<\infty\), let \(f(x) = \sin^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{1}{x}\right)\). Then \(f'(x) =\)
\[ \int_0^{\frac{\pi}{4}} (\tan^3 x + \tan^5 x) \, dx \]
Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:
The value of the limit \(\lim_{x \to 0} \frac{(2 + \cos 3x) \sin^2 x}{x \tan(2x)}\) is equal to:
\[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & \text{if } x \geq 0 \\ x\left( \frac{\pi}{2} - x \right), & \text{if } x < 0 \end{cases} \]
If \( a = \frac{1 + \tan \theta + \sec \theta}{2 \sec \theta} \) and \( b = \frac{\sin \theta}{1 - \sec \theta + \tan \theta} \), then \( \frac{a}{b} \) is equal to:
Let \(f(x) = a^{3x}\) and \(a^5 = 8\). Then the value of \(f(5)\) is equal to:
The area bounded by the parabola \(y = x^2 + 2\) and the lines \(y = x\), \(x = 1\) and \(x = 2\) (in square units) is:
Two circles \(C_1\) and \(C_2\) have radii 18 and 12 units, respectively. If an arc of length \( \ell \) of \(C_1\) subtends an angle 80° at the centre, then the angle subtended by an arc of same length \( \ell \) of \(C_2\) at the centre is:
The critical points of the function \( f(x) = (x-3)^3(x+2)^2 \) are:
The foci of the ellipse \(\frac{x^2}{49} + \frac{y^2}{24} = 1\) are:
The axis of a parabola is parallel to the y-axis and its vertex is at \((5, 0)\). If it passes through the point \((2, 3)\), then its equation is:
The radius of the circle with centre at \((-4, 0)\) and passing through the point \((2, 8)\) is:
The line \(y = 5x + 7\) is perpendicular to the line joining the points \((2, 12)\) and \((12, k)\). Then the value of \(k\) is equal to: