List of practice Questions

If \( y = \log \left[ \tan \left( \sqrt\frac{2x - 1}{2x + 1} \right) \right] \) for \( x>0 \), then find \[ \left( \frac{dy}{dx} \right)_{x = 1}. \]
If \( y = \cos^{-1}\left( \frac{6x^2 - 2x^2 - 4}{2x^2 - 6x + 5} \right) \), then find \( \frac{dy}{dx} \).
The approximate value of \( \sec 59^\circ \) obtained by taking \( 1^\circ = 0.0174 \) and \( \sqrt{3} = 1.732 \) is:
The equation of the normal drawn to the curve \( y^3 = 4x^5 \) at the point \( (4,16) \) is:
A point \( P \) is moving on the curve \( x^3 y^4 = 27 \). The x-coordinate of \( P \) is decreasing at the rate of 8 units per second. When the point \( P \) is at \( (2, 2) \), the y-coordinate of \( P \) is:

If the function \( f(x) = x^3 + ax^2 + bx + 40 \) satisfies the conditions of Rolle's theorem on the interval \( [-5, 4] \) and \( -5, 4 \) are two roots of the equation \( f(x) = 0 \), then one of the values of \( c \) as stated in that theorem is:

If \( x \) and \( y \) are two positive integers such that \( x + y = 24 \) and \( x^3 y^5 \) is maximum, then \( x^2 + y^2 \) is:
Evaluate the integral: \[ \int 4\cos^2 x - 5\sin^2 x \cos x \, dx. \]
Evaluate the integral: \[ \int \frac{4\tan^4 x + 3 \tan^2 x - 1}{\tan^2 x + 4} \, dx. \]
Evaluate the integral: \[ \int \left( \sin^3 x + \cos^2 x \right)^2 \, dx \]
Evaluate the integral: \[ I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\sin^4(4x)}{1 + e^{4x}} \, dx \]
The area of the region enclosed by the curves \( y^2 = 4(x+1) \) and \( y^2 = 5(x-4) \) is:
If \( (h, k) \) is the internal center of similitude of the circles \( x^2 + y^2 + 2x - 6y + 1 = 0 \) and \( x^2 + y^2 - 4x + 2y + 4 = 0 \), then find \( 4h \).
If the focal chord of the parabola \( x^2 = 12y \) drawn through the point \( (3, 0) \) intersects the parabola at the points P and Q, then the sum of the reciprocals of the abscissae of the points P and Q is:
If the normal drawn at the point \( P(9, 9) \) on the parabola \( y^2 = 9x \) meets the parabola again at \( Q(a, b) \), then \( 2a + b = \text{?} \)
The length of the latus rectum of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) \( (a>b) \) is \( \frac{8}{3} \). If the distance from the center of the ellipse to its focus is \( \sqrt{5} \), then \( \sqrt{a^2 + 6ab + b^2} = \text{?} \)
If \(\theta\) is the angle between the vectors \(4\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \( \mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\), then \(\sin 2\theta\) is:
If the probability that a randomly selected student from a college is good at mathematics is 0.6, then the probability of having exactly two students who are good at mathematics in a group of 8 students standing in front of the college is:
(a,b) is the point of concurrency of the lines \(x-3y+3=0\), \(Kx+y+k=0\) and \(2x+y-8=0\). If the perpendicular distance from the origin to the line \(L: ax-by+2k=0\) is \(p\), then the perpendicular distance from the point \((2,3)\) to \(L=0\) is:
A circle \( S \equiv x^2 + y^2 + 2gx + 2fy + 6 = 0 \) cuts another circle \[ x^2 + y^2 - 6x - 6y - 6 = 0 \] orthogonally. If the angle between the circles \( S = 0 \) and \[ x^2 + y^2 + 6x + 6y + 2 = 0 \] is 60°, then the radius of the circle \( S = 0 \) is:
If a circle passing through the point \((1,1)\) cuts the circles \(x^2 + y^2 + 4x - 5 = 0\) and \(x^2 + y^2 - 4x + 3 = 0\) orthogonally, then the center of that circle is:
If \( \frac{2x^3 + 1}{2x^2 - x - 6} = ax + b + \frac{A}{px - 2} + \frac{B}{2x + q} \), then \( 51apB = \):
If \( \tan A + \tan B + \cot A + \cot B = \tan A \tan B - \cot A \cot B \) and \( 0^\circ<A + B<270^\circ \), then \( A + B = \):
The equation that is satisfied by the general solution of the equation \( 4 - 3 \cos \theta = 5 \sin \theta \cos \theta \) is:
If \( \sin^{-1}(4x) - \cos^{-1}(3x) = \frac{\pi}{6} \), then \( x = \):