List of practice Questions

The product of perpendiculars from the two foci of the ellipse $$ \frac{x^2}{9} + \frac{y^2}{25} = 1 $$ on the tangent at any point on the ellipse is: 

The value of \( c \) such that the straight line joining the points \[ (0,3) \quad \text{and} \quad (5,-2) \] is tangent to the curve \[ y = \frac{c}{x+1} \] is:
The descending order of magnitude of the eccentricities of the following hyperbolas is: A. A hyperbola whose distance between foci is three times the distance between its directrices. B. Hyperbola in which the transverse axis is twice the conjugate axis. C. Hyperbola with asymptotes \( x + y + 1 = 0, x - y + 3 = 0 \).
If the plane \[ x - y + z + 4 = 0 \] divides the line joining the points \[ P(2,3,-1) \quad \text{and} \quad Q(1,4,-2) \] in the ratio \( l:m \), then \( l + m \) is:
If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then \( (\alpha, \beta) \) is:
The equation of the plane which bisects the angle between the planes $$ 3x - 6y + 2z + 5 = 0 \quad \text{and} \quad 4x - 12y + 3z - 3 = 0 \quad \text{which contains the origin is:} $$ 
Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}. \]
Evaluate the limit: \[ \lim_{x \to 1} \frac{x + x^2 + x^3 + \dots + x^n - n}{x - 1}. \]
If the function \[ f(x) = \frac{\sqrt{1+x} - 1}{x} \] is continuous at \( x = 0 \), then \( f(0) \) is:
If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:
If \[ \frac{d}{dx} \left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) = ax + b, \] then \( (a,b) \) is:

If $$ y = \sin^{-1} x, $$ then $$ (1 - x^2)y_2 - xy_1 = 0. $$ 

If the percentage error in the radius of a circle is 3, then the percentage error in its area is:
The equation of the tangent to the curve \[ y = x^3 - 2x + 7 \] at the point \( (1,6) \) is:
The distance \( s \) traveled by a particle in time \( t \) is given by: $$ s = 4t^2 + 2t + 3. $$ The velocity of the particle when \( t = 3 \) seconds is: 
If \[ a^2 x^4 + b^2 y^4 = c^6, \] then the maximum value of \( xy \) is:
Evaluate the integral \[ \int \frac{\sin^6 x}{\cos^8 x} \, dx. \]
Evaluate the integral \[ \int \frac{x^5}{x^2 + 1} dx. \]
Evaluate the integral \[ \int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx. \]
When an electron placed in a uniform magnetic field is accelerated from rest through a potential difference \( V_1 \), it experiences a force \( F \). If the potential difference is changed to \( V_2 \), the force experienced by the electron in the same magnetic field is \( 2F \), then the ratio of potential differences \( \frac{V_2}{V_1} \) is:
Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]
Evaluate the integral \[ \int e^x (x+1)^2 dx. \]
Evaluate the integral $$ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. $$
Evaluate the integral \[ I = \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx \]
Evaluate the integral \[ I = \int_{-1}^{1} \left( \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right) \, dx \]