List of practice Questions

The angle between the tangents drawn from a point \( (-3, 2) \) to the ellipse \( 4x^2 + 9y^2 - 36 = 0 \) is:
The number of positive integers less than 10000 which contain the digit 5 at least once is
Let $ A(4, 3), B(2, 5) $ be two points. If $ P $ is a variable point on the same side of the origin as that of line $ AB $ and at most 5 units from the midpoint of $ AB $, then the locus of $ P $ is:
If $ x - y - 3 = 0 $ is a normal drawn through the point $ (5, 2) $ to the parabola $ y^2 = 4x $, then the slope of the other normal that can be drawn through the same point to the parabola is?
If \(x\) is so large that terms containing \(x^{-3}\), \(x^{-4}\), \(x^{-5}\), \ldots can be neglected, then the approximate value of \[ \left(\frac{3x - 5}{4x^2 + 3}\right)^{-4/5} \] is:
If the function $y = g(x)$ represents the slopes of the tangents drawn to the curve $y = 3x^3 - 5x^2 - 12x^2 + 18x - 3$ strictly increasing then the domain of $g(x)$ is
Identify the correct option from the following:
If the circle \(S_1 = x^2 + y^2 + 2gx + 4y + 1 = 0\) bisects the circumference of circle \(x^2 + y^2 - 2x - 3 = 0\), then the radius of circle \(S_1\) is
If the least positive integer \( n \) satisfying the equation \(\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n = -1\) is \( p \) and the least positive integer \( m \) satisfying the equation \(\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^m = \text{cis}\left(\frac{2\pi}{3}\right)\) is \( q \), then \(\sqrt{p^2 + q^2}\) is equal to:
A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and B be the event of getting an even number when the die is thrown second time. Then \(P(A/B) =\)
If the pole of the line \(x + 2by - 5 = 0\) with respect to the circle \(S = x^2 + y^2 - 4x - 6y + 4 = 0\) lies on the line \(x + by + 1 = 0\), then the polar of the point \((b, -b)\) with respect to the circle \(S = 0\) is

The mean deviation about the mean for the following data is 
 

Class Interval0--22--44--66--88--10
Frequency13412
If the values \( x = \alpha, y = \beta, z = \gamma \) satisfy all the 3 equations \(x+2y+3z=4\), \(3x+y+z=3\) and \(x+3y+3z=2\), then \(3\alpha + \gamma = \)
If \(2\sin x - \cos 2x = 1\), then \( (3 - 2\sin^2x) = \)
If \( \left(\frac{\sin 3\theta}{\sin \theta}\right)^2 - \left(\frac{\cos 3\theta}{\cos \theta}\right)^2 = a \cos b\theta \), then \( a : b = \)
\([x]\) denotes the greatest integer less than or equal to x. If \(\{x\}=x-[x]\) and \( \lim_{x\to 0} \frac{\sin^{-1}(x+[x])}{2-\{x\}} = \theta \), then \( \sin\theta + \cos\theta = \)
If \( y = \log(\sec(\tan^{-1}x)) \) for \( x>0 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:
If the normal chord drawn at the point \(\left(\frac{15}{2\sqrt{2}}, \frac{15}{2\sqrt{2}}\right)\) to the parabola \(y^2 = 15x\) subtends an angle \(\theta\) at the vertex of the parabola, then \(\sin \frac{\theta}{3} + \cos \frac{2\theta}{3} - \sec \frac{4\theta}{3} =\) ?
Let \( g(x) = 1 + x - \lfloor x \rfloor \) and \[ f(x) = \begin{cases} -1, & x<0\\ 0, & x = 0 \\ 1, & x>0 \end{cases} \] where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Then for all \( x \), \( f(g(x)) = \)
The set of all real values of \( c \) so that the angle between the vectors
\( \vec{a} = c\hat{i} - 6\hat{j} + 3\hat{k} \) and \( \vec{b} = x\hat{i} + 2\hat{j} + 2c\hat{k} \) is an obtuse angle for all real \( x \), is:
If \( \int \frac{x}{x \tan x + 1} \, dx = \log f(x) + k \), then \( f\left(\frac{\pi}{4}\right) = \)
Let \(z_1 = 3 + 4i\) and \(z_2 = 1 - 2i\). Then the argument of \(\frac{z_1}{z_2}\) is:
If the normal drawn at the point $$ P \left(\frac{\pi}{4}\right) $$ on the ellipse $$ x^2 + 4y^2 - 4 = 0 $$ meets the ellipse again at $ Q(\alpha, \beta) $, then find $ \alpha $.
If \( A \) and \( B \) are both \( 3 \times 3 \) matrices, then which of the following statements are true? \[ \begin{cases}
If a discrete random variable \(X\) has the probability distribution \[ P(X = x) = k \frac{2^{2x+1}}{(2x+1)!}, \quad x=0,1,2,\ldots, \] then find \(k\).
If $\int \left[ 3t^2 \sin^{-1} \left( \frac{1}{-t \cos t} \right) \right] \, dt = f(t) \left( \sin^{-1} \left( \frac{1}{t} \right) \right) + c$, then $f(2) =$
Identify the correct option from the following: