List of practice Questions

A value of \(c\) according to the Lagrange's mean value theorem for \(f(x) = (x - 1)(x - 2)(x - 3)\) in \([0,4]\) is:
The acute angle between the curves \(x^2 + y^2 = x + y\) and \(x^2 + y^2 = 2y\) is:
The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:
If \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) respectively. If \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), then \( k_1 + k_2 = \)
In the interval \([0, 3]\), the function \(f(x) = |x - 1| + |x - 2|\) is:
If \(y = t^2 + t^3\) and \(x = t - t^4\), then \(\frac{d^2y}{dx^2}\) at \(t = 1\) is:
If \(f(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 2ax + bx, & \text{if } x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)
Determine the values of \(a\) and \(b\) for which the function \(f(x)\) defined as: \[ f(x) = \begin{cases} 1 + |\sin x|^{\left(\frac{a}{|\sin x|}\right)} & \text{if } \frac{-\pi}{6} < x < 0, \\ b & \text{if } x = 0, \\ e^{\left(\frac{\tan 2x}{\tan 3x}\right)} & \text{if } 0 < x < \frac{\pi}{6} \end{cases} \] is continuous at \(x = 0\).
Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x} - a - \log(1+x)}{\sin x} = 0, { then find } a. \]
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{4\sqrt{2} - (\cos x + \sin x)^5}{1 - \sin 2x} = \]
The direction cosines of the line of intersection of the planes \( x + 2y + z - 4 = 0 \) and \( 2x - y + z - 3 = 0 \) are:
If \( l_1 \) and \( l_2 \) are the lengths of the perpendiculars drawn from a point on the hyperbola \( 5x^2 - 4y^2 - 20 = 0 \) to its asymptotes, then \( \frac{l_1^2 l_2^2}{100} = \)
If a directrix of a hyperbola centered at the origin and passing through the point \( (4, -2\sqrt{3}) \) is \( \sqrt{5}x = 4 \) and \( e \) is its eccentricity, then \( e^2 = \)
If the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) having \((1,1)\) as its middle point is \( x + \alpha y = \beta \), then:
Equation of a tangent line of the parabola \(y^2 = 8x\), which passes through the point \((1, 3)\) is:
The radical center of the circles \(x^2 + y^2 + 2x + 3y + 1 = 0\), \(x^2 + y^2 - x + y + 3 = 0\), and \(x^2 + y^2 - 3x + 2y + 5 = 0\) is:
If the locus of the mid points of the chords of the circle \(x^2 + y^2 = 25\) that subtend a right angle at the origin is given by \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\), then \(|a| =\)
The circle \(x^2 + y^2 - 8x - 12y + \alpha = 0\) lies in the first quadrant without touching the coordinate axes. If \((6, 6)\) is an interior point to the circle, then the range of \(\alpha\) is:
The equation of the straight line passing through the point of intersection of the lines represented by \(x^2 + 4xy + 3y^2 - 4x - 10y + 3 = 0\) and the point \((2,2)\) is:
If the line \(2x - 3y + 5 = 0\) is the perpendicular bisector of the line segment joining \(1, -2\) and \((a, b)\), then \(a + b =\)
The circumcentre of the triangle formed by the lines \(x + y + 2 = 0\), \(2x + y + 8 = 0\) and \(x - y - 2 = 0\) is
The probability that a man failing to hit a target is \(\frac{1}{3}\). If he fires 4 times, then the probability that he hits the target at least thrice is
For a binomial variate \(X \sim B(n, p)\), the difference between the mean and variance is 1 and the difference between their squares is 11. If the probability of \(P(X = 2) = m\left(\frac{5}{6}\right)^n\) and \(n = 36\) then \(m:n\) is
A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is
A box \( P \) contains one white ball, three red balls and two black balls. Another box \( Q \) contains two white balls, three red balls and four black balls. If one ball is drawn at random from each one of the two boxes, then the probability that the balls drawn are of different color is