List of practice Questions

Let \( n \in (0, \infty) \). If for all the curves \( y = x^n \log x \) for distinct values of \( n \), we have \( y = x - 1 \) as the tangent at a fixed point \((\alpha, \beta)\), then \(\alpha + \beta = \):
If \(\int \frac{dx}{1 + \sin x} = \tan \left( \frac{x}{2} - \theta \right) + C\), then \(\theta =\):
Let \( S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots + n \text{ terms} -\frac{1}{3}<x<\frac{1}{3} \). If \( f(x) = S_n \), then \( f(x) \) is discontinuous at the point \( x = \):
The locus of the point on the curve \( y = \sin x \) where the tangent drawn at that point always passes through the point \( (0, \pi) \) is:
\( f(x) \) and \( g(x) \) are differentiable functions such that \( \frac{f(x)}{g(x)} = \) a non-zero constant. If \( \frac{f'(x)}{g'(x)} = \alpha(x) \) and \( \left( \frac{f(x)}{g(x)} \right)' = \beta(x) \), then \( \frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = \):
If all the normals drawn to the curve \( y = \frac{1 + 3x^2}{3 + x^2} \) at the points of intersection of \( y = \frac{1 + 3x^2}{3 + x^2} \) and \( y = 1 \) pass through the point \( (\alpha, \beta) \), then \( 3\alpha + 2\beta = \):
If A and B are the points of intersection of the circles \( x^2 + y^2 - 4x + 6y - 3 = 0 \) and \( x^2 + y^2 + 2x - 2y - 2 = 0 \), then the distance between A and B is:
A parabola having its axis parallel to the Y-axis passes through the points \(\left(0, \frac{2}{5}\right)\), \((4, -2)\), and \(\left(1, \frac{8}{5}\right)\). Then the point that lies on this parabola is:
Let the eccentricity of the ellipse \(2x^2 + ay^2 - 8x - 2ay + (8 - a) = 0\) be \(\frac{1}{\sqrt{3}}\). If the major axis of this ellipse is parallel to the Y-axis, then the equation of the tangent to this ellipse with slope 1 is:
The equation of the pair of asymptotes of the hyperbola \(\frac{(x - 3)^2}{3} - \frac{(y - 2)^2}{2} = 1\) is:
Let \( S = 0 \) be the circle passing through the points \((2, 0)\), \((1, -2)\), and \((-1, 1)\). Then the point \((1, 2)\):
If a cubical die is thrown, then the mean and variance of the random variable \( X \), giving the number on the face that shows up, are respectively:
If the image of the point \( (3, 8) \) in the line \( x + 3y = 7 \) is \( (\alpha, \beta) \), then \( \alpha + \beta = \)
The square of the distance from the origin to the point of intersection of the pair of lines \( ax^2 + 2hxy - ay^2 + 2gx + 2fy + c = 0 \) is:
If the vectors \( \vec{a} = 2\vec{i} + 3\vec{j} - \vec{k} \), \( \vec{b} = 4\vec{i} - \vec{j} + 3\vec{k} \), and \( \vec{c} = p\vec{i} + \vec{j} - \vec{k} \) are coplanar, then: \[ |\vec{a} \times \vec{c}| = ? \]
Students of two sections A and B of a class show the following results in a test conducted for 100 marks. Then, the comparison of variability between Section A and Section B is:
Students of two sections A and B of
If \( A \) and \( B \) are any two events of a random experiment and \( P(B) \neq 1 \), then: \[ P(A|B^C) = ? \]
A bag contains 10 similar balls, of which 4 are blue and 6 are red. Three balls are taken at random from the bag one after the other without replacement. The probability that all the three balls drawn are red is:
It is observed that there will be 25 blood specimens of normal persons, if 100 blood samples are tested. If 10 specimens are sent to a laboratory for testing, then the probability of having at least two specimens of normal persons is:
In \( \triangle ABC \), if \( A, B, C \) are in arithmetic progression, then \( \frac{c}{a} \sin 2A + \frac{a}{c} \sin 2C = \):
Let \( x \) be a real number and \( -2<x<2 \). When \( \frac{x+1}{(x+3)(x-2)} \) is expanded in powers of \( x \), then the coefficient of \( x^3 \) is:
If \( \vec{a} = t\vec{b} \) where \( t<0 \) is a scalar, then:
If \( \vec{a} \) and \( \vec{b} \) are two unit vectors with \( (\vec{a}, \vec{b}) = 0 \) and \( |\vec{a} - \vec{b}| = 1 \), then: \[ 2|\vec{a} + \vec{b}| \cos\theta = \frac{2}{|\vec{a} + \vec{b}| \cos\theta} \]
Evaluate: \( (1 + \cos\frac{\pi}{8})(1 + \cos\frac{2\pi}{8})(1 + \cos\frac{3\pi}{8}) \ldots (1 + \cos\frac{7\pi}{8}) = \)
If \(3\sin^4 x + 2\cos^4 x = \frac{6}{5}\) and \(x\) is an acute angle, then \(\tan 2x =\)