List of practice Questions

On a line with direction cosines l, m, n, \( A(x_1, y_1, z_1) \) is a fixed point. If \( B=(x_1+4kl, y_1+4km, z_1+4kn) \) and \( C=(x_1+kl, y_1+km, z_1+kn) \) (\(k>0\)) then the ratio in which the point B divides the line segment joining A and C is
If \( \sqrt{x-xy} + \sqrt{y-xy} = 1 \), then \( \frac{dy}{dx} = \)
If the line of intersection of the planes \(2x+3y+z=1\) and \(x+3y+2z=2\) makes an angle \( \alpha \) with the positive x-axis, then \( \cos \alpha = \)
If g is the inverse of the function f(x) and \( g(x) = x + \tan x \) then, \( f'(x) = \)
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} a - \frac{\sin[x-1]}{x-1} & , \text{if } x>1
1 & , \text{if } x = 1
b - \frac{\sin([x-1] - [x-1]^3)}{([x-1]^2)} & , \text{if } x<1 \end{cases} \] where \([t]\) denotes the greatest integer less than or equal to t. If f is continuous at \(x=1\), 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 the distance between the foci of a hyperbola H is 26 and distance between its directrices is \( \frac{50}{13} \), then the eccentricity of the conjugate hyperbola of the hyperbola H is
If the circles \( x^2+y^2-2\lambda x - 2y - 7 = 0 \) and \( 3(x^2+y^2) - 8x + 29y = 0 \) are orthogonal, then \( \lambda = \)
If Q \( (\alpha, \beta, \gamma) \) is the harmonic conjugate of the point P(0,-7,1) with respect to the line segment joining the points (2,-5,3) and (-1,-8,0), then \( \alpha - \beta + \gamma = \)
Let \( A_1 \) be the area of the given ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Let \( A_2 \) be the area of the region bounded by the curve which is the locus of mid point of the line segment joining the focus of the ellipse and a point P on the given ellipse, then \( A_1 : A_2 = \)
If the perpendicular distance from the focus of a parabola \(y^2=4ax\) to its directrix is \( \frac{3}{2} \), then the equation of the normal drawn at \( (4a, -4a) \) is
If the equation of the tangent of the hyperbola \( 5x^2 - 9y^2 - 20x - 18y - 34 = 0 \) which makes an angle \( 45^\circ \) with the positive X-axis in positive direction is \( x+by+c=0 \) then \( b^2+c^2 = \)
A circle passing through the point (1,0) makes an intercept of length 4 units on X-axis and an intercept of length \(2\sqrt{11}\) units on Y-axis. If the centre of the circle lies in the fourth quadrant, then the radius of the circle is
If the equation of the circle lying in the first quadrant, touching both the coordinate axes and the line \( \frac{x}{3} + \frac{y}{4} = 1 \) is \( (x-c)^2+(y-c)^2=c^2 \), then c =
If \( \left(\frac{1}{10}, \frac{-1}{5}\right) \) is the inverse point of a point (-1, 2) with respect to the circle \( x^2+y^2-2x+4y+c=0 \) then c =
If two sides of a triangle are represented by \( 3x^2 - 5xy + 2y^2 = 0 \) and its orthocentre is (2,1), then the equation of the third side is
If the point of contact of the circles \( x^2+y^2-6x-4y+9=0 \) and \( x^2+y^2+2x+2y-7=0 \) is \( (\alpha, \beta) \), then \( 7\beta = \)
If \( ax^2 + 2hxy - 2ay^2 + 3x + 15y - 9 = 0 \) represents a pair of lines intersecting at (1,1), then ah =
A line \(L_1\) passing through the point of intersection of the lines \(x-2y+3=0\) and \(2x-y=0\) is parallel to the Line \(L_2\). If \(L_2\) passes through origin and also through the point of intersection of the lines \(3x-y+2=0\) and \(x-3y-2=0\), then the distance between the lines \(L_1\) and \(L_2\) is
If the lines \(x+y-2=0\), \(3x-4y+1=0\) and \(5x+ky-7=0\) are concurrent at \((\alpha, \beta)\), then equation of the line concurrent with the given lines and perpendicular to \(kx+y-k=0\) is
If A(1,0), B(0,-2), C(2,-1) are three fixed points, then the equation of the locus of a point P such that area of \( \triangle \text{PAB} \) is equal to area of \( \triangle \text{PAC} \) is
The transformed equation of \( 3x^2 - 4xy = r^2 \) when the coordinate axes are rotated about the origin through an angle of \( \tan^{-1}(2) \) in positive direction is
If the probability distribution of a discrete random variable X is given by \( P(X=k) = \frac{2^{-k(3k+1)}{2^c} \), k = 0, 1, 2, ..., \( \infty \) then P(X \( \le \) c) = } (The expression seems to be \( \frac{2^{-k}(3k+1)}{K} \) where K is a constant, or \(2^c\) is part of the constant. Assuming \(2^c\) is the normalization constant \(K\).)
In a binomial distribution, if \(n=4\) and \( P(X=0) = \frac{16{81} \), then \( P(X=4) = \)}
An urn A contains 4 white and 1 black ball; urn B contains 3 white and 2 black balls; urn C contains 2 white and 3 black balls. One ball is transferred randomly from A to B; then one ball is transferred randomly from B to C. Finally, a ball is drawn randomly from C. Find the probability that it is black.