List of practice Questions

The eccentricity of the ellipse \( 9x^2+16y^2=144 \) is

The equation of the line which is a normal to the curve \(3x^2-xy+y^2=3\) at \((1,1)\) is

The length of the latus rectum of the parabola \( y^2-4x-4y+8=0 \) is

The locus of the point of intersection on the line \( \sqrt{3}x - y - 4\sqrt{3}k = 0 \) and \( \sqrt{3}kx + ky - 4\sqrt{3} = 0 \) for different real values of k is a hyperbola H. If e is the eccentricity of H then \(4e^2 = \)

If \( ax^2 - 34xy - 5y^2 + 2x + 26y - 5 = 0 \) represents a pair of straight lines, then the value of a is

Suppose ABOC is a rhombus in the first quadrant with O being the origin. If the vertices B and C of \(\triangle ABC\) lie respectively on \(y=\frac{x}{\sqrt{3}}\) and \(y=0\), and the side BC passes through \((\frac{2}{3}, \frac{2}{3})\), then the midpoint of BC is

The equation of circle with centre (2, -3) and touching x-axis is

The equations of the lines perpendicular to \(x^2 - 5xy + 4y^2 = 0\) and passing through (2, 1) is

The ratio of the areas of the greatest and the smallest circles touching \((x \pm 1)^2 + (y \pm 1)^2 = 1\) is (The equation \((x \pm 1)^2 + (y \pm 1)^2 = 1\) represents four circles with radius 1, centered at (1,1), (1,-1), (-1,1), (-1,-1).)

The straight line touching the circle \(x^2+y^2-2x-3=0\) and remaining normal to the circle \(x^2+y^2-4y-6=0\) is

A point P on a line is at a distance of 4 units from the origin (0, 0). If the line makes 60\(^\circ\) with the negative direction of the x-axis, then P is

Suppose the hypotenuse and its opposite vertex of an isosceles right angled triangle are \(3x+4y-4=0\) and \((2,2)\) respectively. Then, which of the following is another side of the triangle?

A bag contains 3 white, 2 blue and 5 red balls. One ball is drawn at random from this bag. Then, the probability that the ball drawn is not red is

A person P speaks truth in 75% cases and another person R in 80% cases. Then the probability that they likely to contradict each other in narrating the same event is

A point P(x, y) is such that the sum of squares of its distances from (a, 0) and (--a, 0) is \(2b^2\). The equation representing the locus of P is

During the winter months, in a certain village in Scotland, the probability of a day having severe fog is 0.6. The probability that in a given week there will be exactly two days with severe fog is

From a well shuffled pack of 52 cards, two cards are drawn at random. Then, the probability of both the cards being kings is

If X is a Poisson random variate with mean 3, then \(P(|X-3|<2)\) =

Person A can solve 90% of the problems given in the book and Person B can solve 70%. Then the probability that atleast one of them will solve the problem selected at random from the book is

If \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = 2\hat{i} + 3\hat{j} + \hat{k} \), \( \vec{c} = 8\hat{i} + 13\hat{j} + 9\hat{k} \) and \( x\vec{a} + y\vec{b} + z\vec{c} = \vec{0} \), then \( \frac{xy}{z^2} = \)

Let \( \vec{a} = \hat{i} + x\hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \) and \( |\vec{a}+\vec{b}| = |\vec{a}| + |\vec{b}| \) then

The set of real values of \( \lambda \) for which the vectors \( \lambda\hat{i} - 3\hat{j} + 5\hat{k} \) and \( 2\lambda\hat{i} - \lambda\hat{j} + \hat{k} \) are perpendicular to each other is

The standard deviation of first 10 multiples of 4 is

The vector \( \vec{x} \) is perpendicular to the vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 18\hat{i} - 22\hat{j} - 5\hat{k} \) and makes an obtuse angle with \( \hat{j} \). If \( |\vec{x}|=14 \), then \( \vec{x} = \)

\( \vec{a}, \vec{b} \) are two vectors such that \( |\vec{a}|=\sqrt{3}, |\vec{b}|=\sqrt{2} \). If \( \vec{x} \) is a unit vector satisfying \( \vec{x} \times \vec{a} = \vec{b} \) then \( \vec{x} = \)