List of practice Questions

A die is thrown. Find the probability of getting:
(A) A number greater then 6 and
(B) A number less than 3.

What does ICDS stand for?

Let \(y_c:\R \rightarrow(0,\infin)\) be the solution of the Bernoulli’s equation
\(\frac{dy}{dx}-y+y^3=0,\ \ \ \ \ \ \ y(0)=c \gt 0.\)
Then, for every 𝑐 > 0, which one of the following is true ?

Let A be a $n \times n$ matrix such that $| A |=2$ If the determinant of the matrix$\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 A ^{-1}\right)\right) \cdot$ is $2^{84}$, then $n$ is equal to ____

A bullet from a gun is fired on a rectangular wooden block with velocity u. When the bullet travels 24 cm through the block along its length horizontally, velocity of bullet becomes \(\frac{u}{3}\). Then it further penetrates into the block in the same direction before coming to rest exactly at the other end of the block. The total length of the block is:

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by \(\vec{E}\)=30(2𝑥̂ + 𝑦̂)sin [2𝜋 (5×10¹⁴𝑡 −\(\frac{10^7}{3}z\))] Vm⁻¹. Which of the following option(s) is(are) correct? [Given: The speed of light in vacuum, 𝑐 = 3 × 10⁸ ms⁻¹]

Let the circle \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \) cut the circles \( x^2 + y^2 - 2x + 2y - 2 = 0 \) and \( x^2 + y^2 + 4x - 6y + 9 = 0 \) orthogonally. If the centre of the circle \( S = 0 \) lies on the line \( 2x + 3y - 2 = 0 \), then \( 2g + f = \)

If the line passing through the points \( (5,1,a) \) and \( (3,b,1) \) crosses the YZ plane at the point \( \left( 0, \frac{17}{2}, \frac{-13}{2} \right) \), then \( a + b = \):

In \( \triangle ABC \), the coordinates of the vertices are \( A(1,8,4) \), \( B(0,-11,4) \), and \( C(2,-3,1) \). If \( D \) is the foot of the perpendicular drawn from \( A \) to \( BC \), then the coordinates of \( D \) are:

The distance between two parallel planes \( ax + by + cz + d_1 = 0 \) and \( ax + by + cz + d_2 = 0 \) is given by \( \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \). If the plane \( 2x - y + 2z + 3 = 0 \) has the distances \( \frac{1}{3} \) units from the planes \( 4x - 2y + 4z + \lambda = 0 \) and \( 2x - y + 2z + \mu = 0 \) respectively, then the maximum value of \( \lambda + \mu \) is:

Let \( S \) be the circumcircle of the triangle formed by the line \( x - 2y - 4 = 0 \) with the coordinate axes. If \( P(-2, -4) \) is a point in the plane of the circle \( S \) and \( Q \) is a point on \( S \) such that the distance between \( P \) and \( Q \) is the least, then \( PQ = \)

If the coordinates of the point of contact of the circles \( x^2 + y^2 - 4x + 8y + 4 = 0 \) and \( x^2 + y^2 + 2x = 0 \) is \( (a, b) \), then \( a + 2b \) is:

If the chord of contact of the point \( P(h, k) \) with respect to the circle \( x^2 + y^2 - 4x - 4y + 8 = 0 \) meets the circle in two distinct points and it also makes an angle \( 45^\circ \) with the positive X-axis in the positive direction, then \( (h, k) \) cannot be:

The equation of the pair of tangents drawn from the point \( (1, 1) \) to the circle \( x^2 + y^2 + 2x + 2y + 1 = 0 \) is:

If eight coins are tossed simultaneously, then the probability of getting at least six heads is:

The locus of a point which is at a distance of 2 units from the line \( 2x - 3y + 4 = 0 \) and at a distance of \( \sqrt{13} \) units from a point \( (5,0) \), is:

The combined equation of the lines passing through the point \( (3, 4) \) and each making an angle of 45° with the line \( x + y + 1 = 0 \) is:

The equal sides of an isosceles triangle are given by equations \( 7x - y + 3 = 0 \) and \( x + y - 3 = 0 \). If the slope \( m \) of the third side is an integer, then \( m = \):

In \( \triangle ABC \), the coordinates of the vertex \( A \) are \( (-3, 1) \). If the equation of the median through \( B \) is \( 2x + y - 3 = 0 \) and the equation of the bisector of angle \( C \) is \( 7x - 4y - 1 = 0 \), then the equation of the side \( BC \) is:

If the lines given by \( (x^2 + y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 \) are perpendicular to each other, then \( \sin^2 \alpha + \tan^2 \alpha = \):

If the area of the triangle formed by the lines \( y = x + c \) and \( 2x^2 + 5xy + 3y^2 = 0 \) is \( \frac{1}{20} \) sq. units, then \( c = \)

Let \( \overline{OA} = 2\overline{i} - 3\overline{j} + \overline{k} \), \( \overline{OB} = \overline{i} - 4\overline{j} - 3\overline{k} \), and \( \overline{OC} = -3\overline{i} + \overline{j} + 2\overline{k} \) be the position vectors of three points A, B, C respectively. If G is the centroid of triangle ABC, then find: \[ BC^2 + CA^2 + AB^2 + 9(OG)^2 \]

The variance of 20 observations is 5. If each one of the observations is multiplied by 2, then the variance of the resulting observations is:

In a game, a pair of dice is rolled 24 times. If a person wins the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is:

Bag \( B_1 \) contains 4 white and 2 black balls. Bag \( B_2 \) contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is: