List of top Mathematics Questions

Let \( S = \{(x, y, z) \in \mathbb{R}^3 \setminus \{(0,0,0)\} : z = -(x + y)\} \). Denote:

\[ S^\perp = \{(p, q, r) \in \mathbb{R}^3 : px + qy + rz = 0 \text{ for all } (x, y, z) \in S\}. \]

Then which one of the following options is correct?

Among the following four statements about countability and uncountability of different sets, which is the correct statement?

Let \( f : [0, \infty) \to [0, \infty) \) be a differentiable function with \( f(x)>0 \) for all \( x>0 \), and \( f(0) = 0 \). Further, \( f \) satisfies} \[ (f(x))^2 = \int_{0}^{x} \left( (f(t))^2 + f(t) \right) \, dt, \, x>0. \] {Then which one of the following options is correct?

The following empirical relationship describes how the number of trees \( N(t) \) in a patch changes over time \( t \): \[ N(t) = -2t^2 + 12t + 24 \] where \( t = 0 \) is when the number of trees were first counted. Given this relationship, the maximum number of trees that occur in the patch is

In a lake with a large population of fish, there are 4 blue fish for every red fish. Every fish has an equal probability of being caught. If you dip a net into this lake and pick up 4 individuals at random, the probability that you will get 2 fish of each colour is:

There are 9 different rock lizard species, each with a unique colour. Lizards of 2 different species sit on a rock at any given time. The number of possible colour combinations of rock lizards seen together on a rock is:

A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is:

Consider a probability distribution given by the density function \( P(x) \).

\[ P(x) = \begin{cases} Cx^2, & \text{for } 1 \leq x \leq 4, \\ 0, & \text{for } x<1 \text{ or } x>4. \end{cases} \]
The probability that \( x \) lies between 2 and 3, i.e., \( P(2 \leq x \leq 3) \), is ______. (rounded off to three decimal places)

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ___________. (rounded off to three decimal places)

A box contains 5 coins: 4 regular coins and 1 fake coin.

- When a regular coin is tossed, the probability \( P(\text{head}) = 0.5 \).
- For a fake coin, \( P(\text{head}) = 1 \).

You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is ______.

If the direction cosines of a line are \( \sqrt{3}k, \sqrt{3}k, \sqrt{3}k \), then the value of \( k \) is:

Assertion (A): Zeroes of a polynomial \(p(x) = x^2 − 2x − 3\) are -1 and 3.
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).

If one of the zeroes of the quadratic polynomial \((\alpha - 1)x^2 + \alpha x + 1\) is \(-3\), then the value of \(\alpha\) is:

The graph of a polynomial intersects the y-axis at one point and the x-axis at two points. The number of zeroes of this polynomial are :

Assertion (A): Zeroes of a polynomial \(p(x) = x^2 − 2x − 3\) are -1 and 3.
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).

If one of the zeroes of the quadratic polynomial \((\alpha - 1)x^2 + \alpha x + 1\) is \(-3\), then the value of \(\alpha\) is:

For what value of \(k\), the product of zeroes of the polynomial \(kx^2 - 4x - 7\) is 2?