List of top Mathematics Questions

The shadow of a tower on level ground is $30\ \text{m}$ longer when the sun's altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower. (Use $\sqrt{3}=1.732$.)
 

The following table shows the literacy rate (in percent) of 35 cities. Find the mean literacy rate.
\[\begin{array}{|c|c|c|c|c|c|} \hline \text{Literacy rate (in \%)} & 45-55 & 55-65 & 65-75 & 75-85 & 85-95 \\ \hline \text{Number of cities} & 3 & 10 & 11 & 8 & 3 \\ \hline \end{array}\]

If \(\left(\frac{1-i}{1+i}\right)^{n/2} = -1\), where \(i = \sqrt{-1}\), one possible value of n is
If the system of linear equations
x + my + az = 0
2x + ay + mz = 0
ax + 2y - z = 0
with m and a as non-zero constants, admits a non-trivial solution, then which one of the following conditions is correct?
In Cartesian coordinates, consider the functions \(u(x, y) = \frac{1{2}(x^2 - y^2)\) and \(v(x,y) = xy\). If \((r, \theta)\) are the polar coordinates, the Jacobian determinant \(|\frac{\partial(u,v)}{\partial(r,\theta)}|\) is}
Given a function \(f(x, y) = \frac{x}{a}e^y + \frac{y}{b}e^x\), where \(x = at\) and \(y = bt\) (a and b are non-zero constants), the value of \(\frac{df}{dt}\) at \(t = 0\) is
Which one of the following figures represents the vector field \(\mathbf{A} = y\hat{i}\)?
(\(\hat{i}\) is the unit vector along the x-direction)
Two point-particles having masses \(m_1\) and \(m_2\) approach each other in perpendicular directions with speeds \(v_1\) and \(v_2\), respectively, as shown in the figure below. After an elastic collision, they move away from each other in perpendicular directions with speeds \(v'_1\) and \(v'_2\), respectively.
The ratio \(\frac{v'_1}{v_1}\) is
The correct option for \( x \) which satisfies the following equation is \[ \begin{vmatrix} x & 2 & 3 \\ 4 & x & 6 \\ x & 8 & 9 \end{vmatrix} = \begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{vmatrix} \]
Consider 10 balls each having different colors including a blue ball. If 6 balls are selected randomly, the probability of the blue ball being selected is
Maximum value of the function \( f(r) = r^2 e^{-r} \), when \( 0<r<\infty \) is
The ages of Ajay and Vijay differ by 22.5 years. If \(6\frac{1}{2}\) years ago, Vijay was \(3\frac{1}{2}\) times as old as Ajay, find their present ages.
Consider the matrix \( A = \begin{bmatrix} 4 & -1 \\ 12 & -3 \end{bmatrix} \).
The value of the determinant of \( A^5 \) is .......... (rounded off to two decimal places).
You are tossing a fair coin repeatedly until a ‘Head’ appears. The expected number of tosses required for a ‘Head’ to appear is ...........
The value of the derivative of the function \( y = x e^x \) at \( x = 1 \) is ........... (rounded off to two decimal places).
Anubhab faces a multiple-choice question (MCQ) with four alternative choices, where only one is the right choice. There is no negative mark for making a wrong choice. Also, assume that the probability that he knows the answer is 0.50. The probability of making the correct choice is 0.25, if he does not know the answer. He got full marks in this question.
The probability that he knew the answer is ......... (rounded off to one decimal place).
A fair coin is tossed 3 times in succession. The probability of the event that ‘both first and second toss result in head’ is (rounded off to two decimal places).
Two cards are drawn from a full pack of 52 cards at random. The probability of getting a heart and a diamond is (rounded off to two decimal places).
The correlation coefficient between \(x\) and \(y\) using the following information is (rounded off to two decimal places). \[ \sum_{i=1}^{100} x_i = 280, \quad \sum_{i=1}^{100} y_i = 60, \quad \sum_{i=1}^{100} x_i^2 = 2384, \quad \sum_{i=1}^{100} y_i^2 = 117, \quad \sum_{i=1}^{100} x_i y_i = 438 \]
Which of the following is/are NOT TRUE?
In the context where sample mean of the dependent variable \( Y \) lies closer to the observed \( Y_1 \) than its least square predictor \( \hat{Y}_1 \), ............
Power of a statistical test is defined as
In statistical hypothesis testing, the area of non-rejection is defined as
Let \( M = (m_{ij}) \) be a \( 3 \times 3 \) real, invertible matrix and \( \sigma \in S_3 \) be the permutation defined by \( \sigma(1) = 2, \sigma(2) = 3 \) and \( \sigma(3) = 1 \). The matrix \( M_\sigma \) is defined by \( n_{ij} = m_{i\sigma(j)} \) for all \( i,j \in \{1, 2, 3\} \). Then, which one of the following is TRUE?
For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \frac{3n}{n^3} \quad \text{and} \quad y_n = \left(4n^3 + (-1)^n 3n^3 \right)^{1/n}. \] Then, which one of the following is TRUE?