List of top Mathematics Questions

One out of 9 ships is likely to sink when they are set on sail. When 6 ships are set on sail, the probability that exactly 3 of them will not arrive safely is:
The centre of a square of side 4 units length is \( (3, 7) \), and one of the diagonals is parallel to the line \( y = x \). If \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) are the vertices of this square, then \[ \frac{y_1 y_2 y_3 y_4}{x_1 x_2 x_3 x_4} = ? \]
If \( X \) is a random variable such that \[ P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = \frac{1}{6}, \quad P(X = 0) = \frac{1}{3}, \] then the mean of \( X \) is:
If \( t \in \mathbb{R} \setminus \{-1\} \), then the locus of the point \[ \left( \frac{3at}{1 + t^3}, \frac{3at^2}{1 + t^3} \right) \] is:
Let origin be the centroid of an equilateral triangle ABC and one of its sides is along the straight line \( x + y = 3 \). If \( R \) and \( r \) are its circumradius and inradius respectively, then \( R + r = \):
The area (in square units) of the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( 3x + 4y = 12 \) is:
A and B are mutually exclusive events of a random experiment and \( P(B^c) \ne 1 \), then \[ P(A | B^c) = ? \]
In a test, a student either guesses, copies, or knows the answer to a multiple-choice question with four choices. The probability that he makes a guess is \( \frac{1}{3} \), and the probability that he copies is \( \frac{1}{6} \). The probability that his answer is correct, given that he guessed it, is \( \frac{1}{4} \), and the probability that he copied it and it is correct is \( \frac{1}{8} \). The probability that he knew the answer to the question, given that he answered it correctly, is:
If \( \vec{p} = 4\vec{i} - \vec{j} + \vec{k} \) is a point and \( \vec{q} = 9\vec{i} - 2\vec{j} + 6\vec{k} \) is a vector, then the perpendicular distance of origin from the plane passing through \( \vec{p} \) and perpendicular to \( \vec{q} \) is:
Let \( \vec{a} = 3\vec{i} + \vec{j} - 2\vec{k} \), \( \vec{b} = -5\vec{i} + 7\vec{j} \), and \( \vec{c} = 3\vec{i} + y\vec{j} \) be three vectors such that \( |\vec{a} - \vec{b} + \vec{c}| = \sqrt{141} \). If \( y_1 \) and \( y_2 \) are the values of \( y \) satisfying the given condition, then \( |y_1 - y_2| = \)
An analysis of monthly wages paid to the workers of two jute mills A and B gives the following data:
\begin{tabular}{|c|c|c|} \hline & Mill - A & Mill - B
\hline No. of workers & 500 & 600
Average daily wage (in rupees) & 186 & 175
Variance of distribution of wages & 81 & 100
\hline \end{tabular} Then:
A, B, C are three horses participating in a race. The probability of horse A to win is twice that of B, and the probability of B to win is twice that of C. Then the probabilities of A, B, and C winning the race are respectively:
Let ‘O’ be the origin. A and B be two points with position vectors \( -3\vec{i} - 3\vec{j} + 4\vec{k} \) and \( 4\vec{i} - 4\vec{j} - 3\vec{k} \) respectively. Let \( P \) be a point such that the line drawn through \( P \) parallel to \( OB \) meets \( OA \) in \( L \), and another line through \( P \) parallel to \( OA \) meets \( OB \) in \( M \). If \( L \) divides \( OA \) in the ratio 2:3 and \( M \) divides \( OB \) in the ratio 3:2, then the distance from \( O \) to \( P \) is:
For a positive real number \( \lambda \), if the vector \( \vec{a} = \lambda \vec{i} - 5\vec{j} + 6\vec{k} \) satisfies the equation \[ \left[ \vec{i} \times (\vec{a} \times \vec{i}) + \vec{j} \times (\vec{a} \times \vec{j}) + \vec{k} \times (\vec{a} \times \vec{k}) \right]^2 = 440, \] then \( \lambda = \)
In \( \triangle ABC \), if \( r_1 - r = \frac{a}{3} \) and \( r_2 - r = \frac{b}{3} \), then \( r_1 + r_2 - r \) is:
In \( \triangle ABC \), if \( (a - b)(s - c) = (b - c)(s - a) \), then \( r_1, r_2, r_3 \) are in:
If \( \tan A = \tan \alpha \coth x = \cot \beta \tanh x \), then \( \tan(\alpha + \beta) = \):
If the position vectors of the points \( A \) and \( B \) are \( 2i + 3j - k \) and \( i - j + 2k \) respectively, then the unit vector along \( \overrightarrow{BA} \) and in the direction of \( AB \) is:
In \( \triangle ABC \), if \( a, b, c \) are 5, 12, and 13 respectively, then \( b^2 \sin 2C + c^2 \sin 2B \) is:
If \( \frac{6x^3 + 7x^2 - 14x + 11}{6x^3 + x^2 - 10x + 3} = \frac{a}{x + p} + \frac{b}{qx + 3} + \frac{c}{3x + p} \), then \( a + b \) is:
Evaluate \( \left[ 1 + \sec 2\theta \right] \left[ 1 + \sec 40^\circ \right] \):
The number of arrangements of the word KANGAROO in which the A's do not appear together is:
If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is not in the first quadrant, then \( 15 \sin 2\theta - 20 \cos 2\theta - 7 \tan 2\theta \) is:
In \( \triangle ABC \), if \( \cos^2 A + \cos^2 B + \cos^2 C = 1 \), then \( \triangle ABC \) is:
If \( A = \{ (a, b) : 4a = 5b, a \in \{1, 2, 3, \dots, 30\} \}, \) then the number of such ordered pairs \( (a, b) \) is: