List of top Mathematics Questions

If the distance of a variable point \(P\) from a fixed point \((3,-4)\) is \(\dfrac23\) times the distance of \(P\) from the fixed line \(x-y+2=0\), then the locus of \(P\) is \[ ax^2+4by+by^2-62x+80y+c=0, \] then \(2c=\)

If the point \(M\) is the foot of the perpendicular drawn from the point \(P(0,9)\) to the straight line \[ 2x-5y+16=0, \] and \(Q=(12,8)\), then the orthocentre of \(\triangle MPQ\) is

Two persons A and B are playing a game of rolling 2 dice alternately. The person who first gets the sum of the numbers appearing on the dice as a prime number will win the game. If A starts the game, then the probability of B winning the game is

If the vectors \[ \vec a=4\hat{i}+6\hat{j}+\lambda\hat{k},\qquad \vec b=\hat{i}-2\hat{j}-3\hat{k}, \qquad \vec c=4\lambda\hat{i}+\hat{j}-3\hat{k} \] are coplanar and \(\lambda\in\mathbb{Z}\), then \(\vec a\cdot\vec c=\)

Three persons A, B and C planned to have a running race among themselves. If the probability that A wins the race is three times that of B and the probability that B wins the race is \(\dfrac32\) times that of C, then the difference in probabilities of A and C to win the race is

A dealer bought a fixed number of laptops from 3 different companies A, B and C. Among these laptops, \(28\%\) are bought from A, \(32\%\) are bought from B and \(40\%\) are bought from C. \(2.5\%\) of laptops bought from A, \(1.5\%\) bought from B and \(1\%\) bought from C are likely to be defective. If a customer found that the laptop bought by him is defective, then the probability that it was from B is

If \(\vec a\) is a vector perpendicular to the plane containing the vectors \[ 2\hat i-\hat j+3\hat k \quad\text{and}\quad -\hat i+3\hat j+2\hat k, \] then the magnitude of the projection of the vector \[ 3\hat i+2\hat j-\hat k \] on \(\vec a\) is

In a triangle \(ABC\), \[ 2\sqrt{r_1r_2+r_2r_3+r_3r_1} = \]

In a triangle \(ABC\), if \(s=\dfrac{15}{2},\; a=3,\; b=5\), then \[ \frac{\sin \dfrac{B}{2}}{\sin \dfrac{A}{2}}= \]

If \[ \vec a=\hat i+p\hat j-3\hat k,\qquad \vec b=2\hat i-3\hat j+q\hat k,\qquad \vec c=\hat i+2\hat j+2\hat k\;(p0) \] are three vectors such that the magnitude of projection of \(\vec a\) on \(\vec c\) is \(3\) and the magnitude of projection of \(\vec b\) on \(\vec c\) is \(2\), then the magnitude of projection of \(\vec a\) on \(\vec b\) is

The straight line given by the equation \[ \vec r=(4\hat{i}+5\hat{j}+\hat{k})+s(4\hat{i}+6\hat{j}+2\hat{k}) \] is coplanar with the straight line given below. Choose the correct option.

In a triangle \(ABC\), if \[ \overrightarrow{AB}=\hat{i}-2\hat{j}+3\hat{k}, \qquad \overrightarrow{BC}=3\hat{i}+2\hat{j}-2\hat{k}, \] then the triangle is

For \( A = \frac{\pi}{24} \), if \( \frac{\sin 2A + \sin 3A + \sin 4A}{\cos 2A + \cos 3A + \cos 4A} = k \), then \( (k+1)^2 = \)

If \( \theta \) is an acute angle, \( x = \sum_{n=0}^{\infty} \cos^{2n} \theta \), \( y = \sum_{n=0}^{\infty} \sin^{2n} \theta \), then \( \frac{1}{x^2} + \frac{1}{y^2} = \)

The sum of all the values of \( \theta \in \left[\frac{\pi}{2}, 2\pi\right] \) satisfying the equation \( \sin^2\theta \tan\theta + \cos^2\theta \cot\theta = \sin 2\theta \) is:

If \[ \sin^{-1}(2x)-2\cos^{-1}\sqrt{1-x^2}=\frac{\pi}{2}, \] then \(\tan^{-1}(2x+1)=\)

\(\coth^{-1}2=\)

If \( a \) is the coefficient of \( x^7 \) and \( b \) is the term independent of \( x \) in the expansion of \( \left( \frac{3x^4}{4} - \frac{4}{3x^3} \right)^7 \), then \( \frac{b}{a} = \)

By using all the letters of the word 'LETTER', all possible 6-letter words (with or without meaning) are formed. If all these words are arranged in dictionary order, then the rank of the word 'TRELET' is:

If all the letters of the word 'ASSOCIATION' are permuted in all possible ways to form all 11-letter words (with or without meaning), then among these words, the number of words in which the identical letters are always together but no two such groups of identical letters are together is:

The number of ways of distributing 5 identical things to 4 persons so that any person may get at most 5 things is:

If two roots of the equation \( x^5 - 9x^4 + 27x^3 - 23x^2 - 24x + 36 = 0 \) are repeated roots of multiplicity 2 and all the roots are integers then the sum of the squares of all different roots of the equation is:

If \( \alpha, \beta, \gamma, \delta \) are the roots of the equation \( x^4 - 10x^3 + 35x^2 - 50x + 24 = 0 \) and \( \alpha^2, \beta^2, \gamma^2, \delta^2 \) are the roots of the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \), then \( a = \)

If \( \alpha, \beta \in \mathbb{R} \) and \( \alpha \leq \frac{x^2 - 3x + 4}{x^2 - 4x + 5} \leq \beta \) for all \( x \in \mathbb{R} \), then \( (2\alpha - 3)^2 + (2\beta - 3)^2 = \)

If the sum of the squares of two consecutive positive odd integers is 1354, then one of them is: