List of top Mathematics Questions

Let \( p_n \) denote the total number of triangles formed by joining the vertices of an \( n \)-side regular polygon. If \( p_{n+1} - p_n = 66 \), then the sum of all distinct prime divisors of \( n \) is:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - 3x + r = 0 \), and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 + 3x + r = 0 \). If the roots of the equation \( x^2 + 6x = m \) are \( 2\alpha + \beta + 2r \) and \( \alpha - 2\beta - \frac{r}{2} \), then \( m \) is equal to:

Let the circles \( C_1 : |z| = r \) and \( C_2 : |z - 3 - 4i| = 5, z \in \mathbb{C} \), be such that \( C_2 \) lies within \( C_1 \). If \( z_1 \) moves on \( C_1 \), \( z_2 \) moves on \( C_2 \) and \( \min |z_1 - z_2| = 2 \), then \( \max |z_1 - z_2| \) is equal to:

If the system of equations} \[ x + 5y + 6z = 4 \] \[ 2x + 3y + 4z = 7 \] \[ x + 6y + az = b \] has infinitely many solutions, then the point \( (a, b) \) lies on the line}

If \[ \alpha=\int_{0}^{2\sqrt{3}} \log_2(x^2+4)\,dx + \int_{2}^{4} \sqrt{2^x-4}\,dx, \] then \(\alpha^2\) is equal to _____.

Let a circle \(C\) have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of \(C\) on the line \(x+y=1\) is \(\sqrt{14}\), then the square of the radius of \(C\) is _____.}

Let \(a,b,c \in \{1,2,3,4\}\). If the probability that \[ ax^2 + 2\sqrt{2}\,bx + c>0 \quad \text{for all } x \in \mathbb{R} \] is \( \frac{m}{n} \), where \(\gcd(m,n)=1\), then \(m+n\) is equal to _____.

Let \([\,]\) denote the greatest integer function. Then the value of} \[ \int_{0}^{3}\left(\frac{e^x+e^{-x}}{[x]!}\right)dx \] is:

If the domain of the function} \[ f(x)=\sqrt{\log_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)} \] is \((-\infty,a] \cup \{b\} \cup [c,d) \cup (e,\infty)\), then the value of \(a+b+c+d+e\) is _______.}

Let \(y=y(x)\) be the solution curve of the differential equation} \[ (1+\sin x)\frac{dy}{dx}+(y+1)\cos x=0,\qquad y(0)=0. \] If the curve passes through the point \( \left(\alpha,-\frac12\right) \), then a value of \( \alpha \) is:

If \[ \sum_{k=1}^{n} a_k = 6n^3, \] then} \[ \sum_{k=1}^{6}\left(\frac{a_{k+1}-a_k}{36}\right)^2 \] is equal to _______.

If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation} \[ dy=y(2+\log_e x)\,dx,\quad x>0, \] then \(f(e)\) is equal to:

Let a line \(L\) passing through the point \((1,1,1)\) be perpendicular to both the vectors \(2\hat{i}+2\hat{j}+\hat{k}\) and \(\hat{i}+2\hat{j}+2\hat{k}\). If \((a,b,c)\) is the foot of perpendicular from the origin on the line \(L\), then the value of \(34(a+b+c)\) is:

If \(|\vec a|=2\) and \(|\vec b|=3\), then the maximum value of \[ 3\left|\left(\vec a+2\vec b\right)\right| + 4\left|\left(3\vec a-2\vec b\right)\right| \] is:

Let \[ S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\; a\in\mathbb{Z}\}. \] Then \(n(S)\) is equal to:

If the point of intersection of the lines \[ \frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} \] \[ \frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7} \] lies on the \(xy\)-plane, then the value of \(a+b\) is:

If \[ \lim_{x\to 2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m, \] then \(a+b+m\) is equal to:

The number of elements in the set} \[ S=\left\{(r,k): k\in \mathbb{Z} \text{ and } {^{36}C_{r+1}}=\frac{6\left({^{35}C_r}\right)}{k^2-3}\right\} \] is:

If \[ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)=K, \] then the value of \[ \sin\left(\frac{10K\pi}{3}\right) \] is:

The number of seven-digit numbers that can be formed by using the digits \(1,2,3,5,7\) such that each digit is used at least once, is:

Let an ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a<b \] pass through the point \((4,3)\) and have eccentricity \( \frac{\sqrt5}{3} \). Then the length of its latus rectum is:

Let \(A\) be the set of first \(101\) terms of an A.P., whose first term is \(1\) and the common difference is \(5\), and let \(B\) be the set of first \(71\) terms of an A.P., whose first term is \(9\) and the common difference is \(7\). Then the number of elements in \(A \cap B\), which are divisible by \(3\), is:

Let \[ A= \begin{bmatrix} 1 & 2\\ 1 & \alpha \end{bmatrix} \quad \text{and} \quad B= \begin{bmatrix} 3 & 3\\ \beta & 2 \end{bmatrix}. \] If \(A^2-4A+I=O\) and \(B^2-5B-6I=O\), then among the following statements: (S1): \[ [(B-A)(B+A)]^T= \begin{bmatrix} 13 & 15\\ 7 & 10 \end{bmatrix} \] (S2): \[ \det(\operatorname{adj}(A+B))=-5 \] Choose the correct option:

Let \( \alpha, \alpha + 2 \in \mathbb{Z} \) be the roots of the quadratic equation} \[ x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \cdots + (x+n-1)(x+n+1) = 4n \] for some \( n \in \mathbb{N} \). Then \( n + \alpha \) is equal to:

Find the coordinates of the points of trisection of the line segment joining the points \( A(-1, 4) \) and \( B(-3, -2) \).