List of top Mathematics Questions

Let O be the vertex of the parabola \(x^2=4y\) and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2:3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:

If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.

Let $(h,k)$ lie on the circle $C:x^2+y^2=4$ and the point $(2h+1,\,3k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\dfrac{5}{e^2}$ is equal to

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to :

The value of \[ \frac{{}^{100}C_{50}}{51} + \frac{{}^{100}C_{51}}{52} + \cdots + \frac{{}^{100}C_{100}}{101} \] is:

The value of \[ \binom{100}{50} + \binom{100}{51} + \binom{100}{52} + \dots + \binom{100}{100} \] is:

In the binomial expansion of \( (ax^2 + bx + c)(1 - 2x)^{26} \), the coefficients of \( x, x^2 \), and \( x^3 \) are -56, 0, and 0 respectively. Then, the value of \( (a + b + c) \) is

The coefficient of \( x^{48} \) in the expansion of \[ 1 + (1+x) + 2(1+x)^2 + 3(1+x)^3 + \dots + 100(1+x)^{100} \] is

If in the expansion of \( (1 + x^2)^2(1 + x)^n \), the coefficients of \( x \), \( x^2 \), and \( x^3 \) are in arithmetic progression, then the sum of all possible values of \( n \) (where \( n \geq 3 \)) is:

If three vectors are given as shown. If the angle between vectors \( \mathbf{p} \) and \( \mathbf{q} \) is \( \theta \) where \( \cos \theta = \frac{1}{\sqrt{3}} \), \( |\mathbf{p}| = 2 \), and \( |\mathbf{q}| = 2 \), then the value of \( |\mathbf{p} \times (\mathbf{q} - 3\mathbf{r})|^2 - 3|\mathbf{r}|^2 \) is:

If \( \vec{a}, \vec{b}, \vec{c} \) are three vectors such that \[ \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c}), \] \( |\vec{a}| = 1,\; |\vec{b}| = 4,\; |\vec{c}| = 2 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( 60^\circ \), then find \( |\vec{a} \cdot \vec{c}| \):

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

Given that \[ \vec a=2\hat i+\hat j-\hat k,\quad \vec b=\hat i+\hat j,\quad \vec c=\vec a\times\vec b, \] \[ |\vec d\times\vec c|=3,\quad \vec d\cdot\vec c=\frac{\pi}{4},\quad |\vec a-\vec d|=\sqrt{11}, \] find $\vec a\cdot\vec d$.

Let the lines \[ L_1:\ \vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k),\ \lambda\in\mathbb R \] \[ L_2:\ \vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k),\ \mu\in\mathbb R \] intersect at the point $R$. Let $P$ and $Q$ be the points lying on the lines $L_1$ and $L_2$ respectively, such that \[ |PR|=\sqrt{29}\quad \text{and}\quad |PQ|=\sqrt{\frac{47}{3}}. \] If the point $P$ lies in the first octant, then find $27(QR)^2$.

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

If three vectors are given as shown. If the angle between vectors \( \vec{p} \) and \( \vec{q} \) is \( \theta \), where \[ \cos \theta = \frac{1}{\sqrt{3}}, \quad |\vec{p}| = 2\sqrt{3}, \quad |\vec{q}| = 2, \] then find the value of \[ \left| \vec{p} \times (\vec{q} - 3\vec{r}) \right|^{2} - 3|\vec{r}|^{2}. \]

For given vectors \( \mathbf{a} = -\hat{i} + \hat{j} + 2\hat{k} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \), where \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \), then the value of \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} \) is:

\[ \left(\frac{1}{^{15}C_0}+\frac{1}{^{15}C_1}\right) \left(\frac{1}{^{15}C_1}+\frac{1}{^{15}C_2}\right) \cdots \left(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}\right) = \frac{\alpha^{13}}{^{14}C_0\cdot {}^{14}C_1\cdot {}^{14}C_2\cdots {}^{14}C_{12}} \] If so, then find the value of \(30\alpha\).

Let \( S \) be the number of 4-digit numbers \( abcd \), where \[ a>b>c>d \] and let \( P \) be the number of 5-digit numbers \( abcde \), where the product of digits is 20. Find \( S + P \):

If \( A = \{ 1, 2, 3, 4, 5, 6 \}, B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \), then the number of strictly increasing functions from \( A \to B \) such that \( f(i) \neq i \) for \( i = 1, 2, 3, 4, 5, 6 \) is