List of top Mathematics Questions

Let the sixth term in the binomial expansion of $({\sqrt{2}^{log_{2}}(10-3^{x})+\sqrt[5]{2^{(x-2)log_{2}{3}}}})^{m}$, in the increasing powers of $2^{(x-2)log_{2}3}$, be 21 If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an AP, then the sum of the squares of all possible values of x is

If the term without $x$ in the expansion of $\left(x^{\frac{2}{3}}+\frac{\alpha}{x^3}\right)^{22}$is 7315 , then $|\alpha|$ is equal to ___

The point of intersection $C$ of the plane $8 x+y+2 z=0$ and the line joining the points $A (-3,-6,1)$ and $B (2,4,-3)$divides the line segment $AB$ internally in the ratio$k : 1 \ If a , b , c (| a |,| b |, | c |$are coprime) are the direction ratios of the perpendicular from the point $C$on the line $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$, then $| a + b + c |$is equal to ___

The sum of the common terms of the following three arithmetic progressions$3,7,11,15, \ldots , 399$, $2,5,8,11, \ldots , 359$and $2,7,12,17, \ldots , 197,$ is equal to _____

If the x-intercept of a focal chord of the parabola $y^2=8 x+4 y+4$ is 3 , then the length of this chord is equal to ___

Let $\alpha x+\beta y+y z=1$ be the equation of a plane passing through the point$(3,-2,5)$and perpendicular to the line joining the points $(1,2,3)$ and $(-2,3,5)$ Then the value of $\alpha \beta y$is equal to ____

The line $x=8$ is the directrix of the ellipse $E : \frac{x^2}{ a ^2}+\frac{y^2}{b^2}=1$with the corresponding focus $(2,0)$ If the tangent to $E$at the point $P$ in the first quadrant passes through the point $(0,4 \sqrt{3})$and intersects the$x$-axis at $Q$, then $(3 PQ )^2$is equal to ____

Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2 x+y+3 z=16$ at the point $P$ Let the point $Q$ be the foot of perpendicular from the point $R(1,-1,-3)$ on the line $L$ If $\alpha$ is the area of triangle $P Q R$, then $\alpha^2$ is equal to

Let 5 digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers Then the serial number of the number 42923 is

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

Let for $x \in R$ $f(x)=\frac{x+|x|}{2} \text { and } g(x)=\begin{cases}x, & x<0 \\x^2, & x \geq 0\end{cases} $

Then area bounded by the curve $y=(f \circ g)(x)$ and the lines $y=0,2 y-x=15$ is equal to

The number of seven digits odd numbers, that can be formed using all the seven digits $1,2,2,2,3,3,5$ is_______

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$ If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3 \alpha^2$ is equal to_______

Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$ Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to ______

The vertices of a hyperbola $H$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$ Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2} x+y=2 \sqrt{2}$If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis then $d^2$ is equal to

Let the equation of the plane passing through the line $x-2 y-z-5=0=x+y+3 z-5$ and parallel to the line $x+y+2 z-7=0=2 x+3 y+z-2$ be $a x+b y+c z=65$ Then the distance of the point $(a, b, c)$ from the plane $2 x+2 y-z+16=0$ is _______

It the area enclosed by the parabolas $P_1: 2 y=5 x^2$ and $P_2: x^2-y+6=0$ is equal to the area enclosed by $P_1$ and $y=\alpha x, \alpha>0$, then $\alpha^3$ is equal to ____ .

for some a,b,c ∈ $\N$, let f(x) = ax-3 and g(x)=xb+c, x ∈ $\R$. If (fog)^-1(x) = $(\frac{x-7}{2})^{\frac{1}{3}}$ then (fog) (ac) + (gof) (b) is equal to _________ .

If the sum of all the solutions of $\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3},-1 < x < 1, x \neq 0$, is $\alpha-\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to ___

Let $A_1, A_2, A_3$ be the three $A P$ with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively, Let $a, b, c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\begin{vmatrix}a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{vmatrix}+70=0$If $a=29$, then the sum of first 20 terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$ Then, the number of ways of choosing $x$ and $y$, such that $x+y$ is divisible by 5 , is ____

Let $S =\{1,2,3,5,7,10,11\}$ The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 , is _____

The constant term in the expansion of $\left(2 x+\frac{1}{x^7}+3 x^2\right)^5$ is_____

Let $S=\left\{\alpha: \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2\right\}$ Then the maximum value of $\beta$ for which the equation $x^2-2\left(\displaystyle\sum_{\alpha \in s} \alpha\right)^2 x+\displaystyle\sum_{\alpha \in s}(\alpha+1)^2 \beta=0$ has real roots, is _____

Let $\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha>$ If the minimum value of the scalar triple product $[\vec{u}\,\, \vec{v}\,\, \vec{w}]$ is $-\alpha \sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to _____