Two forces having magnitude $$A$$ and $$\frac{A}{2}$$ are perpendicular to each other. The magnitude of their resultant is:

The angle of elevation of the top $$\mathrm{P}$$ of a tower from the feet of one person standing due South of the tower is $$45^{\circ }$$ and from the feet of another person standing due west of the tower is $$30^{\circ }$$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

From the top $$\mathrm{A}$$ of a vertical wall $$\mathrm{AB}$$ of height $$30\mathrm{m}$$, the angles of depression of the top $$\mathrm{P}$$ and bottom $$\mathrm{Q}$$ of a vertical tower $$\mathrm{PQ}$$ are $$15^{\circ }$$ and $$60^{\circ }$$ respectively, $$\mathrm{B}$$ and $$\mathrm{Q}$$ are on the same horizontal level. If $$\mathrm{C}$$ is a point on $$\mathrm{AB}$$ such that $$\mathrm{CB}=\mathrm{PQ}$$, then the area (in $${\mathrm{m}}^2$$ ) of the quadrilateral $$\mathrm{BCPQ}$$ is equal to :

Let $x=(8\sqrt{3}+13{)}^{13}$ and $y=(7\sqrt{2}+9{)}^9$. If $[t]$ denotes the greatest integer $\le t$, then :

If the coefficient of $$x^{15}$$ in the expansion of $${\left(\mathrm{a}{x}^3+\frac{1}{\mathrm{b}{x}^{1/3}}\right)}^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $${\left(a{x}^{1/3}-\frac{1}{b{x}^3}\right)}^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$(\mathrm{a},\mathrm{b})$$ :

The coefficient of $$x^{301}$$ in $$(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+...+x^{500}$$ is :

Let K be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x{)}^{99}$$. Let $$a$$ be the middle term in the expansion of {\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}. If {{{}^{200}{C_{99}}K} \over a} = {{{2^l}m} \over n}, where m and n are odd numbers, then the ordered pair $$(l,\mathrm{n})$$ is equal to

If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1+x{)}^{10}$$, then \sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}} is equal to

If {({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}} then $$\alpha$$ is equal to :

The value of $$\sum _{r=0}^{22}{}^{22}{C}_r{}^{23}{C}_r$$ is

Let ${\left(a+bx+c{x}^2\right)}^{10}=\sum _{i=0}^{20}{p}_i{x}^i,a,b,c\in \mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to :

The coefficient of $$x^5$$ in the expansion of $${\left(2x^3-\frac{1}{3x^2}\right)}^5$$ is :

Fractional part of the number $$\frac{4^{2022}}{15}$$ is equal to

If $$\frac{1}{n+1}{}^n{\mathrm{C}}_n+\frac{1}{n}{}^n{\mathrm{C}}_{n-1}+\dots +\frac{1}{2}{}^n{\mathrm{C}}_1+{}^n{\mathrm{C}}_0=\frac{1023}{10}$$ then $$n$$ is equal to :

The sum, of the coefficients of the first 50 terms in the binomial expansion of $$(1-x{)}^{100}$$, is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of $$(1+\mathrm{x}{)}^{\mathrm{n}+2}$$, which are in the ratio $$1:3:5$$, is equal to :

If the $$1011^{\text{th }}$$ term from the end in the binominal expansion of $${\left(\frac{4x}{5}-\frac{5}{2x}\right)}^{2022}$$ is 1024 times $$1011^{\text{th }}$$R term from the beginning, then $$|x|$$ is equal to

Let the number $$(22{)}^{2022}+(2022{)}^{22}$$ leave the remainder $$\alpha$$ when divided by 3 and $$\beta$$ when divided by 7. Then $$\left({\alpha }^2+{\beta }^2\right)$$ is equal to :

If the coefficients of $$x$$ and $$x^2$$ in $$(1+x{)}^{\mathrm{p}}(1-x{)}^{\mathrm{q}}$$ are 4 and $$-$$5 respectively, then $$2p+3q$$ is equal to :

If the coefficient of $$x^7$$ in {\left( {ax - {1 \over {b{x^2}}}} \right)^{13}} and the coefficient of $$x^{-5}$$ in {\left( {ax + {1 \over {b{x^2}}}} \right)^{13}} are equal, then $$a^4{b}^4$$ is equal to :