Maths

The least positive value of ' $$ $a$ ' for which the equation, $2x^2+(a-10)x+\frac{33}{2}=2a$ ${}^{}\frac{}{}$ has real root is

(Quadratic Equations)

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that the most three of them are red is(Combinations)

Let the normal at a point $P$ $$ on the curve ${}^{}{}^{}$ $y^2-3x^2+y+10=0$ intersect the $$ $y-axis$ at $\left(0,\frac{3}{2}\right)$ $\left(\frac{}{}\right)$ . If $m$ $$ is the slope of the tangent at $P$ $$ to the curve, then $$ $\left|m\right|$ is equal to(2D)

Let $f\left(x\right)={\left(\mathrm{s}\mathrm{i}\mathrm{n}?\left({\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?x\right)+\mathrm{s}\mathrm{i}\mathrm{n}?\left({\mathrm{c}\mathrm{o}\mathrm{t}}^{-1}?x\right)\right)}^2-1,\left|x\right|>1$ ${\left(\left({}^{}\right)\left({}^{}\right)\right)}^{}$ . If $\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left({\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}?\left(f\right(x\left)\right)\right)$ $\frac{}{}\frac{}{}\frac{}{}\left({}^{}\right)$ and $\text{exception 25:}\frac{}{}$ $y\left(\sqrt[]{3}\right)=\frac{\pi }{6}$ , then $y(-\sqrt[]{3})$ $\text{exception 25:}$ is equal to

Let the line $y=mx$ $$ and the ellipse $2x^2+y^2=1$ ${}^{}{}^{}$ intersect at a point $$ $P$ in the first quadrant. If the normal to this ellipse at $P$ $$ meets the co - ordinate axes at $\left(\frac{}{\text{exception 25:}}\right)$ $\left(-\frac{1}{3\sqrt[]{2}},0\right)$ and $$ $(0,\beta )$ then $\beta$ $$ is equal to:

Let $f:R\to R$ $$ be such that for all $x\in R\left(2^{1+x}+2^{1-x}\right),f\left(x\right)$ $\left({}^{}{}^{}\right)$ and $\left(3^x+3^{-x}\right)$ $\left({}^{}{}^{}\right)$ are in A.P., then the minimum value of $f\left(x\right)$ $$ is:

Let $A,B,C$ and $D$ be four non - empty sets. The contrapositive statement of "If $A\subseteq B$ and $B\subseteq D$ , then $A\subseteq C$  " is:

If $a,b$ $$ and $c$ $$ are the greatest values of ${}^{}{}_{}{}^{}{}_{}$ ${}^{19}{\mathrm{C}}_{\mathrm{p}},{}^{20}{\mathrm{C}}_{\mathrm{q}}$ and ${}^{21}{\mathrm{C}}_{\mathrm{r}}$ ${}^{}{}_{}$ respectively, then:

If $\mathrm{c}$ $$ is a point at which Rolle's theorem holds for the function, ${}_{}\left(\frac{{}^{}}{}\right)$ $f\left(x\right)={\mathrm{l}\mathrm{o}\mathrm{g}}_e?\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $\left[\mathrm{3,4}\right]$ $$ , where $$ , $\alpha \in R$ then ${}^{}$ $f^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(c\right)$ is equal to:

The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1-x{)}^8+\dots \dots +x^{10}$ is: