Maths

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:

If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $\left(\mathrm{0,2}\pi \right)-\left\{\pi \right\}$ which satisfy the equation, $2{\mathrm{c}\mathrm{o}\mathrm{t}}^2\theta -\frac{5}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }+4=0$ , then ${\int }_{{\theta }_1}^{{\theta }_2}?{\mathrm{c}\mathrm{o}\mathrm{s}}^2?3\theta d\theta$ is equal to:

In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ $$ and then reduced by $$ $q$ , where $p=0$ $$ and $q\ne 0$ $$ . If the new mean and new s.d. become half of their original values, then $q$ $$ is equal to:

Let the vectors a,b,c be such that |a|=2, |b|=4 and |c|=4. If the projection of b on a is equal to the projection of c on a and b is perpendicular to c, then the value of |a+b-c| is