Maths

Required area

$A=2{\displaystyle {\int }_{}^{}\left(\left(sinx+cosx\right)-\left(cosx-sinx\right)\right)dx}$

$=2{\displaystyle {\int }_0^{\pi /4}sinxdx}$

$=2\sqrt[]{2}\left(\sqrt[]{2}-1\right)$    

Total ways = 6!

Ways satisfying g (3) = 2g (1) is 3

Number of onto function 3 * 4!

Probability = $\frac{1}{10}$

$b^2{x}^2+a^2{y}^2=a^2{b}^2$

$\left(b^2{x}^2+a^2\left(ab-x^2\right)\right)=a^2{b}^2$

$x^2=\frac{b{a}^2\left(b-a\right)}{b^2-a^2},y^2=\frac{a{b}^2}{a+b}$

points of intersection

$\left(a\sqrt[]{\frac{b}{a+b}},b\sqrt[]{\frac{a}{a+b}}\right)$

$tan\theta =\left|\frac{m_1-m_2}{1+m_1.m_2}\right|$

$=\left|\frac{a-b}{\sqrt[]{ab}}\right|$

$\Delta =0\Rightarrow a=3,4$  

 For a = 3, no solution

For a = 4, no solution

n (S₁) = 2, n (S₂) = 0,

Equation of intersection of line

$\frac{x-0}{1}=y=\frac{z-0}{-1}$

Let $\overrightarrow{r}=\widehat{i}-\widehat{k}$

Direction ratio of $\overrightarrow{PQ}$

$\Rightarrow \left(\lambda +1\right)\left(1\right)+\left(-2\right)\left(0\right)+\left(2-\lambda \right)\left(-1\right)=0$

$\lambda =\frac{1}{2}\Rightarrow Q\left(\frac{1}{2},0,\frac{-1}{2}\right)$

$PQ=\frac{\sqrt[]{34}}{2}$

$f\left(x\right)=lo{g}_{\sqrt[]{5}}\left(3+\sqrt[]{2}\left(cosx-sinx\right)\right)$

$-\sqrt[]{2}\le cosx-sinx\le \sqrt[]{2}$

$\Rightarrow 0\le f\left(x\right)\le 2$

$\left|\begin{array}{ccc}\hfill \alpha \hfill & \hfill \beta \hfill & \hfill 1\hfill \\ \hfill 5\hfill & \hfill 6\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right|=24$

$4\alpha -2\beta -8=\pm 24$

$4\alpha -2\beta =24+8,4\alpha -2\beta =-24+8$

$2\left(2\alpha -\beta \right)=322\alpha -\beta =-8$

Distance of origin

$D=\sqrt[]{{\alpha }^2+{\left(2\alpha +8\right)}^2}$

$\alpha =-\frac{16}{5}$

$D=\sqrt[]{{\left(\frac{-16}{5}\right)}^2+{\left(\frac{8}{5}\right)}^2}$

if 2 - = 16

$D=\frac{16}{\sqrt[]{5}}$

Point of intersection of

$\frac{x^2}{9}+y^2=1and{x}^2+y^2=3inthe1stquadrantis\left(\frac{3}{2},\frac{\sqrt[]{3}}{2}\right)$                

$m_1=-\frac{1}{3\sqrt[]{3}},m_2=-\sqrt[]{3}$                

$tan\theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\frac{2}{\sqrt[]{3}}$                

$S_{20}={\displaystyle \underset{n=1}{\overset{20}{?}}\frac{1}{d}\left(\frac{1}{a_n}?\frac{1}{a_{n+1}}\right)}$

$=\frac{1}{d}\left(\frac{1}{a_1}?\frac{1}{a_{21}}\right)$

$?a\left(a+20d\right)=45.......\left(i\right)$

$?a+10d=9.........\left(ii\right)$

$\left(i\right)\&\left(ii\right)?d^2=\frac{36}{100}$

Yes! The NCERT Notes are more than enough to get full marks in the board exams. You want to know why these notes are enough?

• NCERT revision notes cover each chapter in clear, easy-to-read summaries along with detailed concepts.
• These notes cover the entire CBSE syllabus in simple language, including important definitions, formulas, and examples for the exams.
• Our subject matter experts have designed information in a genuine and concise way.
• These notes are perfect guides for both quick revision and thorough conceptual understanding.