Ncert Solutions Maths class 11th

Let z = α ± iβ be roots of the equation
So 2α = -b and α² + β² = 45, (α+1)² + β² = 40
So (α+1)² - α² = -5 ⇒ 2α+1 = -5 ⇒ 2α = -6
So b=6
Hence b² - b = 30

D = |0, 2, 1; 1, -1, 1; x', y', 1| = 0

Let lim (x→0) [ (3x²+2)/ (7x²+2)]^ (1/x²) = e^ [lim (x→0) (1/x²) * (3x²+2)/ (7x²+2) - 1)] =

e^ [lim (x→0) (1/x²) * (-4x²)/ (7x²+2)] = e^ (-4/2) = e? ²

$\mathrm{}{f}^{\mathrm{\text{'}}}\left(x\right)=a(x+1)(x-1)x^2$

$f^{\mathrm{\text{'}}}\left(x\right)=a{x}^4-x^{2}f$

$f\left(x\right)=\frac{a{x}^5}{5}-\frac{a{x}^3}{5}+C$
$?f\left(0\right)=0\Rightarrow c=0$

${\mathrm{l}\mathrm{i}\mathrm{m}}_{x\to 0}?\frac{f\left(x\right)}{x^3}=2$

$\Rightarrow \mathrm{a}=-6$

Minima at $\therefore f^{\mathrm{\text{'}}}\left(x\right)=-6\left(x^2-1\right)\left(x^2\right)$

Maxima at $x=-1$

$3+4+8+9+13+14+\dots .$ . upto 40 terms

$\Rightarrow 7+17+27+\dots \dots ..20$ terms

$\mathrm{S}=\frac{20}{2}[2*7+19*10]$

$=102*20=102\mathrm{m}$

$\therefore \mathrm{m}=20$

$y=-\frac{3x}{4}+3\sqrt[]{2}$ line is tangent to ellipse

$\therefore {\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2$

$18=\frac{9{\mathrm{a}}^2}{16}+9$

${\mathrm{a}}^2=16$

$\therefore {\mathrm{e}}^2=1-\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$

$\mathrm{e}=\frac{\sqrt[]{7}}{4}$

Distance between focii $=2\mathrm{a}\mathrm{e}$

$=2\sqrt[]{7}$

$\mathrm{}6\cdot {}^{35}{\mathrm{C}}_{\mathrm{r}}=\left({\mathrm{k}}^2-3\right)\cdot \frac{36}{\mathrm{r}+1}{}^{35}{\mathrm{C}}_{\mathrm{r}}$

$\Rightarrow {\mathrm{k}}^2-3=\frac{\mathrm{r}+1}{6},{\mathrm{k}}^2-3>0$

(i) $k=\pm 2$ gives $r=5$

(ii) $k=\pm 3$ gives $r=35$

4 ordered pairs

Re (z²) = x²-y². 2 (Im (z)² = 2y². 2Re (z) = 2x.
x²-y²+2y²+2x=0 ⇒ x²+y²+2x=0.
(x+1)²+y²=1. Center (-1,0).
Parabola: x²-6x+9 = y-13+9 ⇒ (x-3)²=y-4. Vertex (3,4).
Line through (-1,0) and (3,4): slope = (4-0)/ (3- (-1) = 1.
y-0 = 1 (x+1) ⇒ y=x+1.
y-intercept is 1.

2log? (2? -5) = log?2 + log? (2? -7/2).
(2? -5)² = 2 (2? -7/2) = 2*2? -7.
Let t=2? (t-5)² = 2t-7.
t²-10t+25=2t-7 ⇒ t²-12t+32=0 ⇒ (t-4) (t-8)=0.
t=4 or t=8.
2? =4 ⇒ x=2. log? (4-5) undefined.
2? =8 ⇒ x=3. log? (8-5)=log?3=1. log? (8-3.5)=log?4.5.
2 (1) = log?2+log?4.5 = log?9=2. Correct.
x=3.