Class 11th

Ratio of masses on two pistons of the hydraulic lift equals to that of their cross- section area.

$\frac{A_1}{A_2}=\frac{100}{m}$            

Now,     $\frac{4^2{A}_1}{A_2/4^2}=\frac{M}{m}?M=256\frac{A_1}{A_2}.m=25600kg.$

Given $P_n={\alpha }^n+{\beta }^n,P_{n-1}=11\&P_{n+1}=29$

    $P_n={\alpha }^{n-2}.{\alpha }^2+{\beta }^{n-2}.{\beta }^2.........\left(i\right)$                          

Now quadratic equation having roots α & β will be x² – (α + β)x + αβ = 0

i.e.         x² – x – 1 = 0,     put x = α and put x = β

So          α² = α + 1           & β² = β + 1

(i)     $P_n={\alpha }^{n-2}\left(\alpha +1\right)+{\beta }^{n-2}\left(\beta +1\right)$

$P_n=P_{n-1}+P_{n-2}$          

= > ${P_n}^2=234$  

Given sequence is -16, 8, -4, 2, .

are in G.P. with first term a = -16 & common ratio r =  $\frac{-1}{2}$

Now $t_p=a{r}^{P-1}=-16{\left(-\frac{1}{2}\right)}^{p-1}\&t_q=a{r}^{q-1}=-16{\left(-\frac{1}{2}\right)}^{q-1}$

So A.M. = -8   $\left[{\left(-\frac{1}{2}\right)}^{p-1}+{\left(\frac{-1}{2}\right)}^{q-1}\right]=\alpha \left(let\right)$

Given equation is 4x² – 9x + 5 = 0 gives $x=1,\frac{5}{4}$

From roots we get possible value of b = 1 so

$16{\left(-\frac{1}{2}\right)}^{\frac{P+q-2}{2}}=1OR{\left(-\frac{1}{2}\right)}^{\frac{p+q-2}{2}}=\frac{1}{16}-{\left(-\frac{1}{2}\right)}^4$

$\Rightarrow \frac{p+q-2}{2}=4\Rightarrow p+q=10$

W = 4.5 g

M.W = 90

V = 250 ml

Using M = $\frac{W}{M.W*V}*1000$  

$M=\frac{4.5}{90*250}*1000=0.2$  

M = 2 * 10^-1 M

So, x = 2

Mass of Na⁺ in 50 ml = 70 * 50 = 3500 mg

23000 mg of Na⁺ is present in 85000 mg of NaNO₃ (1 mole NaNO₃ contains 1 mole Na⁺)

$\therefore 3500$ mg Na⁺ will be present in   $\frac{85000}{23000}*3500$

= 12934.78 mg

= 12.93478 gm

$\cong 13$        

pH = $\frac{1}{2}\left[p{K}_w+p{K}_a-p{K}_b\right]$ $\frac{}{}{}_{}{}_{}{}_{}$

$=\frac{1}{2}\left[14+4.75-5.23\right]$

$=6.76\cong 7$

Uncertainty in speed = 90 $*\frac{5}{100}=4.5m/s=\Delta V$ $\frac{}{}$

$\Delta x=\frac{h}{4\pi m\Delta V}=\frac{6.63*10^{-34}}{4*3.14*10^{-2}*4.5}=1.173*10^{-33}$

the nearest integer is 1

A (g) -> B (g)        K_P = 100

$\Delta G$ at 300 K and 1 atm

Using

$\Delta G=-RTlnK{}_P$           

$\Delta G=-R*300ln100$            

=-R * 300 * 2 * 2.3

$\Delta G=-1380R$            

So; x = 1380.

Methane gas is produced during anaerobic degradation of vegetation that leads to global warming so considered as greenhouse gas like CO₂. Also it causes cancer.