Class 12th

Given curves are y = x² - 1 and y = 1 - x² so intersection points are (±1,0). Bounded area =
4∫? ¹ (1 - x²)dx = 4 [x - x³/3]? ¹
= 4 (1 - 1/3) = 8/3 sq. units

Let y = (ex)?
ln y = x ln (ex) = x [1 + ln x]
(1/y) (dy/dx) = (1) (1 + ln x) + x (1/x) = 2 + ln x
⇒ dy = (ex)? (2 + ln x)dx
∫? ² (ex)? (2 + log? x)dx = [ (ex)? ]? ² = (2e)² - (1e)¹ = 4e² - e

For concentration cell E? cell = 0
Anode: Cu (s) → Cu²? (aq)?
Cathode: Cu²? (aq)? → Cu (s)
Overall: Cu²? (aq)? → Cu²? (aq)?
As ΔG = −nFEcell
If ΔG = -ve than Ecell is positive
Ecell = E? cell − (0.059/2) log (C? /C? )
Ecell = − (0.059/2) log (C? /C? )
Ecell > 0 ⇒ C? > C?

Relative lowering in vapour pressure depends on no. of mole of solute greater the no. of mole of solute greater in RLVP and smaller will be vapour pressure. So order of vapour pressure is B > C > A.

Clearly, ∫ [0 to n] {x}dx = n/2
∫ [0 to n] [x]dx = 1 + 2 + 3 . . . n − 1
= n (n-1)/2
∴ (n (n-1)/2)² = n/2 {10n (n-1)}
Solving, n = 21

The chemical structure shown is 5-Amino-4- (hydroxymethyl)-2-nitrobenzene carbaldehyde.

Let a? = xî + y? + zk
î * (a? * î) = (î·î)a? – (a? ·î)î = y? + zk
Similarly? * (a? *? ) = xî + zk and k? * (a? * k? ) = xî + yk
|î * (a? * î)|² + |? * (a? *? )|² + |k? * (a? * k? )|²
= |y? + zk|² + |xî + zk|² + |xî + y? |² = 2|a? |² = 2 (9) = 18

Absolute refractive index is the ratio of speed of light in air/vacuum to speed of light in any medium. It is closely related to refractive index, except that it is relative to vacuum and not to any other medium. This is different from relative refractive index because it is the ratio of speed of light in one medium compared to the speed of light in another medium.