Class 12th

∫? ^ (π/2) sin³x e? sin²? dx
=∫? ^ (π/2) (1−cos²x)sinx e? (¹? cos²? )dx
=2∫? ^ (π/2) (1−cos²x)sinx e? (¹? cos²? )dx
Let cos²x=t⇒sin2xdx=−dt
=−2∫? (1−t)e? (¹? ) dt/ (−2cosx)
=1/e ∫? e? dt −∫? te? dt
=1/e [e? ]? − [te? −e? ]?
=2e−3∫? ¹ √t e? dt
⇒α=2, β=3
α+β=5

f' (x)=g (x)
f" (x)=g' (x)
⇒g (x).g' (x)=f' (x).f" (x)
f (x) has five roots
⇒f' (x) has atleast 4 roots.
And f" (x) has atleast of 3 roots
⇒g (x).g' (x)=0 has atleast 7 roots in (a, b)

General form Σ? ? C? (1/2)? < 1/2
Σ? ? C? < 2?
⇒? C? +? C? +.? C? < 128
⇒256− (? C? +? C? +.? C? ) < 128
⇒? C? +? C? +.? C? > 128
n≥5

A? =B? (i)
A³B²=A²B³. (ii)
Subtract (i) & (ii)
⇒A³ (A²−B²)=B³ (B²−A²)
⇒ (A²−B²) (A³+B³)=0
A²−B² is invertible matrix
∴A²−B²≠0
⇒A³+B³=0
∴? A³+B³? =0

The direction of the net magnetic force is always perpendicular to the plane of current-carrying conductors. The direction of the magnetic force is determined through the right-hand rule. 

The right-hand thumb rule states that if you wrap a wire in a way that the thumb points towards the flow of current. Then your curled fingers represent the direction of the field lines of the magnetic field

The "ampere" was defined based on the magnetic interaction. The scientific community used the magnetic force between two parallel current-carrying conductors placed at a distance. Below is the definition of 1 ampere, based on this concept.

The exact formula for the magnetic force per unit length between two parallel current-carrying conductors:

For more information, check: Force between parallel conductors

3 [Co (en)? Cl? ]Cl + 3AgNO? (excess) → 3AgCl↓ (ppt) + 3 [Co (en)? Cl? ] NO?
Secondary valency of complex = 6