Class 12th

While approaching: v = v? (c / (c - vcosθ)
While receding: v = v? (c / (c + vcosθ)

Required area = 4 [∫? ¹/² 2y²dy + ½*½*1] = 5/6.

α+β=64; αβ=256=2?
(α³/β? )¹/? + (β³/α? )¹/? = (α+β)/ (αβ)? /? = 64/32=2.

I = ∫ (π/24 to 5π/24) dx/ (1+³√tan2x). Using King's rule.
2I = ∫ (π/24 to 5π/24) dx = 4π/24=π/6. I=π/12.

dy/dx - 1 = xe^ (y-x). Let y-x=t. dt/dx = xe? e? dt=xdx.
-e? = x²/2+C. y (0)=0⇒t=0⇒-1=C.
-e^ (x-y) = x²/2-1. y=x-ln (1-x²/2).
y'=1+x/ (1-x²/2)=0 ⇒ x=-1. Min value at x=-1.
y (-1)=-1-ln (1/2) = -1+ln2. This differs from solution.

f' (x) = 12sin³xcosx+30sin²xcosx+12sinxcosx = 3sin2x (2sin²x+5sinx+2) = 3sin2x (2sinx+1) (sinx+2).
In [-π/6, π/2], sinx+2>0. 2sinx+1>0 except at x=-π/6. sin2x>0 for x∈ (0, π/2), <0 for x (-/6,0).
So f' (x)<0 on (-/6,0) (decreasing) and f' (x)>0 on (0, π/2) (increasing).

Successful collisions help in the formation of new products, whereas in unsuccessful collisions the molecules just bounce back without causing any reaction. In such cases, the reaction fails and no new product or bond is formed.

For a collision to happen, the molecules must collide with each other at a high frequency. But the result can still lead to disappointment. This is because even if molecules collide effectively, they also need to be aligned properly in symmetry. Only this can help in the formation of new bonds. Hence, the molecular orientation is considered important in he concept of collision theory.

Vectors are coplanar. Determinant is zero. Row operations.
This leads to 2a=b+c.