Class 12th

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.

f (x) = ∫? [y]dy. For x∈ [n, n+1), [y]= [x]=n.
f (x) = Σ (k=0 to n-1) ∫? ¹ k dy + ∫? n dy = Σk + n (x-n).
f (x) is continuous at integers. f' (x)=n= [x]. Not differentiable at integers.

Here are a few useful applications of this concept in our day to day lives:

• Preservation and Cooking of Food
• Chemical Manufacturing
• Pharmaceuticals
• Thermometer
• Biology Enzymes
• Electrical Appliances

Endothermic reaction is a reaction which absorbs heat energy from the surrounding, making the environment cooler. The products have higher enthalpy than the reactants in this reaction.

On the other hand, Exothermic reaction is a reaction which releases heat energy into the surrounding, making the environment warmer. The products will have lower enthalpy than the reactants in this reaction.