Class 12th

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.

This happens because although they have the same temeprature constant, some of their other properties can also vary such as activation energies, surface area of the reaction, concentration of the reactant, etc. As a result, the rate of a reaction comes out to be different for the reactions because of their dependence on these factors.

L: x/1=y/0=z/-1. Let a point on L be (r,0, -r).
Direction ratio of PN is (r-1, -2, -r+1).
PN is perpendicular to L, so (r-1) (1)+ (-2) (0)+ (-r+1) (-1)=0 ⇒ 2r-2=0 ⇒ r=1. N (1,0, -1).
Let Q be (s,0, -s). Direction ratio of PQ is (s-1, -2, -s+1).
PQ is parallel to plane x+y+2z=0, so normal to plane is perp to PQ.
(s-1) (1)+ (-2) (1)+ (-s+1) (2)=0 ⇒ s-1-2-2s+2=0 ⇒ -s-1=0 ⇒ s=-1. Q (-1,0,1).
PN = (0, -2,0). PQ= (-2, -2,2).
cosα = |PN.PQ|/|PN|PQ| = 4/ (2√12) = 1/√3.