Class 12th

Normal of plane = |î? k|
|1 0|
|0 1 -1|
n = -î+? +k
D.R.'s = -1,1,1
Plane => -1 (x-1)+1 (y-0)+1 (z-0) = 0
=> x-y-z-1=0
If (x, y, z) is foot of perpendicular of M (1,0,1) on the plane then
(x-1)/-1 = (y-0)/1 = (z-1)/1 = - (1-0-1-1)/3
x=4/3, y=-1/3, z=2/3
α+β+γ = 4/3 - 1/3 + 2/3 = 5/3

x-2y+5z=0
-2x+4y+z=0
-7x+14y+9z=0
2x (i)+ (ii) => z=0
=> x=2y
=> 15 ≤ x²+y²+z² ≤ 150
=> 15 ≤ 4y²+y² ≤ 150
=> 3 ≤ y² ≤ 30
=> y = ±2, ±3, ±4, ±5
=> 8 solutions

For (1,2) of y² = 4x => t=1, a=1
normal => tx+y = 2at + at³
=> x+y=3 intersect x-axis at (3,0)
y = e? => dy/dx = e?
tangent => y-e? = e? (x-c)
at (3,0) => 0-e? = e? (3-c) => c=4

f' (x) = k*x (x+1) (x-1) = k (x³-x)
=> f (x) = k (x? /4 - x²/2) + C
=> f (0) = C
=> f (x) = f (0)
=> k (x? -2x²)/4 + C = C
=> x² (x²-2)=0
=> x = {0, √2, -√2}

Isoenzymes are the different enzymes that catalyse the same biochemical reaction. However, they differ in amino acid composition and kinetic properties. Examples of isoenzymes are hexokinase and lactate dehydrogenase.