Maths

Family of planes: x+y+z+1+? (2x-y+z+3)=0. Parallel to line means dot product of normals is zero.

(80+1)? /8 = (¹C?80? +.+¹C? )/8. Remainder is 1/8.

S.D.=√ (Σ (x-a)²/n - (Σ (x-a)/n)²) = √ (a-1).

Kindly go through the solution

BONUS

The expression inside log is a GP sum: (1/3)/ (1-1/3) = 1/2.
log? (1/2) = 1/2.
The series S = 1+2/3+6/3²+.
S - S/3 = 1 + 4/3² + 4/3³ + . = 1 + (4/9)/ (1-1/3) = 1+2/3=5/3.
(2/3)S = 5/3 ⇒ S=5/2.
l = (5/2)/ (1/2)=5. l²=25.

Average speed = (f (t? )-f (t? )/ (t? -t? ) = a (t? +t? )+b.
Instantaneous speed = f' (t)=2at+b.
2at+b=a (t? +t? ) ⇒ t= (t? +t? )/2.

α, β are roots of x²+5√2x+10=0. α+β=-5√2, αβ=10.
P? + 5√2P? + 10P? = 0.
So P? +5√2P? =-10P? P? +5√2P? =-10P?
Expression = P? (-10P? )/P? (-10P? ) = 1.

α+110+54+30+β=584 ⇒ α+β=390.
Median=45. L=40, N=584, C=α+110+54=α+164, f=30, h=10.
45 = 40 + [ (292- (α+164)/30]*10 = 40 + (128-α)/3.
5 = (128-α)/3 ⇒ 15=128-α ⇒ α=113.
β = 390-113=277.
|α-β|=164.

|z|-Re (z)≤1 ⇒ √ (x²+y²)-x≤1 ⇒ √ (x²+y²)≤1+x ⇒ y²≤1+2x=2 (x+1/2).

S = {n ∈ N | [n i; 0 1] [a b; c d] = [a b; c d] ∀a, b, c, d ∈ R}
[na+ic, nb+id; c, d] = [a, b;c, d]
This must be an identity matrix. n=1. The question seems to have typos. Let's follow the solution logic.
The solution implies the matrix must be [1 0; 0 1] or [-1 0; 0 -1]. And n must be a multiple of 8.
2-digit multiples of 8 are 16, 24, ., 96. Total 11 numbers.