Maths

λ=- (sin? θ+cos? θ) = - (sin²θ+cos²θ)²-2sin²θcos²θ)
λ = - (1-½sin²2θ) = ½sin²2θ-1
sin²2θ ∈
λ ∈ [-1, -1/2]

A? A = I
⇒ a²+b²+c²=1 and ab+bc+ca=0
Now, (a+b+c)²=1 ⇒ a+b+c=±1
So, a³+b³+c³-3abc = (a+b+c) (a²+b²+c²-ab-bc-ca) = (±1) (1-0)=±1
⇒ 3abc = 2±1 = 3,1
⇒ abc = 1, 1/3

1. Let t? denotes r+1th term of (? x? +? x? )¹?
   t? = ¹? C? (? x? )¹? (? x? )? = ¹? C? ¹? x? ¹?
   If t? is independent of x
   90-15r=0? r=6
   This differs from the solution.
   Let's follow the solution's powers.
   (10-r)/9 - r/6 = 0? r=4
   maximum value of t? is 10K (given)
   ? ¹? C? is maximum
   By AM? GM (for positive numbers)
   (? ³/2+? ³/2+? ²/2+? ²/2)/4? (? /16)¹/?
   ? ? ? 16
   So, 10K = ¹? C?16
   ? K=336

∫? ² |x-1|-x|dx
Let f (x)=|x-1|-x|
= {|1-2x|, x≤1; 1, x≥1}
A = 1/2+1=3/2
∫? ¹/² (1-2x)dx+∫? /? ¹ (2x-1)+∫? ²1dx
= [x-x²]? ¹/²+ [x]? ² = 3/21dx
= [x-x²]? ¹/²+ [x]? ² = 3/2

Circle x²+y²-2x-4y+4=0
⇒ (x-1)²+ (y-2)²=1
Centre: (1,2) radius=1
line 3x+4y-k=0 intersects the circle at two distinct points.
⇒ distance of centre from the line < radius
⇒ |3*1+4*2-k|/√ (3²+4²) < 1
⇒ |11-k|<5
⇒ 6⇒ k∈ {7,8,9, .,15} since k∈I
Number of K is 9.

lim (x→1) (x+x²+.+x? -n)/ (x-1) = 820
⇒ lim (x→1) (x-1)/ (x-1)+ (x²-1)/ (x-1)+.+ (x? -1)/ (x-1) = 820
⇒ 1+2+.+n = 820
⇒ n (n+1)=2*820
⇒ n (n+1)=40*41
Since n∈N, so n=40

|a|=|b|=|c|=1
|a-b|²+|a-c|²=8
⇒|a|²+|b|²-2a.b+|a|²+|c|²-2a.c=8
⇒4-2 (a.b+a.c)=8
⇒a.b+a.c=-2
|a+2b|²+|a+2c|²
=|a|²+4|b|²+4a.b+|a|²+4|c|²+4a.c
=10+4 (a.b+a.c)
=10-8=2

MOTHER
1? E
2? H
3? M
4? O
5? R
6? T
So position of word MOTHER in dictionary
2*5!+2*4!+3*3!+2!+1
=240+48+18+2+1
=309

Kindly consider the following figure

|x|<1, |y|<1, x? y
(x+y) + (x²+xy+y²) + (x³+x²y+xy²+y³) + .
By multiplying and dividing by x-y:
(x²-y²)+ (x³-y³)+ (x? -y? )+.)/ (x-y)
= (x²+x³+x? +.)- (y²+y³+y? +.)/ (x-y)
= (x²/ (1-x)- (y²/ (1-y)/ (x-y)
= (x²-y²-xy (x-y)/ (1-x) (1-y) (x-y)
= (x+y-xy)/ (1-x) (1-y)                                         

Let B? be the event where Box-I is selected and B? →where box-II selected
P (B? )=P (B? )=1/2
Let E be the event where selected card is non prime.
For B? : Prime numbers: {2,3,5,7,11,13,17,19,23,29}
For B? : Prime numbers: {31,37,41,43,47}
P (E)=P (B? )*P (E/B? )+P (B? )P (E/B? )
= 1/2*20/30+1/2*15/20
Required probability:
P (B? /E) = (P (E/B? )P (B? )/P (E) = (1/2*20/30)/ (1/2*20/30+1/2*15/20) = (2/3)/ (2/3+3/4) = 8/17