Class 12th

Solubility product of A? X = 4S? ³
Where S? is the solubility of salt A? X.
Solubility product of MX = S? ²
Where S? is the solubility of MX.
Given 4S? ³ = 4 * 10? ¹² ⇒ S? = 10? M
Given S? ² = 4 * 10? ¹² ⇒ S? = 2 * 10? M
So, S? / S? = 10? / (2 * 10? ) = 50

Solve sin? ¹ (3x/5) + sin? ¹ (4x/5) = sin? ¹x.
Using the formula sin? ¹a + sin? ¹b = sin? ¹ (a√ (1-b²) + b√ (1-a²):
sin? ¹ ( (3x/5)√ (1 - (4x/5)²) + (4x/5)√ (1 - (3x/5)²) ) = sin? ¹x
(3x/5) * √ (1 - 16x²/25) + (4x/5) * √ (1 - 9x²/25) = x
x * [ (3/5) * √ (25-16x²)/5 + (4/5) * √ (25-9x²)/5 - 1 ] = 0
So, x = 0 is one solution.
For the other part:
3√ (25-16x²) + 4√ (25-9x²) = 25
Let's check integer solutions. If x = 1:
3√ (9) + 4√ (16) = 33 + 44 = 9 + 16 = 25. So x = 1 is a solution.
If x = -1:
3√ (9) + 4√ (16) = 25. So x = -1 is a solution.
The solutions are x = 0, 1, -1.

Edge length in bcc, a? = 27 Å
Let, Edge length in fcc be a? Å
Now, the same element crystallises in bcc as well as fcc.
For bcc: 4r = √3 a? ⇒ r = (√3 / 4) a?
For fcc: 4r = √2 a? ⇒ r = a? / (2√2)
So, (√3 / 4) a? = a? / (2√2)
(√3 / 4) * 27 = a? / (2√2)
a? = 33.13 Å
The nearest integer is 33.

Antihistamines are antacids and antiallergics

Roasting is a process in which sulphur is removed as SO? gas from sulphide ores on heating in excess of oxygen.

Kindly go through the solution 

R-CONH? + Br? + 4NaOH → R-NH? + 2NaBr + Na? CO? + 2H? O
This reaction is the Hoffmann bromamide degradation, in which an amide is converted to a 1° amine.

The shortest distance D between two skew lines is given by the formula:
D = | (a? - a? ) ⋅ (b? x b? )| / |b? x b? |
Line L? : (x-1)/2 = (y-2)/3 = (z-4)/4
Line L? : (x-2)/3 = (y-4)/4 = (z-5)/5

Here, a? = I + 2j + 4k, b? = 2i + 3j + 4k
a? = 2i + 4j + 5k, b? = 3i + 4j + 5k

a? - a? = I + 2j + k
b? x b? = | I j k |
| 2 3 4 |
| 3 4 5 |
= I (15-16) - j (10-12) + k (8-9) = -i + 2j - k

D = | (i + 2j + k) ⋅ (-i + 2j - k)| / √ (-1)² + 2² + (-1)²)
= |-1 + 4 - 1| / √ (1 + 4 + 1)
= 2 / √6

Kindly consider the following Image 

The equation of the plane is given as x + y + z = 42. It is also mentioned that x³ + y³ + z³ = 3xyz.
From the identity, if x³ + y³ + z³ - 3xyz = 0, then x + y + z = 0 or x = y = z.
Given the expression:
3 + (x³ + y³ + z³ - 3xyz) / (xyz)²

Since x³ + y³ + z³ = 3xyz, the expression simplifies to:
3 + 0 = 3