Science MCQs

Trigonometry Problem: Prove the Identity Question: Prove that: sec x = 2 √ 2 + √ 2 + 2 cos 4x Solution / Proof: We will solve the R.H.S. (Right Hand Side) and prove that it is equal to sec x . Step 1: Simplify the innermost square root First, look at the term inside the inner root: (2 + 2 cos 4x) . We can take '2' as common: = 2 (1 + cos 4x) Formula Used: 1 + cos 2θ = 2 cos² θ So, for 4x, it becomes: = 2 (2 cos² 2x) = 4 cos² 2x Now, putting this back into the square root: √(4 cos² 2x) = 2 cos 2x Step 2: Update the expression Now our main expression becomes: = 2 √ 2 + 2 cos 2x ...

Calculus Problem: Differentiation & Integration Question: If y = x 5 + 3x 2 , find the derivative dy/dx and the integral ∫ y dx . Solution: Given Function: y = x 5 + 3x 2 Step 1: Differentiation Differentiating with respect to x : dy/dx = (d/dx)(x 5 ) + (d/dx)(3x 2 ) Using the power rule (d/dx)(x n ) = nx n-1 : ∴ dy/dx = 5x 5-1 + 3(2x 2-1 ) = 5x 4 + 3(2x) Answer: dy/dx = 5x 4 + 6x Step 2: Integration Now, integrating with respect to x : ∫ y dx = ∫ (x 5 + 3x 2 ) dx Separating the terms: = ∫ x 5 dx + ∫ 3x 2 dx Using the integration rule ∫ x n dx = x n+1 / (n+1) : = x 5+1 / (5+1) + 3 · x 2+1 / (2+1) = x 6 /6 + 3 · x 3 /3 + C Here, the 3 cancels out. Answer: ∫ y dx = x 6 /6 + x 3 + C *(Where C is the constant of integration)*