Master Class 10 Math Ex 1.3 The Substitution Secret and MCQs
Ex 1.3 Unlocked: Solve Complex Equations Like a Pro + Practice MCQs
Mastering the Art of Substitution - Full Step-by-Step Guide
Classroom Talk: The Hidden Truth of Ex 1.3
Assalam-o-Alaikum Students! Have you ever met someone wearing a mask? In Exercise 1.3, we are going to deal with equations that are wearing masks! At first glance, they don't look like quadratic equations because they have powers like $x^4$ or are hidden in fractions, but they follow the same rules.
Our secret weapon today is the "Substitution Method." We simplify these scary-looking equations by temporary naming them 'y'. Think of it like a nickname. Once we solve for 'y', we switch back to 'x' and get our final answer. It’s like solving a puzzle!
I have broken down the 5 major types into small, bite-sized steps. By the end of this page, you’ll be solving these "monster" equations like a pro. Ready? Let's go!
Type 1: Solve 2x⁴ − 11x² + 5 = 0
Step: Let $x^2 = y$. The equation becomes $2y^2 - 11y + 5 = 0$.
Solving for y: $y = 5, y = 1/2$.
Back-Substitute: $x^2 = 5 \rightarrow x = \pm \sqrt{5}$ and $x^2 = 1/2 \rightarrow x = \pm 1/\sqrt{2}$.
MCQ 1: What is the substitution for x⁴? (A) y (B) y² Answer
(B) y²
MCQ 2: How many roots will this equation have? (A) 2 (B) 4 Answer
(B) 4
Type 2: Solve 5x^{1/2} = 7x^{1/4} − 2
Step: Let $x^{1/4} = y$. Then $x^{1/2} = y^2$.
Equation: $5y^2 - 7y + 2 = 0$.
Solving: $y = 1, y = 2/5$. Back-substitute to find $x$.
MCQ 3: Which power is smaller? (A) 1/2 (B) 1/4 Answer
(B) 1/4
MCQ 4: To find x, we take the power ___ of y. (A) 4 (B) 2 Answer
(A) 4
Type 3: Reciprocal Eq (x⁴ − 2x³ − 2x² + 2x + 1 = 0)
Step: Divide by $x^2$. We get $(x^2 + 1/x^2) - 2(x - 1/x) - 2 = 0$.
Substitution: Let $x - 1/x = y$.
MCQ 5: Reciprocal eq remains unchanged if x is replaced by: (A) 1/x (B) -x Answer
(A) 1/x
MCQ 6: After dividing x³ by x², we get: (A) x (B) 1/x Answer
(A) x
Type 4: Solve 4.2^{2x+1} − 9.2^x + 1 = 0
Step: Let $2^x = y$. Equation becomes $8y^2 - 9y + 1 = 0$.
Roots: $y = 1, y = 1/8 \rightarrow x = 0, x = -3$.
MCQ 7: In exponential equations, variables are in: (A) Base (B) Power Answer
(B) Power
MCQ 8: $2^x = 1$ means x is: (A) 0 (B) 1 Answer
(A) 0
Type 5: (x+1)(x+3)(x-5)(x-7) = 192
Step: Group them: $(x+1)(x-5)$ and $(x+3)(x-7)$ because $1-5 = 3-7 = -4$.
Result: $(x^2-4x-5)(x^2-4x-21) = 192$. Let $x^2-4x = y$.
MCQ 9: We group factors based on constant ___: (A) Sum (B) Product Answer
(A) Sum
MCQ 10: $x^2-4x$ is called: (A) Linear (B) Quadratic part Answer
(B) Quadratic part
