question of the day cbse
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Samadhan Academy
Mathematics Excellence Series: Step-by-Step Solutions
1. Solve the Rational Equation
Solve for \(x\): \( \); where \( x \neq -4, 7 \)
Step 1: Find a common denominator:
\( \frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30} \)
\( \frac{(x - 7) - (x + 4)}{(x + 4)(x - 7)} = \frac{11}{30} \)
Step 2: Simplify the numerator:
\( \frac{-11}{x^2 - 3x - 28} = \frac{11}{30} \)
\( \frac{-11}{x^2 - 3x - 28} = \frac{11}{30} \)
Step 3: Cross-multiply and solve the quadratic:
\( x^2 - 3x + 2 = 0 \)
\( x^2 - 3x + 2 = 0 \)
Answer: \( x = 1 \) or \( x = 2 \)
2. Linear Equations: Infinitely Many Solutions
Find \(k\) for: \( kx + y = k^2 \) and \( x + ky = 1 \)
Condition: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \implies \frac{k}{1} = \frac{1}{k} = \frac{k^2}{1} \)
From \( k^2 = 1 \), we get \( k = \pm 1 \). Verifying with the third ratio, only \( k = 1 \) satisfies all parts.
Answer: \( k = 1 \)
3. Geometry: Surface Area of a Cone
Assertion (A): CSA of a cone (r=3.5, l=4) is 44 sq cm.
Reason (R): CSA formula is \( \pi rl \).
Calculation: \( CSA = \frac{22}{7} \times 3.5 \times 4 = 11 \times 4 = 44 \).
Conclusion: Both (A) and (R) are true; (R) is the correct explanation.
4. Y-axis Intersection
Equation: \( 2y - x = 4 \)
Set \( x = 0 \): \( 2y - 0 = 4 \implies y = 2 \).
Answer: (C) (0, 2)
5. Graphical Representation
Lines: \( 8x - 4y + 12 = 0 \) and \( 2x - y + 5 = 0 \)
Compare ratios: \( \frac{8}{2} = 4 \), \( \frac{-4}{-1} = 4 \), \( \frac{12}{5} = 2.4 \).
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines never meet.
Answer: (C) Parallel
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