To help you understand the logic behind the math, here are the step-by-step explanations for the key questions from the mock paper.

Section A: Key MCQs

1. HCF of $a = x^3 y^2$ and $b = xy^3$

  • Concept: HCF is the product of the lowest power of each common prime factor.

  • For $x$: The powers are $3$ and $1$. Lowest is $x^1$.

  • For $y$: The powers are $2$ and $3$. Lowest is $y^2$.

  • Result: $xy^2$.

3. No solution for $x + 2y = 3$ and $5x + ky + 7 = 0$

  • Condition: For no solution (parallel lines), $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

  • $\frac{1}{5} = \frac{2}{k}$

  • Cross-multiply: $k = 10$.

12. Perimeter and Area of a circle are numerically equal

  • Equation: $2\pi r = \pi r^2$.

  • Divide both sides by $\pi r$: $2 = r$.

  • Result: $2$ units.

Section B & C: Logical Steps

24. Finding A and B from Trig Ratios

  • $\sin(A - B) = \frac{1}{2} \implies A - B = 30^\circ$ (since $\sin 30^\circ = 1/2$)

  • $\cos(A + B) = \frac{1}{2} \implies A + B = 60^\circ$ (since $\cos 60^\circ = 1/2$)

  • Adding the two equations: $2A = 90^\circ \implies A = 45^\circ$.

  • Substituting $A$: $45^\circ + B = 60^\circ \implies B = 15^\circ$.

26. Relationship between Zeroes and Coefficients

  • For $x^2 + 7x + 10$, we split the middle term: $x^2 + 5x + 2x + 10 = 0$.

  • $(x+5)(x+2) = 0 \implies$ Zeroes ($\alpha, \beta$) are $-5$ and $-2$.

  • Verification:

    • Sum $(\alpha + \beta) = -5 + (-2) = -7$. Formula: $-b/a = -7/1 = -7$. (Matches)

    • Product $(\alpha \beta) = (-5)(-2) = 10$. Formula: $c/a = 10/1 = 10$. (Matches)

Section D: Long Answer Concepts

32. Motor Boat (Upstream/Downstream)

  • Let speed of stream = $x$ km/h.

  • Speed Upstream = $(18 - x)$; Speed Downstream = $(18 + x)$.

  • Time Equation: $\text{Time Upstream} - \text{Time Downstream} = 1 \text{ hour}$.

  • $\frac{24}{18-x} - \frac{24}{18+x} = 1$.

  • Solving the quadratic equation $x^2 + 48x - 324 = 0$ gives $x = 6$ (speed cannot be negative).

34. Surface Area of a Tent

  • Total Canvas = CSA of Cylinder + CSA of Cone.

  • $\text{CSA Cylinder} = 2\pi rh = 2 \times \pi \times 2 \times 2.1 = 8.4\pi$.

  • $\text{CSA Cone} = \pi rl = \pi \times 2 \times 2.8 = 5.6\pi$.

  • $\text{Total Area} = 14\pi = 14 \times \frac{22}{7} = 44$ $m^2$.

  • Cost = $44 \times 500 = 22,000$.

Section E: Case Study Logic

38. TV Production (Arithmetic Progression)

  • $a_3 = 600 \implies a + 2d = 600$

  • $a_7 = 700 \implies a + 6d = 700$

  • Subtracting gives $4d = 100 \implies d = 25$ (yearly increase).

  • $a + 2(25) = 600 \implies a = 550$ (1st year production).

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