Past Paper 2025 12th Class Karachi Board Mathematics Subjective (Science General Group)

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xiii)Obtain the differential equation by eliminating arbitrary constant from the relation y = C1ex + C2e-x.

xiv) The volume of the cone is given by formula V = 13πr2h.

Differentiate V w.r.t their independent variable. (where r and h are its independent variables).

SECTION “C” (Detailed Answer Questions)(40)

NOTE: Attempt any FIVE question from this section.

All questions carry equal marks.

3. Evaluate any two of the following:

*∫ 2x sec-1x dx

(by using integration by parts)

*∫ cos x dx / sin x (2 + sin x)

(by using partial fraction)

*∫ 2√3 x³ dx / √(x³+4)

(by using trigonometric substitution)

4. Differentiate any two of the following:

*y = tan (esin h-1x)

*y = sin-1 √(1 - cos x2)

*y = cos2x w.r.t sin2x

5. Find the right-angled triangle of maximum area with a hypotenuse of length h.

6. Find the equation of a line through the intersection of the lines 3x – 4y + 1 = 0, 5x + y – 1 = 0 and cutting off equal intercepts from the axes.

7. Find the equation of tangent(s) to the circle x2 + y2 = 25 which is perpendicular to 3x + 4y + 1 = 0.

8. Find foci, eccentricity, vertices and equation of directrices of the hyperbola 16(y + 2)2 – 9(x – 2)2 = 144.

9. Solve the differential equation: (x – y – 1)dx – (x – y – 5)dy = 0

—OR—Solve the problem by differential equation, a ball is thrown upward vertically with velocity 49m/s and g = –9.8 m/s².

Find: i) time when the body at maximum height,

ii) height with t = 3 seconds

iii) maximum height.

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