Past Paper 2025 12th Class Karachi Board Mathematics Subjective (Science Pre-Engineering Group)

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SECTION 'B' (Short-Answer Questions) (40)

NOTE: Attempt any Ten part questions in all. All questions carry equal marks. (i.e. 4 marks of each part).

2.

(i) If f(x) = x+13 and g(x) = 3x + 5 are two functions then show that (f o g)−1 = g−1 o f−1.

(ii) Evaluate any one of the following:

*limx→0 1−cos3xsin2x

*limx→e ln x−1x−e

(iii) Find the derivative of f(x) = sin x by ab-initiol/ first principle method.

(iv) Obtain first four terms of Taylor's series for f(x) = ln x centered at a = 1.

(v) Show that the function r⃗(t) = sin2t î + tan t ĵ + 1t k̂ is continuous at t = π4.

(vi) Compute the definite integral ∫π/2π cos x dx by using basic properties.

(vii) Find the area, above x-axis under the curve y = 3x2 + 2 between the ordinates x = 1 and x = 2.

(viii) Find the equation of the line which passes through (5, 6) and y-intercept is twice that of x-intercept.

(ix) If A(2, 5), B(3, 7) and C(0, 8) are the vertices of a triangle then find equation of median through A.

(x) Find the equation of circle containing the points (0, 0), (0, 3) and (−4, 0).

(xi) Find the equation of parabola whose vertex and focus are (0, 0) and (0, −2) respectively.

(xii) Find the eccentricity of ellipse if axes are 32 and 24 respectively.

(xiii) Show that y = Ae2x + Be3x is the general solution of d2ydx2 − 5dydx + 6y = 0.

(xiv) Verify Euler's theorem for the homogentous function f(x, y) = cos (x/y).

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:

*∫ 5x−2(x−3)(x+7) dx (by using partial fraction)

*∫π/6π/2 x sin 2x dx (by using integration by parts)

*∫ x2√(3 − x2) dx (by using trigonometric substitution)

4. Differentiate any two of the following:

*y = tan−1 sin 2x1+cos 2x

*y = x cosh−1 x − √(x2 − 1)

*x √(1 + y) + y √(1 + x) = 0

5. Find the maximum and minimum value of the f(x) = x3 − 9x2 + 15x + 3

6. If a point P(k, 7) divides the line segment joining A(8, 9) and B(1, 2) in a ratio m:n then find ratio m:n and value of k.

7. Find the condition that the line 3x + 4y = c may touch the circle x2 + y2 = 8x.

8. Find foci, eccentricity, vertices and latus rectum of hyperbola (x−3)264 − (y+4)236 = 1

9. Solve the differential equation: (x2 + y2) dx = 2xy dy

—OR— The population of a certain town is directly proportional to the square root of the present population at any time initially is 20,000. How much the population after 10 years and after how much time the population be doubled.

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