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AZHAR UNSOLVED
20 Equation of latus rectum of the parabola x² = 4ay is:
Ⓐ x = a
Ⓑ y = a
Ⓒ x = -a
Ⓓ y = -a
UNSOLVED
Mathematics
Inter Part-II, 2025
Lahore Board
Group - II
Time: 2.30 hrs.
Essay Type
Marks : 80
SECTION - I
Q.2 Write short answers to any EIGHT (8) questions: 16
i Show that the parametric equations x = a sec θ , y = b tan θ represent the equation of hyperbola x²/a² - y²/b² = 1
ii Without finding the inverse, state the domain and range of f⁻¹, where f(x) = 2 + √(x - 1)
iii Define continuity of a function at a number C.
iv Evaluate Lim θ→0 (1 - cos θ)/sin θ
v Find fog(x) and gof(x), when f(x) = 2x + 1 and g(x) = 3/(x - 1), x ≠ 1
vi Find dy/dx, if x = at² and y = 2at
vii Prove that d/dx (tan⁻¹ x) = 1/(1 + x²), ∀x ∈ R
viii If x = y sin y, find dy/dx
ix Find y₂, if y = ln ((2x + 3)/(3x + 2))
x Apply the Maclaurin Series expansion to prove that cos x = 1 - x²/2! + x⁴/4! - x⁶/6! + - - -
xi Determine the intervals in which f is increasing or decreasing, f(x) = cos x; x ∈ (-π/2, π/2)
xii Differentiate x² - 1/x² w.r.t x⁴
Q.3 Write short answers to any EIGHT (8) questions: 16
i Find δy of the function y = x² - 1 when x changes from 3 to 3.02.
ii Evaluate ∫ (sin θ dθ)/(1 + cos² θ)
iii Find ∫ x sin x dx
iv Evaluate the integral ∫ dx/(x ln x)
v Find the definite integral ∫ (x dx)/(x² + 2) from 1 to e^(tan⁻¹ x)
vi Evaluate ∫ e^(tan⁻¹ x) dx/(1 + x²)
vii Solve the differential equation (x - 1)dx + y dy = 0
viii Define inclination of a line.
ix Check whether the lines 4x - 3y - 8 = 0, 3x - 4y - 6 = 0 and x - y - 2 = 0 are concurrent.
x Find an equation of line through A (-6, 5) having slope 7.
xi Convert 15y - 8x + 3 = 0 into normal form of line.
xii Define homogeneous function.
Q.4 Write short answers to any NINE (9) questions: 18
i Write the procedure for graphing a linear inequality in two variables.
ii Draw a graph of linear inequality x + 2y < 6
iii Show that the equation 5x² + 5y² + 24x + 36y + 10 = 0 represents a circle. Also find its centre and radius.
iv Check the position of point (5, 6) with respect to circle 2x² + 2y² + 12x - 8y + 1 = 0.
v Write an equation of parabola with focus (1, 2) and vertex (3, 2).
vi Find foci and eccentricity of ellipse 9x² + y² = 18
vii Find the centre and directrix of hyperbola x²/4 - y²/9 = 1
viii Find a unit vector in the direction of the vector y = 2i + 6j
ix Find the vector from the point A to the origin where AB = 4i - 2j and B is the point (-2, 5)
x Find a and b so that the vectors 3i - j + 4k and ai + bj - 2k are parallel.
xi Find the projection of a along b when a = 3i + j - k, b = 2i - j + k
xii Find a vector perpendicular to vectors a = 2i + j + k, b = 4i + 2j - k
xiii Differentiate between scalar quantity and vector quantity.
SECTION - II
Note: Attempt any THREE questions.
Q.5(a) If f(x) = { 3x if x ≤ -2
x² - 1 if -2 < x < 2
3 if x ≥ 2
Discuss continuity at x = 2 and x = -2
(b) Differentiate (x² + 1)/(x² - 1) w.r.t. (x - 1)/(x + 1)
Q.6(a) Find the extreme values of the function defined by f(x) = x³ - 6x² + 9x
(b) Evaluate the integral ∫ (5x + 8)/((x + 3)(2x - 1)) dx
Q.7(a) Evaluate ∫ from 0 to π/4 cos⁴ t dt
(b) Maximize f(x,y) = 2x + 3y subject to the constraints 2x + y ≤ 8, x + 2y ≤ 14, x ≥ 0, y ≥ 0
Q.8(a) Find a joint equations of the lines through the origin and perpendicular to the lines x² - 2xy tan α - y² = 0
(b) Find an equation of circle which passes through the points A (5,10), B (6, 9) and C (-2, 3)
Q.9(a) If a = 4i + 3j + k and b = 2i - j + 2k, find a unit vector perpendicular to both a and b. Also find the sine of angle between them.
(b) Show that an equation of the parabola with focus at (a cos α, a sin α) and directrix x cos α + y sin α + a = 0 is (x sin α - y cos α)² = 4a(x cos α + y sin α).
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