Mathematics IB — Important Questions
High-frequency SAQ and LAQ questions from TS Inter 1st Year Mathematics IB — Locus, Straight Lines, Pair of Lines, Limits, Differentiation, and Applications. Based on TGBIE exam patterns 2025-26.
Ch. 1 & 2 Locus & Transformation — Important Questions
- 1Find the equation of locus of a point equidistant from A(2, 3) and B(−1, 4).
- 2Find the locus of a point P such that PA²/PB² = 3/4 where A = (1,0) and B = (3,0).
- 3When the origin is shifted to (2, −3), the equation of a curve becomes x² + y² = 9. Find the original equation.
- 4Find the transformed equation of x² + y² − 4x + 6y − 12 = 0 when the origin is shifted to (2, −3).
Ch. 3 The Straight Line — Important Questions (SAQ & LAQ)
- 1Find the equation of the straight line passing through (−2, 3) with slope 4. Also find x-intercept and y-intercept.
- 2A straight line makes intercepts a and b on the axes such that a + b = 5 and ab = 6. Find the equation.
- 3Find the equation of the line passing through the intersection of 3x + 2y = 5 and x − 2y = 3 and perpendicular to 4x − 5y = 0.
- 4Show that the lines 3x − 4y + 5 = 0, 4x + 3y − 6 = 0, 3x − 4y − 5 = 0, and 4x + 3y + 4 = 0 form a rhombus. Find its area.
- 5The distance of a point (a, b) from the line 3x − 4y = 0 is 5. Find the values of a and b if a − b = 3.
Ch. 4 Pair of Straight Lines — Important Questions (LAQ)
- 1Show that the pair of lines ax²+2hxy+by²=0 are perpendicular to each other if a + b = 0.
- 2If the equation ax²+2hxy+by²+2gx+2fy+c=0 represents a pair of parallel lines, prove that h² = ab and find the distance between them.
- 3Find the angle between the pair of lines represented by x² − 5xy + 4y² + x + 2y − 2 = 0.
- 4Show that the equation 2x² − 13xy − 7y² + x + 23y − 6 = 0 represents a pair of straight lines. Find their equations.
- 5Find p if the lines 3x² + 7xy + py² = 0 make equal angles with the coordinate axes.
Ch. 8 Limits — Important Questions
- 1Evaluate: lim(x→2) (x³−8)/(x²−4).
- 2Evaluate: lim(x→0) (√(1+x) − 1)/x.
- 3Evaluate: lim(x→0) (e^(3x) − 1)/x.
- 4Evaluate: lim(x→∞) (3x² + 5x + 2)/(2x² − 3x + 1).
- 5Show that f(x) = 2x + 3 is continuous at x = 2.
- 6Find k if f(x) = kx + 1 for x ≤ 5 and f(x) = 3x − 5 for x > 5 is continuous at x = 5.
Ch. 9 Differentiation — Important Questions
- 1Find dy/dx if y = (x² + 2x + 3)(3x − 1).
- 2Find the derivative of f(x) = (sin x − cos x)² from first principles.
- 3If y = sin(log x), find dy/dx.
- 4If x = a(θ − sinθ), y = a(1 − cosθ), find dy/dx.
- 5Find d²y/dx² if y = e^(ax)·sin(bx).
- 6If y = (tan⁻¹ x)², prove that (1+x²)²y₂ + 2x(1+x²)y₁ = 2.
Ch. 10 Applications of Derivatives — Important Questions (LAQ)
- 1Find the equations of tangent and normal to the curve y = x² − 4x + 2 at (4, 2).
- 2Find the angle between the curves y = x² and y = x³ at their intersection point (1,1).
- 3Verify Rolle's theorem for f(x) = x² − 5x + 6 on [2, 3].
- 4Using LMVT, show there exists c ∈ (1, 4) such that f'(c) = 1/2 where f(x) = √x.
- 5Find the maximum and minimum values of f(x) = 2x³ − 3x² − 36x + 10 on [−3, 5].
- 6A closed cylindrical can is to hold 1 litre of liquid. Find the dimensions that minimise the total surface area.