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FBISE Examinations Mathematics (HSSC) Syllabus

The Mathematics (HSSC) syllabus for FBISE Examinations in full: 8 chapters and 25 topics, every one listed below, plus 50 Mathematics (HSSC) flashcards with answers.

8Chapters
25Topics
50Flashcards

Mathematics (HSSC) syllabus: all 8 chapters

Expand any chapter to see its topics and sub-topics.

  1. Functions and Limits

     3 topics
    • Types of Functions
    • Limit of a Function
    • Continuity

  2. Differentiation

     4 topics
    • Derivative from First Principle
    • Rules of Differentiation
    • Higher Order Derivatives
    • Maxima and Minima

  3. Integration

     3 topics
    • Indefinite Integrals
    • Definite Integrals
    • Area Under the Curve

  4. Trigonometric Identities and Equations

     3 topics
    • Sum and Difference Formulae
    • Double and Half Angle Identities
    • Solution of Trigonometric Equations

  5. Matrices and Determinants

     3 topics
    • Algebra of Matrices
    • Determinants and Properties
    • Solution of Linear Equations

  6. Conic Sections

     3 topics
    • Circle
    • Parabola
    • Ellipse and Hyperbola

  7. Vectors

     3 topics
    • Vectors in Plane and Space
    • Scalar and Vector Product
    • Scalar Triple Product

  8. Sequences and Series

     3 topics
    • Arithmetic Progression
    • Geometric Progression
    • Sum to Infinity

FBISE Examinations Mathematics (HSSC) flashcards — 50 questions with answers

  1. What is a one-to-one (injective) function?

    A function f in which distinct elements of the domain map to distinct elements of the codomain: if f(x1)=f(x2) then x1=x2. No two different inputs share the same output.

  2. What is an onto (surjective) function?

    A function f: A→B in which every element of the codomain B is the image of at least one element of A. The range equals the codomain.

  3. What is a bijective function?

    A function that is both one-to-one (injective) and onto (surjective). It establishes a one-to-one correspondence between domain and codomain, and is invertible.

  4. Define an even function and a key symmetry it has.

    A function with f(-x)=f(x) for all x in its domain. Its graph is symmetric about the y-axis. Example: f(x)=x^2, cos x.

  5. Define an odd function and a key symmetry it has.

    A function with f(-x)=-f(x) for all x in its domain. Its graph is symmetric about the origin. Example: f(x)=x^3, sin x.

  6. What is the formal (epsilon-delta intuition) meaning of lim x→a f(x)=L?

    As x gets arbitrarily close to a (from both sides, but x≠a), the value f(x) gets arbitrarily close to L. Formally: for every ε>0 there is a δ>0 such that |f(x)-L|<ε whenever 0<|x-a|<δ.

  7. State the value of lim x→0 (sin x)/x (x in radians).

    lim x→0 (sin x)/x = 1.

  8. State the value of lim x→0 (1-cos x)/x and lim x→0 (1-cos x)/x^2.

    lim x→0 (1-cos x)/x = 0, and lim x→0 (1-cos x)/x^2 = 1/2.

  9. State the limit definition of the number e involving (1+1/n)^n.

    lim n→∞ (1+1/n)^n = e ≈ 2.718. Equivalently lim x→0 (1+x)^(1/x) = e.

  10. What is lim x→0 (a^x - 1)/x?

    lim x→0 (a^x - 1)/x = ln a. In particular, lim x→0 (e^x - 1)/x = 1.

  11. State the limit of (x^n - a^n)/(x - a) as x→a.

    lim x→a (x^n - a^n)/(x - a) = n·a^(n-1).

  12. What are the three conditions for a function f to be continuous at x=a?

    (1) f(a) is defined; (2) lim x→a f(x) exists; (3) lim x→a f(x) = f(a). All three must hold.

  13. What distinguishes a removable discontinuity from a jump (non-removable) discontinuity?

    Removable: the two-sided limit exists but does not equal f(a) (or f(a) undefined), so it can be 'fixed'. Jump: the left-hand and right-hand limits exist but are unequal, so no value of f(a) makes it continuous.

  14. State the definition of the derivative of f at x from first principles.

    f'(x) = lim h→0 [f(x+h) - f(x)] / h, provided the limit exists.

  15. Using first principles, what is the derivative of f(x)=x^2?

    f'(x) = lim h→0 [(x+h)^2 - x^2]/h = lim h→0 (2xh + h^2)/h = lim h→0 (2x + h) = 2x.

  16. Using first principles, what is the derivative of f(x)=√x?

    f'(x) = lim h→0 (√(x+h)-√x)/h = lim h→0 1/(√(x+h)+√x) = 1/(2√x).

  17. State the power rule for differentiation.

    d/dx (x^n) = n·x^(n-1) for any real number n.

  18. State the product rule for differentiation.

    d/dx [u·v] = u'·v + u·v', where u and v are differentiable functions of x.

  19. State the quotient rule for differentiation.

    d/dx [u/v] = (u'·v - u·v') / v^2, provided v ≠ 0.

  20. State the chain rule for differentiation.

    If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Equivalently dy/dx = (dy/du)(du/dx).

Show the other 30 Mathematics (HSSC) flashcards

  1. Give the derivatives of sin x, cos x, and tan x.

    d/dx(sin x)=cos x; d/dx(cos x)=-sin x; d/dx(tan x)=sec^2 x.

  2. Give the derivatives of sec x, cosec x, and cot x.

    d/dx(sec x)=sec x tan x; d/dx(cosec x)=-cosec x cot x; d/dx(cot x)=-cosec^2 x.

  3. Give the derivatives of e^x, a^x, and ln x.

    d/dx(e^x)=e^x; d/dx(a^x)=a^x ln a; d/dx(ln x)=1/x.

  4. What is a higher order derivative, and how is the second derivative written?

    A derivative obtained by differentiating a derivative repeatedly. The second derivative is d^2y/dx^2 or f''(x), obtained by differentiating f'(x) once more.

  5. What does the sign of the second derivative tell you about concavity?

    If f''(x)>0 the curve is concave up (convex); if f''(x)<0 the curve is concave down. A point where f'' changes sign is a point of inflection.

  6. What is a critical (stationary) point of a function?

    A point in the domain where f'(x)=0 or f'(x) does not exist. Local maxima and minima can only occur at critical points (for differentiable functions, where f'(x)=0).

  7. State the First Derivative Test for a maximum and a minimum.

    At a critical point x=c: if f' changes from + to - as x increases through c, it is a local maximum; if f' changes from - to +, it is a local minimum; no sign change means neither.

  8. State the Second Derivative Test for local extrema.

    If f'(c)=0: when f''(c)<0, f has a local maximum at c; when f''(c)>0, a local minimum at c; when f''(c)=0 the test is inconclusive.

  9. What are the steps to find the absolute maximum/minimum of a continuous function on a closed interval [a,b]?

    Find critical points in (a,b) where f'(x)=0 or is undefined, evaluate f at these points and at the endpoints a and b, then compare values: the largest is the absolute max and the smallest the absolute min.

  10. What is an indefinite integral (antiderivative)?

    ∫f(x)dx = F(x) + C, where F'(x)=f(x). It represents the family of all antiderivatives of f, and C is the arbitrary constant of integration.

  11. State the power rule for indefinite integration.

    ∫x^n dx = x^(n+1)/(n+1) + C, valid for n ≠ -1. For n=-1, ∫(1/x)dx = ln|x| + C.

  12. Give the integrals of sin x, cos x, and sec^2 x.

    ∫sin x dx = -cos x + C; ∫cos x dx = sin x + C; ∫sec^2 x dx = tan x + C.

  13. Give the integrals of e^x, a^x, and 1/x.

    ∫e^x dx = e^x + C; ∫a^x dx = a^x/ln a + C; ∫(1/x)dx = ln|x| + C.

  14. State the integration by parts formula.

    ∫u dv = uv - ∫v du. Used to integrate products; choose u so that du simplifies (often by the LIATE priority).

  15. What is the substitution method (u-substitution) used for in integration?

    To simplify ∫f(g(x))g'(x)dx by letting u=g(x), so du=g'(x)dx, transforming the integral into ∫f(u)du, which is easier to evaluate.

  16. State the Fundamental Theorem of Calculus for evaluating a definite integral.

    If F is an antiderivative of f on [a,b], then ∫(a→b) f(x)dx = F(b) - F(a).

  17. How does a definite integral differ from an indefinite integral?

    A definite integral ∫(a→b)f(x)dx has limits a and b and yields a definite number (with no constant C). An indefinite integral ∫f(x)dx yields a family of functions F(x)+C.

  18. State two key properties: the value of ∫(a→a)f(x)dx and the effect of swapping limits.

    ∫(a→a)f(x)dx = 0; and ∫(a→b)f(x)dx = -∫(b→a)f(x)dx (swapping limits changes the sign).

  19. State the additivity property of definite integrals over an interval.

    ∫(a→b)f(x)dx = ∫(a→c)f(x)dx + ∫(c→b)f(x)dx, for any c (typically a≤c≤b).

  20. How is the area under a curve y=f(x) above the x-axis from x=a to x=b found?

    Area = ∫(a→b) f(x) dx, provided f(x) ≥ 0 on [a,b].

  21. How do you handle the area when the curve f(x) lies below the x-axis on [a,b]?

    The definite integral is negative, so the area is |∫(a→b)f(x)dx| = -∫(a→b)f(x)dx. If the curve crosses the axis, split the interval at the crossing points and add the absolute values of each piece.

  22. What is the formula for the area bounded between two curves y=f(x) (upper) and y=g(x) (lower) from a to b?

    Area = ∫(a→b) [f(x) - g(x)] dx, where f(x) ≥ g(x) on [a,b].

  23. State the sum and difference formulae for sine.

    sin(A+B) = sin A cos B + cos A sin B; sin(A-B) = sin A cos B - cos A sin B.

  24. State the sum and difference formulae for cosine.

    cos(A+B) = cos A cos B - sin A sin B; cos(A-B) = cos A cos B + sin A sin B.

  25. State the sum and difference formulae for tangent.

    tan(A+B) = (tan A + tan B)/(1 - tan A tan B); tan(A-B) = (tan A - tan B)/(1 + tan A tan B).

  26. State the double angle formula for sin 2A.

    sin 2A = 2 sin A cos A.

  27. State the three forms of the double angle formula for cos 2A.

    cos 2A = cos^2 A - sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A.

  28. State the double angle formula for tan 2A.

    tan 2A = (2 tan A)/(1 - tan^2 A).

  29. State the half-angle formulas for sin(A/2) and cos(A/2).

    sin(A/2) = ±√[(1 - cos A)/2]; cos(A/2) = ±√[(1 + cos A)/2]. The sign depends on the quadrant of A/2.

  30. Express sin^2 A and cos^2 A in terms of cos 2A (power-reduction).

    sin^2 A = (1 - cos 2A)/2; cos^2 A = (1 + cos 2A)/2.

FBISE Examinations Mathematics (HSSC) FAQ

What is in the FBISE Examinations Mathematics (HSSC) syllabus?

FBISE Examinations Mathematics (HSSC) is split into 8 chapters: Functions and Limits, Differentiation, Integration, Trigonometric Identities and Equations, Matrices and Determinants, Conic Sections, Vectors and Sequences and Series. Together they contain 25 topics.

How many chapters are in FBISE Examinations Mathematics (HSSC)?

8 chapters, containing 25 topics. The largest chapter is Differentiation, with 4 topics.

Are there flashcards for FBISE Examinations Mathematics (HSSC)?

Yes — a 50-card Mathematics (HSSC) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.