FBISE Examinations Mathematics (HSSC) Syllabus

Mathematics (HSSC) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (HSSC) in FBISE Examinations, not a summary of it.

1. 1

   Functions and Limits

   3 topics

   • Types of Functions
   • Limit of a Function
   • Continuity
2. 2

   Differentiation

   4 topics

   • Derivative from First Principle
   • Rules of Differentiation
   • Higher Order Derivatives
   • Maxima and Minima
3. 3

   Integration

   3 topics

   • Indefinite Integrals
   • Definite Integrals
   • Area Under the Curve
4. 4

   Trigonometric Identities and Equations

   3 topics

   • Sum and Difference Formulae
   • Double and Half Angle Identities
   • Solution of Trigonometric Equations
5. 5

   Matrices and Determinants

   3 topics

   • Algebra of Matrices
   • Determinants and Properties
   • Solution of Linear Equations
6. 6

   Conic Sections

   3 topics

   • Circle
   • Parabola
   • Ellipse and Hyperbola
7. 7

   Vectors

   3 topics

   • Vectors in Plane and Space
   • Scalar and Vector Product
   • Scalar Triple Product
8. 8

   Sequences and Series

   3 topics

   • Arithmetic Progression
   • Geometric Progression
   • Sum to Infinity

Mathematics (HSSC) flashcards for FBISE Examinations

25 of 50 cards from the Mathematics (HSSC) deck — real questions with worked 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).

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

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