# FSc Part 1 (KPK Boards)

Notes of FSc Part 1 of “A Textbook of Mathematics For Class XI” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.

Author: | Engr. Majid Amin |
---|---|

Type: | Solutions only |

Sender: | Muhammad Kareem |

Format: | PDF Scanned (Handwritten) |

## Chapter 01: Complex Numbers

#### Objectives

After reading this unit the students will be able to:

- know complex numbers, its conjugate and absolute value.
- understand algebraic properties of complex numbers.
- recongnize real and imaginary parts of different types of complex numbers.
- know the solution of simultaneous linear equations with complex co-efficients.
- write the polynomial $P(z)$ as product of linear factors.
- solve quadratic equations in complex variable with real co-efficients.

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## Chapter 02: Matrices and Determinants

#### Objectives

After reading this unit the students will be able to:

- know a matrix and its notations, order of a matrix and equality of two matrices.
- understand types of matrices, algebra of matrices and some properties of matrix addition and scalar multiplication.
- describe determinant of a square matrix and its evaluation using cofactors.
- know adjoint of a square matrix and use of adjoint method to calculate inverse of a square matrix.
- state and prove properties of determinants.
- know elementary row and column operations on matrices.
- recognize echelon and reduced echelon form of a matrix and rank of a matrix.
- solve a system of linear equations of both homogeneous and non-homogeneous equations.

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## Chapter 03: Vectors

#### Objectives

After reading this unit the students will be able to:

- differentiate between scalar and vector quantities.
- give geometrical representation of a vector in a space.
- know the fundamental defintion of vector using geometrical as well as analytical representation.
- use vector to prove simple theorems of descriptive geometry.
- recognize rectangular coordinate system in space.
- define unit vectors i, j and k.
- repeat all fundamental definitions of vector in plane for space.
- know properties of vector addition and cross or vector product.
- define scalar triple product of vectors.
- express scalar triple product of vectors in terms of components (determinantal form).

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## Chapter 4: Sequences

#### Objectives

After reading this unit the students will be able to:

- define a sequence and its terms.
- recognize triangle, factorial and Pascal sequence.
- know the definition of an arithmetic sequence.
- define arithmetic mean and arithmetic series.
- solve real life problems involving arithmetic series.
- define a geometric sequence.
- solve problems involving geometric sequences.
- define geometric mean and geometric series.
- solve real life problems involving geometric series.
- recognize a harmonic sequence and find its nth term.
- define harmonic mean and insert n harmonic means between two numbers.

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## Chapter 5: Miscellaneous Series

#### Objectives

After reading this unit the students will be able to:

- know sigma $(\sum)$ sign and evaluation of $\sum n$, $\sum n^2$ and $\sum n^3$.
- understand arithmetical-geometric series and its sum of $n$ terms.
- know method of differences and its uses.
- use the partial fraction to find the sum to $n$ terms and to infinity of the series of the type $$\frac{a}{a(a+d)}+\frac{a}{(a+d)(a+2d)}+\ldots$$

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## Chapter 6: Permutation, Combination and Probability

#### Objectives

After reading this unit the students will be able to:

- know Kramp's factorial notation to express the product of first n natural numbers by n!.
- recognize the fundamental principle of counting and its illustration by using tree diagram.
- understand the concept of permutation and know the notation $^nP_r$.
- Prove the formula ^nP_r=n(n-1)(n-2)…(n-r+1), its deductions and application to solve relevant problems
- define combination and know the notation $^nC_r=\left(\begin{smallmatrix}n\\ r\end{smallmatrix} \right)=\frac{n!}{r!(n-r)!}$, its deduction and application to solve relevant problems.
- define kind of events.
- recognize the formula $P(E)=\frac{n(E)}{n(S)}$, $0\leq P(E)\leq1$ for probability of occurrence of an event $E$ and to know its application.
- recognize the addition theorem (or law) of probability and its deduction.
- recognize the multiplication theorem (or law) of probability and its deduction.
- Use theorem of addition and multiplication of probability to solve related problems.

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## Chapter 7: Mathematical Induction and Binomial Theorem

#### Objectives

After reading this unit the students will be able to:

- know the principle of mathematical induction.
- apply the principle to prove the statements, identities or formulae.
- state and prove binomial theorem for positive integral index.
- expand $(x+y)^n$ using binomial theorem and find its general form.
- understand pascal's triangle and its use to obtain the coefficients of the binomial expansion $(x+y)^n$ when $n$ is a small number.
- know binomial series and its use to find the sum of the given series.

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## Chapter 8: Functions and Graphs

#### Objectives

After reading this unit the students will be able to

- know linear, quadratic and square root functions.
- define inverse functions and find their domain and range.
- sketch the graph of the function $y=x^n$ for different values of $x$.
- sketch the graph of quadratic function.
- predict function from their graph.
- find the intersecting point of intersecting graphs of a linear functions and coordinate axes, two linear functions and a linear and quadratic function.
- solve graphically appropriate problems from daily life.

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## Chapter 9: Linear Programming

#### Objectives

After reading this unit the students will be able to

- define linear programming (LP) as planning of allocation of limited resources to obtain optimal result.
- understand linear inequalities in one and two variables and its importance in real life problems.
- know the feasible region and identification of feasible region of simple LP problems.
- define optimal solution of an LP problem.
- find optimal solution graphically of LP problems.
- solve real life simple LP problems.

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## Chapter 10: Trigonometric Identities of Sum and Difference of Angles

#### Objectives

After reading this unit the students will be able to

- know the fundamental law of trigonometry and deduction of trigonometric identities from it.
- understand trigonometric ratios and allied angles.
- use fundamental law and its deduction to derive trigonometric ratios of allied angles.
- derive double, half and triple angle identities from fundamental law and its deduction.
- express the product of sines and cosines as sum or differences of sines and cosines.
- express the sums or differences of sines and cosines as product.

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## Chapter 11: Application of Trigonometry

#### Objectives

After reading this unit the students will be able to

- find the solution of right angles triangle.
- understand oblique triangles and find solution of such triangles, using the law of sines, cosines and tangents.
- derive the formula for finding the areas of triangles.
- know circum-circle, in-circle and escribed circle.
- derive the formula for finding circum-radius, in-radius, escribed radii and deduction of different identities.

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## Chapter 12: Graph of Trigonometric and Inverse Trigonometric Functions and Solutions of Trigonometric Equations

#### Objectives

After reading this unit the students will be able to

- know trigonometric functions and their domain and range.
- define periodic, even/odd and translation properties of the graph of $\sin \theta$, $\cos \theta$ and $\tan \theta$.
- solve trigonometric equations of the type $\sin\theta=k$, $\cos\theta=k$ and $\tan\theta=k$.
- solve graphically the trigonometric equations of the type $\sin\theta=\frac{\theta}{2}$, $\cos\theta=\theta$ and $\tan\theta=2\theta$ when $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$.
- define inverse trigonometric functions and their domain and range.
- prove the addition and subtraction formulae of inverse trigonometric functions and know their applications.
- solve general trigonometric equations.