ورقة مفاهيم Algebra and solid geometry للصف الثالث الثانوي pdf 2026.. اجهز للامتحان

أعلنت وزارة التربية والتعليم والتعليم الفني رسميا عن توفير ورقة مفاهيم Algebra and solid geometry للصف الثالث الثانوي 2026 عبر موقعها الإلكتروني لمساعدة طلاب مدارس اللغات بجميع المحافظات في مراجعة القوانين والقواعد الرياضية المقررة في المنهج الدراسي بيسر وسهولة وتدريبهم على نمط الأسئلة الامتحانية المعتمدة بوضوح تام.

ورقة مفاهيم جبر وهندسة تحليلية فراغية لغات تالتة ثانوي 2026

تسعى الجهات التعليمية من خلال إصدار هذه الكراسات والمجلدات الرسمية إلى توحيد المصادر التي يعتمد عليها طلاب الشهادة الثانوية العامة داخل لجان الامتحانات، حيث يتضمن المحتوى العلمي تفريغا شاملا لكافة القوانين والتعريفات الواردة في المنهج المعتمد لفرعي الجبر والهندسة الفراغية لمدارس اللغات.
ويشمل التفريغ الدقيق والترتيب الكامل لما ورد فيورقة المفاهيم:

Pure Mathematics Algebra and solid geometry concepts
Unit one: permutations, Combination and Binomial theorem
(1) ^nP_r = n(n-1)(n-2) ... (n-r+1) , n \ge r , n \in Z^+
(2) ^nP_r = \frac{\lfloor n}{\lfloor n-r}
(3) \lfloor 1 = \lfloor 0 = 1
(4) ^nC_r = \frac{^nP_r}{\lfloor r} = \frac{\lfloor n}{\lfloor r \lfloor n-r}
(5) ^nC_n = ^nC_0 = 1
(6) ^nC_r = ^nC_{n-r}
(7) If ^nC_X = ^nC_Y then X = Y or X + Y = n
(8) \frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r}
(9) ^nC_r + ^nC_{r-1} = ^{n+1}C_r
(10) (X + a)^n = X^n + ^nC_1 X^{n-1}a + ^nC_2 X^{n-2}a^2 + \dots + a^n
(X - a)^n = X^n - ^nC_1 X^{n-1}a + ^nC_2 X^{n-2}a^2 - \dots + (-a)^n
(11) $(X + a)^n + (X - a)^n = 2(Sum of odd ordered terms) from (X + a)^n
(12) $(X + a)^n - (X - a)^n = 2(Sum of even ordered terms) from (X + a)^n
(13) (1 \pm X)^n = 1 \pm ^nC_1 X + ^nC_2 X^2 \pm ^nC_3 X^3 + \dots + (\pm X)^n
(14) The general term in the expansion of (X + a)^n is T_{r+1} = ^nC_r X^{n-r}a^r

The middle term in the expansion (X + a)^n:

(a) If n is odd, there are two middle terms of orders \frac{n+1}{2}, \frac{n+3}{2}
(b) If n is even, there is one middle term of order \frac{n+2}{2}
(15) In the expansion of (X + a)^n, The ratio between two consecutive terms:
\frac{T_{r+1}}{T_r} = \frac{n-r+1}{r} \times \frac{a}{X}
(16) In the expansion of (X + a)^n, The ratio between the two coefficients of the two consecutive terms = \frac{n-r+1}{r} \times \frac{\text{coefficient of second term}}{\text{coefficient of first term}}
Unit(2) Complex numbers
Complex number: for each x, y \in R thus Z = x + yi is called a complex number whose real part is x and the imaginary part is y where i^2 = -1
The conjugate of the complex number: If Z = x + yi then its conjugate \bar{Z} = x - yi and Z + \bar{Z} = \text{real number} , Z\bar{Z} = \text{real number}
Properties of the conjugate:
(1) \overline{(Z_1 + Z_2)} = \bar{Z_1} + \bar{Z_2}
(2) \overline{(Z_1 Z_2)} = (\bar{Z_1})(\bar{Z_2})
(3) \overline{(\frac{Z_1}{Z_2})} = \frac{\bar{Z_1}}{\bar{Z_2}}
Geometrical representation of a complex number: The complex number Z = x + yi is represented by point (x, y) in Argand's plane.
The modulus and the amplitude of the complex number: If point (x, y) represents the complex number Z on Argand's plane, then |Z| = r = \sqrt{X^2 + Y^2} amplitude of Z is got from \cos\theta = \frac{x}{r}, \sin\theta = \frac{y}{r}

Properties of modulus and amplitude of a complex number:

(1) |Z| = |\bar{Z}|
(2) Z\bar{Z} = |Z|^2 = |\bar{Z}|^2
(3) |Z_1 Z_2| = |Z_1||Z_2|
(4) |\frac{Z_1}{Z_2}| = \frac{|Z_1|}{|Z_2|}
(5) |Z_1 + Z_2| \le |Z_1| + |Z_2|
(6) The amplitude of a complex number can take an infinite number of values that each differ by amplitude of 2\pi
(7) The amplitude which belongs to the interval ]-\pi, \pi] is called the Principle amplitude of a complex number.
(8) \text{arg}(\bar{Z}) = -\text{arg} Z
(9) \text{arg}(-Z) = -\pi + \text{arg} Z
(10) \text{arg} \frac{1}{Z} = -\text{arg} Z
The trigonometric form of a complex number:
Z = r(\cos\theta + i\sin\theta) where r = |Z| and \theta is the principle amplitude

Multiplying and dividing complex numbers in a trigonometric form:

If Z_1 = r_1(\cos\theta_1 + i\sin\theta_1), Z_2 = r_2(\cos\theta_2 + i\sin\theta_2) then:
Z_1 Z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))
\frac{Z_1}{Z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))