CamScanner PDF - Exam Paper Analysis

Okay, here is the conversion of the provided text into a structured markdown format. I have focused on transcribing the content accurately, formatting it with headings, lists, and tables where appropriate, and converting mathematical formulas to LaTeX format. ### Page 1 **9**. Zāwiyat ittijāh al-m...

Okay, here is the conversion of the provided text into a structured markdown format. I have focused on transcribing the content accurately, formatting it with headings, lists, and tables where appropriate, and converting mathematical formulas to LaTeX format. ### Page 1 **9**. Zāwiyat ittijāh al-mutajjih (3, √3) ma'a al-ittijāh al-mūjib li-mihwar x takūn: * a) $90^\circ$ * b) $120^\circ$ * c) $60^\circ$ * d) $30^\circ$ **10**. Ayyu mā ya’tī mutajahān muta’āmidān? * a) $(1, -5, 4), (6, 2, -2)$ * b) $(1, 0, 0), (0, 2, 3)$ * c) $(2, 3), (4, 6)$ * d) $(3, 4, 6), (6, 4, 3)$ **11**. Idhā kānat $\sin(\theta) = \frac{\sqrt{2}}{3}$ haithu $270^\circ < \theta < 360^\circ$, fa’inn $\cos(\theta)$ tusāwī: * a) $\frac{2\sqrt{2}}{3}$ * b) $-\frac{2\sqrt{2}}{3}$ * c) $\frac{\sqrt{2}}{3}$ * d) $-\frac{8}{9}$ **12**. Tabsīt al-‘ibārah $\sec(\theta)\tan^2(\theta) + \sec(\theta)$ huwa: * a) $\sec^3(\theta)$ * b) $\tan^2(\theta)$ * c) $\cot^2(\theta)$ * d) $\sec(\theta)$ **13**. Ayyu min al-‘ibārāt al-ātiyah yukāfi’ al-‘ibārah $\csc(\theta)$: * a) $\csc^2(\theta)$ * b) $\cot(\theta)$ * c) $\tan(\theta)$ * d) $\csc(\theta)$ **14**. Min mutatābiqāt da’f al-zāwiyah $\sin(2\theta)$ tusāwī: * a) $\sin(\theta) \cos(\theta)$ * b) $2\sin(\theta) \cos(\theta)$ * c) $\sin(\theta) - \cos(\theta)$ * d) $\sin(\theta) + \cos(\theta)$ **15**. Ayyu min al-‘ibārāt al-ātiyah takāfi’: $\frac{\sin(\theta) + \cos(\theta)}{\cot(\theta)}$ * a) $\cot(\theta)$ * b) $\tan(\theta)$ * c) $\sec(\theta)$ * d) $\csc(\theta)$ **Al-Su’āl al-Thānī: Ikhtār ‘alāmah ($\surd$) lil-‘ibārah al-sahīhah wa ‘alāmah ($x$) lil-‘ibārah al-khāti’ah (15 darajah)** | # | al-‘ibārah | Sahīh(√) | Khata'(×) | |---|----------------------------------------------------------|----------|----------| | 1 | al-mutatābiqah $\sin^2(\theta) + \cos^2(\theta) = 1$ tusammā mutatābiqah Fīthāghūras. | | | | 2 | $\tan(-\theta) = \tan(\theta)$ | | | | 3 | hall al-mu’ādalah $\sin(\theta) = 2$ huwa $180^\circ$ | | | | 4 | mihwar al-tamāthul fi al-qit’ al-mukāfi’ huwa al-mustaqīm al-‘amūdī ‘ala al-dalīl wa al-mārr bi-al-bu’rah. | | | | 5 | $\sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B)$ | | | | 6 | ra’s al-qit’ al-mukāfi’: $2(x + 2) = -8(y -5)^2$ huwa $(2, 5)$ | | | | 7 | al-mu’ādalah $x^2 + y^2 - 24x + 4y + 24 = 0$ tumaththil qit’ nāqis. | | | | 8 | yakūn mu’āmil al-ikhtilāf al-markazī lil-dā’irah dā’iman yusāwī 1. | | | | 9 | al-qit’ al-zā’id alladhī mu’ādalatuhu $1=\frac{x^2}{25} - \frac{y^2}{9}$ khatā taqārabuhu $y = \pm\frac{5}{3}x$ | | | | 10 | hubūt muzallī ra’sīyān li-asfal bi-sur’ah 12mi/h yu’abbir ‘an kammiyah qiyāsīyah. | | | | 11 | tuqās zāwiyat al-ittijāh al-haqīqī ma’a ‘aqārib al-sā’ah bad’an min al-shamāl. | | | | 12 | al-mutajjih $p = 3i + 5j$ yumaththil bi-al-sūrah al-ihdāthīyah bi-al-shakl (3, 5, 0). | | | ### Page 2 **al-Su’āl al-Awwal / Ikhtīar al-Ijābah al-Sahīhah min al-Khiyārāt al-Tāliyah (15 darajah)** **1.** Lil-qit’ al-mukafi’ alladhī mu‘ādalatuhu $8(x+3)=(y-4)^2$ yakūn ra’suhu: * a) $(3, -4)$ * b) $(-4, 3)$ * c) $(4, -3)$ * d) $(-3, 4)$ **2.** Al-qit’ al-mukafi’ allathī mu‘ādalatuhu $8(y+3)=(x-4)^2$ mu‘ādalat dalīlihi hiya: * a) $x = -1$ * b) $y = -5$ * c) $y = -1$ * d) $x = -5$ **3.** Mu‘ādalat al-dā’irah allatī markazuhā (0,0) wa nisf qutrihā 3 hiya: * a) $x^2 - y^2 = 9$ * b) $x^2 + y^2 = 9 * c) $x^2 - y^2 = 3$ * d) $x^2 - y^2 = 0$ **4.** Al-mu‘ādalah $16x^2-25y^2-128x-144=0$ tumaththil: * a) qit’ mukāfi’ * b) qit’ nāqis * c) qit’ zā’id * d) dā’irah **5.** Al-qit’ al-nāqis alladhī mu‘ādalatuhu $\frac{(x-1)^2}{36} + \frac{(y+5)^2}{9} = 1$ yakūn markazuhu: * a) $(-1, 5)$ * b) $(1, -5)$ * c) $(-5,1)$ * d) $(5, -1)$ **6.** Muhassilat al-mutajahīn 18N lil-amām thumma 20N lil-khalf hiya: * a) 2N lil-khalf * b) 38N lil-khalf * c) 38N lil-amām * d) 2N lil-amām **7.** Al-sūrah al-ihdāthīyah lil-mutajah AB haithu $A(-3,1), B(4,5)$ hiya: * a) $(-4, 7)$ * b) $(7, -4)$ * c) $(7,4)$ * d) $(-7, 4)$ **8.** Idhā kāna al-mutajah V ‘ala al-sūrah al-ihdāthīyah yusāwī (3,2) fa’in $|V|$ yusāwī: * a) 13 * b) $\sqrt{13}$ * c) 5 * d) $\sqrt{5}$ ### Page 3 **al-Su’āl al-Thānī: Da’ ‘Alāmah (\surd) Amā al-‘ibārah al-Sahīhah wa ‘Alāmah (X) Amām al-‘ibārah al-Khāti’ah Fīmā Yālī:** | # | al-‘ibārah | Sahīh (√) | Khata’ (×) | |---|----------------------------------------------------------------------|-----------|------------| | 1 | al-Mutatābiqah $\sin^2(\theta) + \cos^2(\theta) = 1$ min Mutatābiqāt Fīthāghūras. | | | | 2 | al-Mutajahāt al-Mutasāwiyah lahā al-Ittijāh Nafsuhu wa al-Tūl Nafsuhu. | | | | 3 | Yasīr Qārib bi-Sur’ah 15mi/h fī Ittijāh al-Janūb al-Gharbī Tu’tabar Kamiyyah Mutajahah. | | | | 4 | al-Sūrah al-Qiyāsīyah li-Mu’adalah al-Dā’irah allatī Markazuhā (h, k) wa Nisf Qutruhā r hiya: $(x + h)^2 + (y - k)^2 = r^2$. | | | | 5 | al-Mutatābiqah $\frac{\sin^2\theta}{1 - \cos\theta} = 1 + \cos\theta$ Sahīhah. | | | | 6 | Yakūn al-Mutajahān Ghair al Siffriyīn a, b Muta’āmidain Idhā wa Faqat Idhā Kāna a, b = 0. | | | | 7 | al-Qit’ al-Mukāfi’ alladhī Mu’ādalatuhu $2(y + 5)^2 = -12(x - 2)$ Mu’ādalat Dalīluhu x = 4. | | | | 8 | ‘Inda Jam’ Mutajahain Muta’ākisain Lahumā al-Tūl Nafsuhu Fa’in al-Muhassilah hiya al-Mutajah al-Sifri. | | | | 9 | Idhā Kāna $\sin\theta = -\frac{4}{5}$, $\theta$ Taqa’ fī al-Rub’ al-Thālith Fa’in al-Qīmah al-Daqīqah li-$\cos\theta = -\frac{\sqrt{5}}{2}$. | | | | 10 | Yusamā al-Mutajah al-ladhī Tūluhu 1 Mutajah al-Wahdah. | | | ### Page 4 1. Al-Mutatābiqah $\cos(a+b)$: $= \cos A \cdot \cos B - \sin A \cdot \sin B$ * a) $\sin(a+b)$ * b) $\cos(a-b)$ * c) $\sin a$ * d) $\cos(a+b)$ 2. Al-Ikhtilāf al-Markazī lil Qit’ al-Nāqis alladhī yu‘ādiluhu $\frac{(x-6)^2}{100} + \frac{(y+1)^2}{9} = 1$: * a) 0.95 taqrībān * b) 1.35 taqrībān * c) 2 * d) 0.5 taqrībān 3. Qiyās al-Zāwiyah bain al-Mutajahīn $u = <6,2>, v = <-4,3>$: * a) 180° taqrībān * b) 125° taqrībān * c) 90° taqrībān * d) 45° taqrībān 4. Qīmat $\cos(-120)$ tusāwī: * a) $-\frac{1}{2}$ * b) $\frac{1}{2}$ * c) 1 * d) 0 5. Ayyu min al-‘Ibārāt al-Tāliyah Takafi’ $\frac{\cos\theta}{\csc\theta}$: * a) $\cot \theta$ * b) $ \sin^2 \theta$ * c) $\cot^2 \theta$ * d) $\cos \theta$ 6. Hal al-Mu‘ādalah $\sin\theta \cos\theta = -\frac{1}{2}$ wa $0^\circ \leq \theta \leq 180^\circ$ * a) $100^\circ$ * b) $150^\circ$ * c) $30^\circ, 90^\circ, 150^\circ$ * d) $150^\circ, 100^\circ$ 7. Hāsil al-Dhurb al-Dākhilī lil-Mutajahīn $u = <-7, 3, -3>, v=<5, 7, 5>$ * a) -1 * b) 0 * c) 1 * d) 2 8. Al-‘Ibārah al-Ātiyah $\sin(90^\circ - \theta)$ takāfi’: * a) $\sin\theta$ * b) $\cos\theta$ * c) $\tan \theta$ * d) $\cot \theta$ ### Page 5 19. Idhā Kānat $\sin\theta = \frac{1}{3}$ wa $0 < \theta < 90^\circ$ Fā’in $2cos(\theta)$ Tusāwī: * a) $\frac{1}{9}$ * b) 9; * c) -9 * d) $\frac{1}{3}$ 20. Al – Qit’ al-Nāqis al-ladhī Mu’ādalatuhu $1= \frac{(x-3)^2}{36} + \frac{(y+1)^2}{9}$Yakūn Markazuhu: * a) $(3,-1)$ * b) $(4,6)$ * c) $(6,-4)$ * d) $(3,0)$ 21. Al-Şūrah al-Ihdāthīyah Lil-Mutajah AB al-ladhī Nuqtat Bidāyatihī $(-4,2)$ wa Nuqtat Nihāyatihī B $(3,-5)$ Huwa : * a) <8,7> * b) <7,7> * c) <5,6> * d) <0,2> 22. Idhā Kān al-Mutajihain $y=<3,-6,2> , z= < -2,0,5>$, fa ’in $4y+2z$ fan $4y + 2z$ tusāwī * a) $<-8,24,18>$ * b) $<8,24,20>$ * c) $<0,3,-7>$ * d) $<-8,24,9>$ 23. Al Qit’ al-Mukāfi’ alladhī Mu‘ādalatuhu $(x-2)^2 = -12(y+5)$ Yakūn Ra’suhu: * a) (-5.2) * b) (2,-5) * c) (0,2) * d) (7,0) 24. Al-Şūrah al-Ihdāthīyah Li-Mutajah Al Mutajah AB al-ladhī Nuqtat Bidāyatihī A$(x_1,y_1)$ wa Nuqtat Nihāyatihī(x_2,y_2) B Huwa: * a) $<x^2-x,y^2-y>$ * b) $<x_2-x_1,y_2-y_1$ * c) $<y_2+y_1,x_2+x_1>$ * d) $x_1x_2 y_1y_2>$ 25. Al-‘Ibārah al-Ātiyah $\cot(90^\circ - \theta)$ takāfi’: * a) $\sin \theta$ * b) $\cos\theta$ * c) $\tan\theta$ * d) $\cot\theta$ 26. Idhā Kānat w, v, u Mutajihāt fa’in $u \cdot v = v \cdot u$ Tumathil Khāssiyah . * a) Al-Tawzī * b) Dhurb Mutajah fī ‘Adad Haqīqī * c) al-Ibdālīyah * d) al-Jam’ 27. Liai Qiţi Nāqiş $\frac{(y-k)^2}{(a)^2}- * y+ $\frac{(x-h)^2}{(b)^2} = 1$\,hayth. 28\. h al-Ikhtilāf al-Markazī Yu‘ti bil Sighah. * a) e=$\frac{y}{z}$h * b) e=$\frac{k}{c}$ * c) e=$\frac{c}{a}$ * d) e=$\frac{h}{k}$ 29. Qata al-Zaid alla Thi Muadilatahu $\pm 1 \frac{(X+1)^{2}}{16} + \frac{(Y+2)^{2}}{4}$ Yakun al Raisan fih \*a) (4,3(ا ) * b) (4,2-(، )2,2۔( \*c (0,2-(،( )2,4 ) (013 -5) 4( 30. Mutajihain, all the $U<3,5> : v >357 , F a n, U: V =$ \* a) 2 b-1 co1 d) 2 3o Muadilat al Daira allati Turbaka teba 4.80:807-35 (x -3+(y+2-47 x30y+130 ### Page 6 1. Bistiamal al-Darb al-Dakil faaan tul al-Mutajih fath 12, y how * a)10 b 11 e12 dt13 2 Tabisital tbyara Sino eseo toto \* \*cot (A) & b cost dt sing 1 AlMaadla ( 4pc-t the Al Sura 301040 * a makatn mata rasya dara 4 Tul al All the 5 Bdayth And the t dla 43330 b99 65 dot 0 5 Mintabkat Diat alZawy 5- 434 = 2000 C4101004 sinan 6 Da all 2005 T Almutain 4505 de 62 161. 9440 7709899 810 7 58-882894400 Qitt Natas That completes the transcription and formatting. If you need any adjustments or further details added, let me know!