Libya Secondary Certificate — Scientific Stream (الشهادة الثانوية - القسم العلمي)

If a = 3i − 4j and b = i + 2j, what is the magnitude of the resultant vector a + b (مقدار المتجه)?

• A5 + √5
• B6
• C2√5
• D2

Expanding along the first row, what is the value of the third-order determinant whose rows are (1, 2, 0), (3, −1, 4) and (2, 0, 1) (محدد من الرتبة الثالثة)?

• A−11
• B9
• C−9
• D7

The region bounded by the curve y = √x, the x-axis and the line x = 4 is rotated through 360° about the x-axis. What is the volume of the solid generated (حجم الجسم الناشئ من الدوران)?

• A16π
• B(16/3)π
• C8
• D8π

Given A = [[1, 2], [3, 0]] and B = [[2, 1], [1, 4]], what is the product AB (ضرب المصفوفات)?

• A[[4, 9], [6, 3]]
• B[[5, 4], [13, 2]]
• C[[2, 2], [3, 0]]
• D[[3, 3], [4, 4]]

If sin(θ) = 3/5 and θ is in the first quadrant, what is the exact value of cos(2θ) (حساب المثلثات)?

• A-7/25
• B24/25
• C7/25
• D16/25

Writing √3 cos θ + sin θ in the form R cos(θ − α), with R > 0 and 0° < α < 90°, gives which expression (المقدار أ جتا θ + ب جا θ)?

• A2 cos(θ + 30°)
• B2 cos(θ − 60°)
• C2 cos(θ − 30°)
• D4 cos(θ − 30°)

Using the compound-angle formula sin(A + B) = sin A cos B + cos A sin B, what is the exact value of sin 75°?

• A(√6 − √2)/4
• B(√6 + √2)/4
• C(√3 + 1)/2
• D(√6 + √2)/2

A function is odd (دالة فردية) when f(−x) = −f(x) for every x in its domain. Which of the following functions is odd?

• Af(x) = x⁴ + x²
• Bf(x) = cos x
• Cf(x) = x³ + 1
• Df(x) = x³ − 4x

What is the inverse function f⁻¹(x) of f(x) = (2x + 3)/5 (الدالة العكسية)?

• Af⁻¹(x) = (5x − 3)/2
• Bf⁻¹(x) = (5x + 3)/2
• Cf⁻¹(x) = 5/(2x + 3)
• Df⁻¹(x) = (2x − 3)/5

A particle moves in a straight line so that its displacement after t seconds is s = t³ − 6t² + 9t metres. At what times is the particle momentarily at rest (السرعة والعجلة)?

• At = 2 s only
• Bt = 0 s and t = 3 s
• Ct = 1 s and t = 3 s
• Dt = 1 s only

What is the first derivative dy/dx of the function y = sin(3x² + 1) (التفاضل)?

• A-6x cos(3x² + 1)
• Bcos(3x² + 1)
• C6x sin(3x² + 1)
• D6x cos(3x² + 1)

Using implicit differentiation on the circle equation x² + y² = 25, find dy/dx at the point (3, 4).

• A3/4
• B-3/4
• C-4/3
• D4/3

The radius of a circular oil spill on the Libyan coast is expanding at a constant rate of 2 cm/s. At what rate is the area of the spill increasing when the radius reaches 5 cm (معدلات التغير المرتبطة)?

• A25π cm²/s
• B10π cm²/s
• C20π cm²/s
• D50π cm²/s

What is the inflection point (نقطة الانقلاب) of the cubic function f(x) = x³ − 6x² + 9x + 2?

• A(2, 0)
• B(1, 6)
• C(3, 2)
• D(2, 4)

An open-top storage box with a square base of side x and vertical height h must hold an exact volume of 32 m³. What dimensions minimize the total surface area S of sheet metal required (تطبيقات القيم العظمى والصغرى)?

• Ax = 2 m, h = 8 m
• Bx = 4 m, h = 2 m
• Cx = 4 m, h = 4 m
• Dx = 8 m, h = 0.5 m

What is the derivative of y = x² sin x (تفاضل حاصل الضرب)?

• A2x cos x
• B2x sin x + x² cos x
• C2x sin x − x² cos x
• Dx² cos x

What is the exact value of the definite integral ∫₀^(π/2) sin(x) cos(x) dx (التكامل المحدد)?

• A0
• B1
• C1/2
• D1/4

What is dy/dx for y = ln(3x² + 1) (تفاضل الدوال اللوغاريتمية)?

• A1/(3x² + 1)
• B6x · ln(3x² + 1)
• C6x/(3x² + 1)²
• D6x/(3x² + 1)

In the passive sentence 'كُتِبَ الدَّرْسُ', what is the grammatical role (إعراب) of 'الدَّرْسُ'?

• Aنائب فاعل مرفوع وعلامة رفعه الضمة (deputy subject of a passive verb)
• Bفاعل مرفوع (subject of an active verb)
• Cمفعول به منصوب (direct object)
• Dمبتدأ مرفوع (subject of a nominal sentence)

In 'سَارَ الجُنْدِيُّ سَيْرَ الأَبْطَالِ', what kind of absolute object (المفعول المطلق) is 'سَيْرَ'?

• Aمؤكد لعامله (it only emphasizes the verb)
• Bمبين للعدد (it states how many times)
• Cمبين للنوع (it specifies the kind of action)
• Dنائب عن المفعول المطلق (a substitute word such as كلّ)

A block of mass 10 kg rests on a rough horizontal surface in a Tripoli physics laboratory (الفيزياء). If the coefficient of kinetic friction is μₖ = 0.2 and g = 9.8 m/s², what horizontal force F is required to move the block at constant velocity?

• A2 N
• B98 N
• C49 N
• D19.6 N

A 5.0 kg crate slides down a smooth, frictionless inclined plane angled at 30° to the horizontal. Taking g = 9.8 m/s², what is the acceleration of the crate down the plane (المستوى المائل)?

• A9.8 m/s²
• B4.9 m/s²
• C8.49 m/s²
• D2.45 m/s²

A force of 20 N acts at 60° above the horizontal (تحليل القوى). What is its horizontal component?

• A17.3 N
• B20 N
• C34.6 N
• D10 N

A ball is thrown vertically upward with an initial speed of 19.6 m/s. Neglecting air resistance and taking g = 9.8 m/s², what maximum height does it reach (الأجسام المقذوفة إلى أعلى)?

• A19.6 m
• B39.2 m
• C9.8 m
• D2.0 m

A stone of mass 2.0 kg is dropped from rest from the top of a 20.0 m high cliff. Neglecting air resistance and taking g = 9.8 m/s², what is its kinetic energy just before striking the ground (حفظ الطاقة)?

• A784 J
• B196 J
• C392 J
• D40 J

An electric pump raises 500 kg of water through a vertical height of 12 m in 60 s. Taking g = 9.8 m/s², what is the useful power output of the pump (القدرة)?

• A58,800 W
• B980 W
• C100 W
• D490 W

Masses of 3.0 kg and 2.0 kg hang on either side of a light, smooth, fixed pulley, joined by a light inextensible string. Taking g = 9.8 m/s², what is the magnitude of their acceleration when released (البكرات)?

• A9.8 m/s²
• B4.9 m/s²
• C1.96 m/s²
• D0.98 m/s²

A steady current of 0.50 A flows through a lamp for 2.0 minutes. How much electric charge passes through it (الشحنة الكهربائية)?

• A1.0 C
• B4.0 C
• C240 C
• D60 C

Resistors of 2 Ω, 3 Ω and 6 Ω are connected in parallel. What is their combined resistance (المقاومات على التوازي)?

• A1 Ω
• B11 Ω
• C3.7 Ω
• D6 Ω

A 2 kW electric heater is used for 3 hours every day. How much electrical energy does it use in 30 days (قياس الطاقة الكهربائية)?

• A6 kWh
• B180 kWh
• C60 kWh
• D180,000 kWh

An electric kettle rated 2,300 W is used on a 230 V supply. Fuses of 3 A, 5 A, 13 A and 30 A are available. Which fuse should be fitted in its plug (الاستخدام الآمن للكهرباء)?

• A3 A
• B5 A
• C13 A
• D30 A

A transformer has 200 turns on its primary coil and 1,000 turns on its secondary coil. A 12 V alternating supply is connected to the primary. What is the secondary voltage, and what type of transformer is it (المحولات الكهربائية)?

• A60 V; step-up
• B2.4 V; step-down
• C12 V; the voltage is unchanged
• D600 V; step-up

In Fleming's left-hand rule (قاعدة فليمنج لليد اليسرى) for a current-carrying conductor in a magnetic field, what does the thumb indicate?

• AThe direction of the magnetic field
• BThe direction of the force (motion) on the conductor
• CThe direction of the conventional current
• DThe direction of the induced electromotive force

Radium-226 (atomic number 88) decays by emitting an alpha particle (جسيم ألفا). What are the mass number and atomic number of the daughter nucleus?

• AMass number 226, atomic number 89
• BMass number 222, atomic number 88
• CMass number 224, atomic number 86
• DMass number 222, atomic number 86

A bar magnet is pushed into a coil connected to a sensitive galvanometer. Which change would produce a larger induced e.m.f. (الحث الكهرومغناطيسي)?

• APushing the magnet into the coil faster
• BHolding the magnet at rest inside the coil
• CUsing a coil with fewer turns
• DPushing the magnet in with its other pole first, at the same speed

A radioactive sample gives a corrected count rate of 800 counts per minute. Its half-life is 2 hours (عمر النصف). What count rate is expected after 6 hours?

• A200 counts per minute
• B267 counts per minute
• C100 counts per minute
• D0 counts per minute

Which of the following words is an intensive form (صيغة مبالغة) related to the verb 'عَلِمَ'?

• Aعالِم
• Bعلّام
• Cمعلوم
• Dعِلْم

Which sentence uses the number correctly according to the rule for the numbers 3 to 10 (العدد من ثلاثة إلى عشرة)?

• Aقرأتُ خمسَ قصصٍ
• Bقرأتُ خمسةَ قصصٍ
• Cقرأتُ خمسَ كتبٍ
• Dقرأتُ خمسةَ كتاباً

Potassium reacts more vigorously with cold water than sodium does. Which explanation is correct (الفلزات القلوية)?

• APotassium has more electrons in its outer shell than sodium
• BPotassium atoms are smaller, so the nucleus holds the water more strongly
• CPotassium has a higher melting point than sodium
• DPotassium's single outer electron is farther from the nucleus, so it is lost more easily

Chlorine water is added to colourless potassium bromide solution and the mixture turns orange-brown. What reaction has taken place (الهالوجينات)?

• ABromine has displaced chlorine from potassium chloride
• BChlorine has displaced bromine: Cl₂ + 2KBr → 2KCl + Br₂
• CNo reaction has occurred; the colour comes from the chlorine water
• DChlorine has displaced potassium to form potassium metal

Using the reactivity series of metals (سلسلة الفاعلية الكيميائية), which metal will NOT displace copper when placed in copper(II) sulfate solution?

• AMagnesium
• BZinc
• CSilver
• DIron

In the blast furnace used to extract iron from haematite (الفرن العالي), which substance is the main reducing agent for iron(III) oxide?

• ACarbon monoxide
• BCalcium carbonate (limestone)
• CHot air (oxygen)
• DSlag (calcium silicate)

Why is aluminium extracted from purified aluminium oxide by electrolysis rather than by heating it with carbon (استخلاص الألومنيوم)?

• AAluminium oxide has a low melting point, so it cannot be heated with carbon
• BCarbon reacts with aluminium to form an explosive compound
• CAluminium is below carbon in the reactivity series
• DAluminium is more reactive than carbon, so carbon cannot remove oxygen from aluminium oxi…

Galvanising protects an iron roof sheet from rusting even when the coating is scratched. What is the reason (صدأ الحديد)?

• AZinc is less reactive than iron, so it keeps oxygen away
• BZinc is more reactive than iron, so it corrodes in preference to the iron (sacrificial pr…
• CZinc absorbs all the water vapour from the air around the iron
• DZinc turns the iron into stainless steel

Ethene molecules (CH₂=CH₂) join together to form poly(ethene). What type of polymerisation is this, and what is the repeating unit (البلمرة)?

• ACondensation polymerisation; –CH₂–CH₂–O–
• BAddition polymerisation; –CH=CH–
• CAddition polymerisation; –CH₂–CH₂–
• DCondensation polymerisation; –CH₂–CH₂–, with water given off

How many structural isomers (المتشكلات) have the molecular formula C₄H₁₀?

• A2
• B1
• C3
• D4

Orange-brown bromine water is shaken separately with ethane and with ethene at room temperature. What is observed (خواص الألكينات)?

• ABoth decolourise it
• BNeither decolourises it
• COnly ethane decolourises it
• DOnly ethene decolourises it