Exhaustive Entrance Exam Study Notes

Mathematical and Quantitative Reasoning

Geometry principles dictate that the sum of the external angles of any polygon, including a 5-sided polygon, is always $360^{\circ}$. This is a constant value regardless of the number of sides $n$. For internal angles, the formula is specifically $180^{\circ}(n-2)$. When dealing with specific polygons like a heptagon, if the internal angles are in the ratio $2:3:5:6:7:4:3$ and the total sum of angles is $900^{\circ}$, the measure of the smallest angle can be calculated by dividing the total sum by the sum of the ratio parts and multiplying by the smallest part. In this specific case, the smallest angle is $60^{\circ}$. Additionally, a regular hexagon possesses exactly $6$ lines of symmetry passing through it.

Algebraic evaluations often involve solving systems of equations or using identity formulas. For instance, if $a+b=10$ and $a-b=6$, one can find the value of $ab$ using the identity $(a+b)^2 - (a-b)^2 = 4ab$. Substituting the values gives $100 - 36 = 64$, which means $4ab = 64$, thus $ab=16$. Linear equations in one variable, such as $3x=10$, represent a vertical line on a coordinate plane. Quadratic equations with specific roots like $2$ and $-3$ can be formed using the structure $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$, resulting in $x^2 + x - 6 = 0$. When roots are $3$ and $0$, the equation becomes $x^2 - 3x = 0$.

Word problems involving movement and rates are common. The average speed of an object is calculated as the total distance divided by the total time. For a boy cycling for $4$ hours at $6\,km/h$ ($24\,km$) and walking $9\,km$ in $7$ hours, the total distance is $33\,km$ and total time is $11$ hours, yielding an average speed of $3.0\,km/h$. Relative speed depends on direction: if two cars are moving at $40\,km/h$ and $30\,km/h$ in the same direction, the relative speed is the difference ($10\,km/h$), whereas if they move in opposite directions, the relative speed is the sum ($70\,km/h$). For work-related problems, if workers A, B, and C can complete a task in $4, 5,$ and $20$ days respectively, their combined rate is $\frac{1}{4} + \frac{1}{5} + \frac{1}{20} = \frac{5+4+1}{20} = \frac{10}{20} = \frac{1}{2}$, meaning they can finish the work together in $2$ days.

Financial and measurement applications include solving for costs and remains. In a system where $2$ copies and $3$ books cost $Rs.\,1300$ and $5$ copies and $1$ book cost $Rs.\,2600$, the cost of a individual copy and book can be determined through simultaneous equations. Profit calculations involve relating sale price to cost price; if a T-shirt sells for $Rs.\,660$ with a $10\%$ profit, the cost price is calculated as $\frac{660}{1.1} = Rs.\,600$. For volume, the volume of a sphere with a radius of $10.5\,cm$ is found using $V = \frac{4}{3}\pi r^3$, which approximates to $4851\,cm^3$.

Statistics and probability cover measures of central tendency and likelihood. The mode is the most frequently occurring number in a data set; for the sequence $2, 3, 4, 4, 5, 6, 8, 8, 6, 5, 2, 3, 8$, the mode is $8$, as it appears three times. The mean of the first six natural numbers ($1, 2, 3, 4, 5, 6$) is $3.5$. In probability, a coin tossed twice has a $1/2$ chance of landing as a doublet ($HH$ or $TT$), as there are $4$ possible outcomes ($HH, HT, TH, TT$).

General Knowledge and Science

Global events and geography include the fact that Argentina was crowned the FIFA World Cup champion in the year $2022$. Geographically, Pakistan is situated in South Asia. China's capital city is Beijing, while Colombo serves as the capital of Sri Lanka. East Pakistan is now known as Bangladesh. In aviation, 'Air Blue' is an airline primarily operating from Pakistan. Historically, Neil Armstrong is recognized as the first person to walk on the moon in $1969$. Doha is a significant seaport located in Qatar.

Biological and physical science topics cover human anatomy and the environment. The human skull provides vital protection for the brain, while the kidneys are responsible for filtering blood and wastes from the body. Tooth decay in humans is primarily caused by the consumption of sugar. On a larger scale, the Earth completes one full revolution around the sun in one year, whereas one full rotation on its axis takes one day. In physics, one kilometer is equivalent to exactly $1000$ meters. Among data storage devices, the Hard disc typically has the capacity to store the maximum amount of data compared to compact discs, floppy discs, or ROM.

Islamiat and Ethics

Islamic history and theology highlight significant figures and events. Hazrat Abu Bakar (R.A.) accompanied Prophet Muhammad (SAW) during the migration (Hijrat) to Madina. The battle of Uhad, the second major battle in Islam, was named after a hill. Sahi-Bukhari, a prestigious collection of Hadith, was authored by Muhammad Bin Ismail. The Prophet Hazrat Adam (A.S.) is referred to by the title Abu-al-Bashar. Essential pillars of Islam include Zakat, which is the 3rd fundamental pillar. Surah Al-Kausar is distinguished as the shortest Surah in the Holy Quran. During the Hajj pilgrimage, the ritual of Qurbani is performed at Mina.

Ethics and moral systems focus on the conduct of individuals and societies. The feeling of remorse is intrinsically connected with a person's conscience. The foremost right allocated to parents is that their children must treat them kindly. In a state, a group that is numerically smaller than the rest of the population is defined as a minority. Ethics are moral principles and values that specifically govern the actions of an individual or a firm\'s behavior. A man who does not follow any religion is commonly termed an atheist.

Pakistan Studies

The history of the Pakistan movement involves the efforts of key leaders. Sir Syed Ahmad Khan established his first school in Moradabad. The All India Muslim League was established in Dacca in $1906$. Quaid-e-Azam Muhammad Ali Jinnah presented his famous Fourteen Points in response to the Nehru Report. Allama Muhammad Iqbal, the national poet, was born in Sialkot. During the presidential elections of $1965$, Miss Fatima Jinnah contested against General Ayub Khan. Zulfiqar Ali Bhutto ascended to the role of Foreign Minister in $1963$.

Constitutional development in Pakistan saw major changes over the decades. In the $1956$ Constitution, the designation of Governor General was changed to President. The $1973$ Constitution declared Urdu as the national language of the country. Under the $18th$ constitutional amendment, the North-West Frontier Province (NWFP) was officially renamed Khyber Pakhtunkhwa. The Senate serves as the upper house of Pakistan's Parliament and specifically represents the provinces. Any bill passed by the Parliament requires the final approval of the President to become law.

Geographic and cultural details of Pakistan include the identification of the Kaghan valley as the "Pearl of the Himalaya." The Indus River system provides the nation with its best agricultural land and irrigation capabilities. The ideological foundation of Pakistan is derived from Islam. Significant religious landmarks include the mausoleum of Sheikh Bahauddin Zakariya, which is located in Multan, and the tomb of Mughal Emperor Jehangir, situated in Lahore.

Physics and Chemistry

Physics principles involve energy, electronics, and thermodynamics. A particle that possesses no charge and no mass is often described as a photon. In electrical systems, a step-down transformer at the end of a transmission line reduces voltage; the relationship between primary and secondary voltage follows the ratio of primary to secondary turns $\frac{V_p}{V_s} = \frac{N_p}{N_s}$. For example, if voltage drops from $2400\,V$ to $120\,V$ and primary turns are $40$, the secondary turns would be $2$. In thermodynamics, a process where the volume of the system remains constant is known as an isochoric process. The gradient of a velocity-time graph describes the acceleration of an object. A device that converts alternating current into pulsating current or voltage is a rectifier.

In Chemistry, analytical techniques such as spectroscopy involve the measurement of wavelength and intensities of electromagnetic radiation absorbed or emitted by atoms. Quantitative chemistry also involves calculating the average power of an individual. For instance, a $70\,kg$ boy running up a hill to a height of $3\,m$ in $2\,s$ generates an average power of $\frac{70 \times 9.8 \times 3}{2} = 1029\,Watts$, which rounds to approximately $1030\,Watt$. The SI unit of magnetic flux is the Weber, which can also be expressed as $Joule/Ampere$. Regarding chemical substances, the formula for minium (red lead) is $Pb_3O_4$.

Verbal Reasoning

Word relationships and analogies are used to test logic. FLOWER is to BUNCH as SHIP is to FLEET, representing a part-to-whole or individual-to-group relationship. BLACKSMITH is to IRON as POTTER is to CLAY, identifying a professional and the material they manipulate. COW is to CALF as HORSE is to FOAL, showing a parent-to-offspring link. Antonyms include WISE to FOOLISH, SINGLE to MANY, OPEN to SHUT, and SLEEP to AWAKE. Synonyms include AUDIENCE to VIEWERS, VITAL to IMPORTANT, and DILIGENTLY to HARD.

Sentence completion and grammar often rely on context and idiom. The phrase "Birds of a feather flock together" is a common proverb. Understanding the word 'shrewd' identifies a fellow who is clever or sharp in judgment. The word 'frantically' is synonymous with working wildly or in a panicked state. In grammar, the instruction "tells" you to switch off equipment before leaving. If a person is 'immune' to something, it means they cannot be harmed by it. To 'hone' something means to improve it through practice or sharpening.

Questions & Discussion

Question 34 (Math): Sum of external angles of a 5-sided polygon is? Response: Correct option is C. $360^{\circ}$.

Question 61 (General): Cybercrime means crimes committed through? Response: Correct option is C. mobile and computer.

Question 62 (Pak Studies): Allama Muhammad Iqbal was born in? Response: Correct option is D. Sialkot.

Question 64 (Pak Studies): Who contested in the presidential elections of 1965 against General Ayub Khan? Response: Correct option is B. Miss Fatima Jinnah.

Question 53 (Islamiat): Which of the following is the shortest surah of the Holy Quran? Response: Correct option is B. Surah Al-Kausar.

Question 17 (Verbal): BLACKSMITH : IRON :: Response: Correct option is C. potter : clay.

Question 31 (Quant): The age of a father is 12 times the age of his son. After 5 years the age of father is 6 times the age of his son. The age of father after 3 years is? Response: If son is $x$ and father is $12x$, then $12x+5 = 6(x+5) \rightarrow 12x+5 = 6x+30 \rightarrow 6x = 25 \rightarrow x = 4.16$. Father is $50$. After $3$ years, he will be $53$. Correct option is D. 53 years.