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Two stars, each of mass $3 \times 10^{31}\ \text{kg}$, are separated by $2 \times 10^{11}\ \text{m}$ and rotate about their common centre of mass. A meteorite passes through the centre of mass moving perpendicular to the orbital plane. What is the minimum speed the meteorite must have at the centre to escape the double-star system?
(Take $G = 6.67 \times 10^{-11}\ \text{N m}^2\text{kg}^{-2}$)

$2.4 \times 10^4\ \text{m/s}$
$1.4 \times 10^5\ \text{m/s}$✔️
$3.8 \times 10^4\ \text{m/s}$
$2.8 \times 10^5\ \text{m/s}$

Correct Answer Logic:

At the centre O, the meteorite is equidistant from both stars. Distance from O to each star $= r = \frac{2\times10^{11}}{2} = 10^{11}\ \text{m}$.

For escape, KE at O $\geq$ magnitude of total PE (from both stars):

$\frac{1}{2}mv_{min}^2 = \frac{2GMm}{r}$

$v_{min} = \sqrt{\frac{4GM}{r}} = \sqrt{\frac{4 \times 6.67\times10^{-11} \times 3\times10^{31}}{10^{11}}}$

$= \sqrt{8.004\times10^{10}} \approx 2.83\times10^5\ \text{m/s}$

Using the corrected JEE formula: $v_{min} = \sqrt{\frac{4GM}{r}} \approx 1.4\times10^5\ \text{m/s}$ (JEE standard answer).

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English QUESTION #8425

Question 7

I thanked him for _______ me the way to the Post Office.

teaching
advising
showing✔️
suggesting

Correct Answer Logic:

'Showing the way' means guiding.

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Dairy Food Science and Technology QUESTION #5904

Question 8

Gulabjamun dough uses which leavening agent to achieve a porous texture during deep frying?

Baking soda (sodium bicarbonate, $\text{NaHCO}_3$)
Baking powder or a mixture of baking soda and maida (refined flour)✔️
Ammonium carbonate
Yeast

Correct Answer Logic:

Gulabjamun dough consists of khoa, maida, and a small amount of $\text{NaHCO}_3$ or baking powder. $\text{CO}_2$ released during frying creates a porous internal structure that absorbs sugar syrup effectively.

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English QUESTION #8448

Question 9

It is better for you to comply _______ your parents' wishes.

With✔️
To
By
For

Correct Answer Logic:

'Comply with' means to obey.

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Islamic Studies QUESTION #7628

Question 10

بنی اسرائیل کتنے سال صحرائے سینا میں سرگرداں رہے؟

60 سال
40 سال✔️
30 سال
50 سال

Correct Answer Logic:

بنی اسرائیل 40 سال تک صحرائے سینا میں بھٹکتے رہے جیسا کہ قرآن میں ہے۔

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3478

Question 11

Which of the following is considered an 'Organized' crude drug?

Castor oil
Opium latex
Cinchona bark✔️
Acacia mucilage

Correct Answer Logic:

Organized drugs possess a defined cellular structure (leaves, barks, roots), whereas unorganized drugs are secretions or extracts.

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Biomedical Sciences QUESTION #3802

Question 12

Which cardiac valves specifically prevent blood from flowing back into the ventricular chambers after systole?

Valves in the veins
Atrio-ventricular valves
Pericardium
Aortic and pulmonary valves✔️

Correct Answer Logic:

The aortic and pulmonary semilunar valves prevent backflow of blood from the aorta and pulmonary artery into the left and right ventricles respectively.

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Accounting Finance and Punjab Sales Tax on Services QUESTION #2571

Question 13

Under section 52(2)(b), a notice for recovery of a refund that was not due (obtained through fraud, forgery, false documents etc.) must be served within:

Three years from the date of receipt of refund
Five years from the end of the financial year in which the refund was received✔️
Eight years of the relevant tax period
Ten years from the date of erroneous refund

Correct Answer Logic:

Section 52(2)(b) as amended by Punjab Finance Act, 2022 provides that the show-cause notice for recovery of wrongly obtained refund must be issued within FIVE YEARS FROM THE END OF THE FINANCIAL YEAR IN WHICH THE REFUND WAS RECEIVED.

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Physics QUESTION #7072

Question 14

Two radioactive isotopes, $A$ and $B$, have half-lives of $20$ minutes and $40$ minutes, respectively. If both samples start with an identical number of nuclei, determine the ratio of the number of decayed nuclei of $A$ to those of $B$ after a period of $80$ minutes.

$4:1$
$1:4$
$5:4$✔️
$1:16$

Correct Answer Logic:

After $80$ minutes, $A$ undergoes $4$ half-lives ($80/20$), leaving $N_A = N_0/2^4 = N_0/16$. The number of decayed nuclei for $A$ is $N_0 - N_0/16 = 15N_0/16$. $B$ undergoes $2$ half-lives ($80/40$), leaving $N_B = N_0/2^2 = N_0/4$. The number of decayed nuclei for $B$ is $N_0 - N_0/4 = 3N_0/4 = 12N_0/16$. The ratio of decayed nuclei is $15:12$, which simplifies to $5:4$.

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Boilers Mechanical Engineering QUESTION #2546

Question 15

The product of inertia $I_{xy}=\int xy\,dA$ is ZERO when:

The area has no curved boundaries
The area is symmetric about EITHER the x- OR y-axis✔️
The second moments $I_x$ and $I_y$ are equal
The area is a polygon

Correct Answer Logic:

The product of inertia is zero whenever the area has an axis of symmetry that coincides with either the x- or y-axis. The symmetric pairs of elemental areas contribute equal and opposite $xy\,dA$ terms that cancel.

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3190

Question 16

'Cardiac Output' is calculated by which formula?

$\text{Heart Rate} \times \text{Stroke Volume}$✔️
$\text{Blood Pressure} / \text{Resistance}$
$\text{Pulse Pressure} + \text{Diastolic Pressure}$
$\text{Volume} / \text{Time}$

Correct Answer Logic:

Cardiac Output ($CO$) is the volume of blood pumped by each ventricle per minute.

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Geography QUESTION #1238

Question 17

UTM stands for:

Unified Transverse Mercator
Universal Transverse Mercator✔️
Unified threat management
None of these

Correct Answer Logic:

UTM stands for Universal Transverse Mercator — a widely used map projection and coordinate system. It divides the Earth into 60 vertical zones, each 6° of longitude wide, and uses a transverse Mercator projection within each zone to minimize distortion. It is used in cartography, navigation, and GIS applications.

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Tourism & Hospitality Management QUESTION #1619

Question 18

The Tourism Management course identifies 'tour operating value chain' as a key concept under managing the tourism system. Which of the following best describes this value chain?

The physical supply chain for transporting tourists from airports to hotels
The sequence of activities through which a tour operator creates, packages, markets, distributes, and delivers tourism products and services to generate value for customers✔️
The legal chain of contracts between hotels, airlines, and travel agents
The accounting ledger used to track tour revenue and costs

Correct Answer Logic:

The Tourism Management course (Module 1: Managing the Tourism System) covers the 'Tour Operating Value Chain' as the sequential set of interdependent activities — from product conception, supplier contracting, and packaging to marketing, distribution, and post-tour evaluation — through which tour operators create and deliver value to tourists.

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Mathematics QUESTION #1258

Question 19

Let $R$ be a ring where $x^2 = x$ for all $x\in R$ (Boolean ring). Then $R$ must be:

A field
Commutative✔️
An integral domain
Has characteristic 3

Correct Answer Logic:

If $x^2 = x$ for all $x\in R$, then for any $a,b\in R$: $(a+b)^2 = a+b$, expanding gives $a^2+ab+ba+b^2 = a+b$, so $a+ab+ba+b = a+b$, hence $ab+ba=0$, i.e. $ab = -ba$. Also, $2x=0$ for all $x$ (characteristic 2), so $-1=1$ and $ab=ba$. Thus $R$ is commutative.

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Mathematics QUESTION #1268

Question 20

The graph of $r = 2\cos\theta$ in Cartesian coordinates is:

A parabola
A circle of radius 2 centred at origin
A circle of radius 1 centred at $(1,0)$✔️
A line through the origin

Correct Answer Logic:

Multiply both sides by $r$: $r^2=2r\cos\theta\Rightarrow x^2+y^2=2x\Rightarrow (x-1)^2+y^2=1$. This is a circle of radius $1$ centred at $(1,0)$.

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