In a circle with center $C$ and radius $6\sqrt{2}$ cm, $PQ$ and $SR$ are two parallel chords separated by one of the diameters. If $\angle PQC = 45^\circ$, and the ratio of the perpendicular distance of $PQ$ and $SR$ from $C$ is $3:2$, then the area, in sq. cm, of the quadrilateral $PQRS$ is:
Show Hint
- \(20(3 + \sqrt{14})\)
- \(20(3\sqrt{2} + \sqrt{7})\)
- \(4(3 + \sqrt{14})\)
- \(4(3\sqrt{2} + \sqrt{7})\)
The Correct Option is A
Solution and Explanation
Step 2: Length of chord $SR$. Let $CN = d_2$ be the perpendicular distance from $C$ to chord $SR$, with $N$ the midpoint of $SR$. Given \[ d_1 : d_2 = 3 : 2,\quad d_1 = 6, \] so \[ \frac{6}{d_2} = \frac{3}{2} \Rightarrow 3d_2 = 12 \Rightarrow d_2 = 4 \text{ cm}. \] Now in right-angled $\triangle CNR$ (with $CR = R$ and $CN = 4$): \[ CR^2 = CN^2 + NR^2 \Rightarrow (6\sqrt2)^2 = 4^2 + NR^2 \Rightarrow 72 = 16 + NR^2 \Rightarrow NR^2 = 56 \Rightarrow NR = \sqrt{56} = 2\sqrt{14}. \] Thus, \[ SR = 2\cdot NR = 4\sqrt{14} \text{ cm}. \]
Step 3: Area of quadrilateral $PQRS$. Since $PQ \parallel SR$ and they lie on opposite sides of the center, $PQRS$ is a trapezium. Its height is the distance between the two chords, equal to: \[ h = d_1 + d_2 = 6 + 4 = 10 \text{ cm}. \] Parallel sides: \[ a = PQ = 12,\quad b = SR = 4\sqrt{14}. \] Area formula for a trapezium: \[ \text{Area} = \frac{1}{2}(a + b)h = \frac{1}{2}(12 + 4\sqrt{14}) \cdot 10 = 5(12 + 4\sqrt{14}) = 60 + 20\sqrt{14}. \] Factor: \[ \text{Area} = 20(3 + \sqrt{14}) \text{ sq. cm}. \] \[ \boxed{20(3 + \sqrt{14})} \]
Top Questions on Geometry
- If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is:
- The $(x, y)$ coordinates of vertices $P$, $Q$ and $R$ of a parallelogram $PQRS$ are $(-3, -2)$, $(1, -5)$ and $(9, 1)$, respectively. If the diagonal $SQ$ intersects the x-axis at $(a, 0)$, then the value of $a$ is:
- In a $\triangle ABC$, points $D$ and $E$ are on the sides $BC$ and $AC$, respectively. $BE$ and $AD$ intersect at point $T$ such that $AD : AT = 4 : 3$, and $BE : BT = 5 : 4$. Point $F$ lies on $AC$ such that $DF$ is parallel to $BE$. Then, $BD : CD$ is:
- Let $ABCDEF$ be a regular hexagon and $P$ and $Q$ be the midpoints of $AB$ and $CD$, respectively. Then, the ratio of the areas of trapezium $PBCQ$ and hexagon $ABCDEF$ is:
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative
Questions Asked in CAT exam
- Find the number of integer pairs (x, y) that satisfy the following system of inequalities:
\[ \begin{cases} x \geq y \geq 3 \\ x + y \leq 14 \end{cases} \]
- CAT - 2025
- Linear & Quadratic Equations
- The given sentence is missing in the paragraph below. Decide where it best fits among the options 1, 2, 3, or 4 indicated in the paragraph.
Sentence: In each of the affected males, the genetic defect was located to the X chromosome in the region of p11-12.
Paragraph: The first suggested evidence of a human genetic mutation associated with aggressive behaviour came from a study in 1993. ____ (1)____. Genetic and metabolic studies were conducted on a large Dutch family in which several of the males has a syndrome of borderline mental retardation and abnormal behaviour. ____ (2)____. The undesirable behaviour included impulsive aggression, arson and exhibitionism. ____ (3)____. A point mutation was identified in the eighth exon of the monoamine oxidase A (MAOA) structural gene which changes glutamine to a termination codon. ____ (4)____.- CAT - 2025
- Para Completion
Imagine a world in which artificial intelligence is entrusted with the highest moral responsibilities: sentencing criminals, allocating medical resources, and even mediating conflicts between nations. This might seem like the pinnacle of human progress: an entity unburdened by emotion, prejudice or inconsistency, making ethical decisions with impeccable precision. . . .
Yet beneath this vision of an idealised moral arbiter lies a fundamental question: can a machine understand morality as humans do, or is it confined to a simulacrum of ethical reasoning? AI might replicate human decisions without improving on them, carrying forward the same biases, blind spots and cultural distortions from human moral judgment. In trying to emulate us, it might only reproduce our limitations, not transcend them. But there is a deeper concern. Moral judgment draws on intuition, historical awareness and context qualities that resist formalisation. Ethics may be so embedded in lived experience that any attempt to encode it into formal structures risks flattening its most essential features. If so, AI would merely reflect human shortcomings; it would strip morality of the very depth that makes ethical reflection possible in the first place.
Still, many have tried to formalise ethics, by treating certain moral claims not as conclusions, but as starting points. A classic example comes from utilitarianism, which often takes as a foundational axiom the principle that one should act to maximise overall wellbeing. From this, more specific principles can be derived, for example, that it is right to benefit the greatest number, or that actions should be judged by their consequences for total happiness. As computational resources increase, AI becomes increasingly well-suited to the task of starting from fixed ethical assumptions and reasoning through their implications in complex situations.
But, what exactly, does it mean to formalise something like ethics? The question is easier to grasp by looking at fields in which formal systems have long played a central role. Physics, for instance, has relied on formalisation for centuries. There is no single physical theory that explains everything. Instead, we have many physical theories, each designed to describe specific aspects of the Universe: from the behaviour of quarks and electrons to the motion of galaxies. These theories often diverge. Aristotelian physics, for instance, explained falling objects in terms of natural motion toward Earth’s centre; Newtonian mechanics replaced this with a universal force of gravity. These explanations are not just different; they are incompatible. Yet both share a common structure: they begin with basic postulates assumptions about motion, force or mass– and derive increasingly complex consequences. . . .
Ethical theories have a similar structure. Like physical theories, they attempt to describe a domain– in this case, the moral landscape. They aim to answer questions about which actions are right or wrong, and why. These theories also diverge, and even when they recommend similar actions, such as giving to charity, they justify them in different ways. Ethical theories also often begin with a small set of foundational principles or claims, from which they reason about more complex moral problems.- CAT - 2025
- Reading Comprehension
In 1982, a raging controversy broke out over a forest act drafted by the Government of India. This act sought to strengthen the already extensive powers enjoyed by the forest bureaucracy in controlling the extraction, disposal and sale of forest produce. It also gave forest officials greater powers to strictly regulate the entry of any person into reserved forest areas. While forest officials justified the act on the grounds that it was necessary to stop the continuing deforestation, it was bitterly opposed by representatives of grassroots organisations, who argued that it was a major violation of the rights of peasants and tribals living in and around forest areas. . . .
The debate over the draft forest act fuelled a larger controversy over the orientation of state forest policy. It was pointed out, for example, that the draft act was closely modelled on its predecessor, the Forest Act of 1878. The earlier Act rested on a usurpation of rights of ownership by the colonial state which had little precedent in precolonial history. It was further argued that the system of forestry introduced by the British—and continued, with little modification, after 1947—emphasised revenue generation and commercial exploitation, while its policing orientation excluded villagers who had the most longstanding claim on forest resources. Critics called for a complete overhaul of forest administration, pressing the government to formulate policy and legislation more appropriate to present needs. . . .
That debate is not over yet. The draft act was shelved, though it has not as yet been formally withdrawn. Meanwhile, the 1878 Act (as modified by an amendment in 1927) continues to be in operation. In response to its critics, the government has made some important changes in forest policy, e.g., no longer treating forests as a source of revenue, and stopping ecologically hazardous practices such as the clearfelling of natural forests. At the same time, it has shown little inclination to meet the major demand of the critics of forest policy—namely, abandoning the principle of state monopoly over forest land by handing over areas of degraded forests to individuals and communities for afforestation.
. . . [The] 1878 Forest Act itself was passed only after a bitter and prolonged debate within the colonial bureaucracy, in which protagonists put forward arguments strikingly similar to those being advanced today. As well known, the Indian Forest Department owes its origin to the requirements of railway companies. The early years of the expansion of the railway network, c. 1853 onwards, led to tremendous deforestation in peninsular India owing to the railway’s requirements of fuelwood and construction timber. Huge quantities of durable timbers were also needed for use as sleepers across the new railway tracks. Inexperienced in forestry, the British called in German experts to commence systematic forest management. The Indian Forest Department was started in 1864, with Dietrich Brandis, formally a Lecturer in Botany, as the first Inspector General of Forests. The early years of the forest department, even as it grew, continued to meet the railway needs for timber and wood. These systems first emerged as part of the needs of the expanding empire.- CAT - 2025
- Reading Comprehension
Over the course of the twentieth century, humans built, on average, one large dam a day, hulking structures of steel and concrete designed to control flooding, facilitate irrigation, and generate electricity. Dams were also lucrative contracts, large-scale employers, and the physical instantiation of a messianic drive to conquer territories and control nature. Some of the results of that drive were charismatic mega-infrastructure—the Hoover on the Colorado River or the Aswan on the Nile—but most of the tens of thousands of dams that dot the Earth’s landscape have drawn little attention. These are the smaller, though not inconsequential, barriers that today impede the flow of water on nearly two-thirds of the world’s large waterways. Chances are, what your map calls a “lake” is actually a reservoir, and that thin blue line that emerges from it once flowed very differently.
Damming a river is always a partisan act. Even when explicit infrastructure goals—irrigation, flood control, electrification—were met, other consequences were significant and often deleterious. Across the world, river control displaced millions of people, threatening livelihoods, foodways, and cultures. In the western United States, dams were often an instrument of colonialism, used to dispossess Indigenous people and subsidize settler agriculture. And as dams slowed the flow of water, inhibited the movement of nutrients, and increased the amount of toxic algae and other parasites, they snuffed out entire river ecologies. Declining fish populations are the most evident effect, but dams also threaten a host of other animals—from birds and reptiles to fungi and plants—with extinction. Every major dam, then, is also a sacrifice zone, a place where lives, livelihoods, and ways of life are eliminated so that new sorts of landscapes can support water-intensive agriculture and cities that sprout downstream of new reservoirs.
Such sacrifices have been justified as offerings at the temples of modernity. Justified by—and for—whom, though? Over the course of the twentieth century, rarely were the costs and benefits weighed thoughtfully and decided democratically. As Kader Asmal, chair of the landmark 2000 World Commission on Dams, concluded, “There have been precious few, if any, comprehensive, independent analyses as to why dams came about, how dams perform over time, and whether we are getting a fair return from our 2 trillion Dollar investment.” A quarter-century later, Asmal’s words ring ever truer. A litany of dams built in the mid-twentieth century are approaching the end of their expected lives, with worrying prospects for their durability. Droughts, magnified and multiplied by the effects of climate change, have forced more and more to run below capacity. If ever there were a time to rethink the mania for dams, it would be now.
There is some evidence that a combination of opposition, alternative energy sources, and a lack of viable projects has slowed the construction of major dams. But a wave of recent and ongoing construction, from India and China to Ethiopia and Canada, continues to tilt the global balance firmly in favor of water impoundment.- CAT - 2025
- Reading Comprehension