Staff Selection Commission Combined Graduate Level Exam
Question : If $\sin \theta+\cos \theta=\frac{\sqrt{3}-1}{2 \sqrt{2}}$, then what is the value of $\tan \theta+\cot \theta$?
Option 1: $8(\sqrt{3}-2)$
Option 2: $12(\sqrt{3}-2)$
Option 3: $12(\sqrt{3}+2)$
Option 4: $8(\sqrt{3}+2)$
Correct Answer: $8(\sqrt{3}-2)$
Solution : Given, $\sin \theta+\cos \theta=\frac{\sqrt{3}-1}{2 \sqrt{2}}$ Squaring both sides, we get, ⇒ $(\sin \theta+\cos \theta)^2=(\frac{\sqrt{3}-1}{2 \sqrt{2}})^2$ ⇒ $\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=\frac{3+1-2\sqrt3}{8}$ ⇒ $1+2\sin\theta\cos\theta=\frac{4-2\sqrt{3}}{8}$ ⇒ $2\sin\theta\cos\theta=\frac{2-\sqrt3}{4}-1$ ⇒ $2\sin\theta\cos\theta=\frac{2-\sqrt3-4}{4}$ ⇒ $\sin\theta\cos\theta=\frac{-2-\sqrt3}{8}$ Now consider, $\tan \theta+\cot \theta$ $=\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}$ $=\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}$ $=\frac{1}{\sin\theta\cos\theta}$ $=\frac{1}{\frac{-2-\sqrt3}{8}}$ $=\frac{-8}{2+\sqrt3}$ Rationalizing it, we get, $=\frac{-8}{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}$ $=\frac{-8(2-\sqrt3)}{2^2-(\sqrt3)^2}$ $=\frac{8(\sqrt{3}-2)}{4-3}=8(\sqrt3-2)$ Hence, the correct
Question : Select the most appropriate synonym of the underlined word.
It was due to treacherous leaders that he decided to leave the party.
Option 1: Devoted
Option 2: Disloyal
Option 3: Transient
Option 4: Ravenous
Correct Answer: Disloyal
Solution : The correct choice is the second option.
Disloyal is the most appropriate synonym for treacherous, as both convey the idea of betrayal and untrustworthiness.
The meanings of the options are as follows:
Devoted: Faithful and loyal.
Transient: Temporary or lasting for only a
Question : Directions: In the following question, select the related word from the given alternatives. Ampere : Electric Current :: Fathom : ?
Option 1: Depth of Water
Option 2: Frequency
Option 3: Sound level
Option 4: Work or Energy
Correct Answer: Depth of Water
Solution : Given: Ampere : Electric Current :: Fathom : ?
The ampere is a unit of measurement used to quantify electric current. Similarly, a fathom is a unit used to measure the depth of water.
Hence, the first option is correct.
Question : Directions: Select the set in which the numbers are related in the same way as the numbers of the following sets. (121, 81, 2) (88, 54, 2) (NOTE: Operations should be performed on the whole numbers without breaking down the numbers into their constituent digits. E.g.13 – operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
Option 1: (13, 5, 9)
Option 2: (78, 6, 45)
Option 3: (31, 56, 98)
Option 4: (77, 18, 5)
Correct Answer: (77, 18, 5)
Solution : Given: (121, 81, 2); (88, 54, 2)
In the above-given set of numbers, divide the first number by 11 and the second number by 9. Then, subtract both the results. Like, (121, 81, 2); (121 ÷ 11) – (81 ÷ 9) = 11
Question : Comprehension:
Read the passage and answer the questions that follow.
The drainage systems of India are mainly controlled by the broad relief features of the subcontinent. Accordingly, the Indian rivers are divided into two major groups: the Himalayan rivers; and the Peninsular rivers.
Apart from originating from the two major physiographic regions of India, the Himalayan and the Peninsular rivers are different from each other in many ways. Most of the Himalayan rivers are perennial. It means that they have water throughout the year. These rivers receive water from rain as well as from melted snow from the lofty mountains. The major Himalayan rivers are the Indus, the Ganga, and the Brahmaputra. These rivers are long and are joined by many large and important tributaries. A river along with its tributaries may be called a river system. The two major Himalayan rivers, the Indus and the Brahmaputra originate from the north of the mountain ranges. They cut through the mountains making gorges in the upper course. The perennial Himalayan rivers have long courses from their source to the sea. They perform intensive erosional activity in their upper courses and carry huge loads of silt and sand with their swift current. In the middle and the lower courses, these rivers form meanders, oxbow lakes, levees, and many other depositional features in their floodplains as their speed slackens. They also have well-developed deltas.
A large number of the Peninsular rivers like the Godavari, Kaveri, and Krishna are seasonal, as their flow is dependent on rainfall. During the dry season when it doesn’t rain, even the large rivers have reduced flow of water in their channels. The Peninsular rivers have shorter and shallower courses as compared to their Himalayan counterparts. However, some of them originate in the central highlands and flow towards the west. Most of the rivers of peninsular India originate in the Western Ghats and flow towards the Bay of Bengal.
Question:
Erosional activity takes place in the upper course of the river because of:
Option 1: the reduced flow of water
Option 2: huge loads of silt and sand
Option 3: the great speed of water
Option 4: the presence of flood plain
Correct Answer: the great speed of water
Solution : The correct choice is the third option.
Explanation: Erosional activity takes place in the upper course of the river due to the great speed of the water. In the passage, it is mentioned that the perennial Himalayan rivers perform intensive erosional
Question : A thief running at a speed of ‘$x$’ km/hr is chased by a policeman running at a speed of 10 km/hr. If the thief is ahead by 100 metres, the policeman catches the thief after 3 minutes. At what speed is the thief running (‘$x$’ being the unknown speed)?
Option 1: 4 km/hr
Option 2: 8 km/hr
Option 3: 10 km/hr
Option 4: 6 km/hr
Correct Answer: 8 km/hr
Solution : Let the speed of the thief be $x$ km/hr. Speed of the policeman = 10 km/hr Time taken by a policeman to catch the thief = 3 minutes = $\frac{1}{20}$ hr Relative distance to be covered by policeman = 100 metres = 0.1 km
Question : Directions: F is the brother of A. C is the daughter of A. K is the sister of F. G is the brother of C. Who is the uncle of G?
Option 1: A
Option 2: C
Option 3: K
Option 4: F
Correct Answer: F
Solution : As per the given information, the family tree will be as follows –
Here, the quadrilateral represents the male, and the circular figure represents the female in the figure.
So, from the above family tree, F is the uncle of G. Hence, the fourth option
Question : A sentence has been given a blank to be filled with an appropriate word. Choose the correct alternative.
To make your own gargle, _______ half a teaspoon of salt in a glass of lukewarm or partly cooled boiled water.
Option 1: cease
Option 2: cast
Option 3: consolidate
Option 4: dissolve
Correct Answer: dissolve
Solution : The fourth option is the correct choice.
When making a gargle solution, you want the salt to be thoroughly mixed or blended into the water, forming a solution. The word, dissolve specifically refers to the process of mixing a solid substance (like salt) into a
Question : The salt concentration (measured as salinity in parts per thousand) is greater than __________ in hyper-saline lagoons.
Option 1: 50
Option 2: 100
Option 3: 150
Option 4: 200
Correct Answer: 100
Solution : The correct option is 100.
Hypersaline lagoons have much higher salinity levels than the open ocean, often exceeding 100 ppt. This high salt content results in unique chemical composition and diverse sediment forms. The average salinity level in ocean water is around 35 ppt,
Question : Select the word segment that substitutes (replaces) the bracketed word segment correctly and completes the sentence meaningfully: Estella had hardly heard the news (after) she began to weep.
Option 1: while
Option 2: well
Option 3: than
Option 4: when
Correct Answer: when
In the sentence, the word after indicates a temporal relationship, suggesting that one action occurred subsequent to another.
In this case, when serves this purpose effectively, indicating that Estella's weeping started immediately or almost immediately upon hearing the
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