Staff Selection Commission Combined Graduate Level Exam
Question : If $x\left(3-\frac{2}{x}\right)=\frac{3}{x}$, then the value of $x^3-\frac{1}{x^3}$ is equal to:
Option 1: $\frac{8}{27}$
Option 2: $\frac{61}{27}$
Option 3: $\frac{62}{27}$
Option 4: $\frac{52}{27}$
Correct Answer: $\frac{62}{27}$
Solution : $x\left(3-\frac{2}{x}\right)=\frac{3}{x}$ ⇒ $3x-2 = \frac{3}{x}$ ⇒ $3x- \frac{3}{x} = 2$ ⇒ $(x- \frac{1}{x}) = \frac{2}{3}$ $\because$ $(x-y)^3 = x^3-y^3-3xy(x-y)$ $(x- \frac{1}{x})^3 = (\frac{2}{3})^3$ ⇒ $x^3-\frac{1}{x^3} - 3(x-\frac{1}{x}) = \frac{8}{27}$ ⇒ $x^3-\frac{1}{x^3} - 3(\frac{2}{3}) = \frac{8}{27}$ ⇒ $x^3-\frac{1}{x^3} = \frac{8}{27}+2$ ⇒ $x^3-\frac{1}{x^3} = \frac{62}{27}$ Hence, the
Question : Directions: From the given alternative words, select the word that cannot be formed using the letters of the given word. PRONOUNCEMENT
Option 1: MOUNT
Option 2: CEMENT
Option 3: PAVEMENT
Option 4: NOUN
Correct Answer: PAVEMENT
Solution : Given: PRONOUNCEMENT
Now, let's compare all the option words with the word PRONOUNCEMENT – First option: MOUNT; All the letters of MOUNT are present in the word PRONOUNCEMENT. Second option: CEMENT; All the letters of CEMENT are present in the word PRONOUNCEMENT. Third option: PAVEMENT;
Question : Select the most appropriate synonym of the underlined word.
During the summer of 1893, Miss. Sullivan and I visited the World's Fair with Dr. Alexander Graham Bell. I recall with unmixed delight those days when a thousand childish fancies became beautiful realities.
Option 1: enchant
Option 2: dismay
Option 3: memory
Option 4: emotion
Correct Answer: enchant
Solution : The correct choice is the first option.
Enchant means to fill with great pleasure. In the sentence, the speaker is recalling days filled with joy, which aligns with the meaning of delight.
The meanings of the other options are as follows:
Question : Directions: Select the correct combination of mathematical signs to sequentially replace the * signs and balance the given equation. 7 * 5 * 4 * 2 * 3 * 7
Option 1: ÷, –, +, ×, =
Option 2: +, –, ÷, –, =
Option 3: ÷, +, ÷, +, =
Option 4: –, ÷, +, ×, =
Correct Answer: +, –, ÷, –, =
Solution : Given: 7 * 5 * 4 * 2 * 3 * 7
Let's check the given options – First option: ÷, –, +, ×, = ⇒ 7 ÷ 5 – 4 + 2 × 3 = 7 Solving the L.H.S. of
Question : If $(x+\frac{1}{x})$ = 5, then the value of $\frac{5x}{x^{2}+5x+1}$ is:
Option 1: $\frac{1}{3}$
Option 2: $\frac{1}{4}$
Option 3: $\frac{1}{2}$
Option 4: $\frac{1}{5}$
Correct Answer: $\frac{1}{2}$
Solution : Given: $(x+\frac{1}{x})$ = 5 ⇒ $x^2+1= 5x$ Now, $\frac{5x}{x^{2}+5x+1}$ = $\frac{5x}{x^{2}+1+5x}$ = $\frac{5x}{5x+5x}$ = $\frac{5x}{10x}$ = $\frac{1}{2}$ Hence, the correct answer is $\frac{1}{2}$.
Question : What is the number of common tangents that can be drawn to two circles that touch each other externally?
Option 1: 2
Option 2: 4
Option 3: 3
Option 4: 1
Correct Answer: 3
Solution : Only three tangents can be drawn passing through two external touching circles. Hence, the correct answer is 3.
Question : Directions: In the following question, some parts of the sentence may have some errors. Find out which part of the sentence has an error and select the appropriate option. If the sentence is free from error, select "No error".
This happened (1) / just exactly (2) / five years ago. (3) / No error (4)
Option 1: (1)
Option 2: (2)
Option 3: (3)
Option 4: (4)
Correct Answer: (2)
Solution : The error lies in the second part of the sentence.
The phrase just exactly is redundant, as just is also an adverb that means exactly, thus making it redundant. Hence, just exactly should be replaced with only exactly.
Therefore, the correct sentence is: "This happened
Question : Directions: In the following question a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series. 1, 3, 7, 13, 21, ?
Option 1: 27
Option 2: 29
Option 3: 31
Option 4: 33
Correct Answer: 31
Solution : Given: 1, 3, 7, 13, 21, ?
Add consecutive even numbers in the previous term to obtain the next term. 1 + 2 = 3; 3 + 4 = 7; 7 + 6 = 13; 13 + 8 = 21; 21 + 10 = 31
Question : The following table shows the shares traded on Mumbai, Rajasthan, Uttar Pradesh and Uttarakhand stock exchanges:
The average of the high rates of the shares in all the four stock exchanges for Kolgate is:
Option 1: INR 34.75
Option 2: INR 51.07
Option 3: INR 52.04
Option 4: INR 63.21
Correct Answer: INR 34.75
Solution : High rate for Kolgate in Mumbai = 34 High rate for Kolgate in Uttarakhand = 60 High rate for Kolgate in Rajasthan = 25 High rate for Kolgate in Uttar Pradesh = 20 Required average = $\frac{34+60+25+20}{4}$ = 34.75 Hence, the correct answer is
Question : What is the difference between the cube and square of the common root of $x^2-8x+15=0$ and $y^2+2y-35=0$
Option 1: 76
Option 2: 100
Option 3: 294
Option 4: 318
Correct Answer: 100
Solution : Here, $x^2-8x+15=0$ ⇒ $(x-5)(x-3)=0$ So, $x=5,3$ $y^2+2y-35=0$ ⇒ $(y-5)(y+7)=0$ So, $y=5,-7$ Here, the common root is 5. So, the difference between the cube and square of 5 $=(5^3-5^2)=125-25=100$ Hence, the correct answer is 100.
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