Staff Selection Commission Combined Graduate Level Exam
Question : If $5\cos\theta+12\sin\theta=13,\ 0^0<\theta<90^0$, then the value of $\sin\theta$ is:
Option 1: $\frac{5}{13}$
Option 2: $–\frac{12}{13}$
Option 3: $\frac{6}{13}$
Option 4: $\frac{12}{13}$
Correct Answer: $\frac{12}{13}$
Solution : Given: $5\cos\theta+12\sin\theta=13,\ 0^0<\theta<90^0$ Here, $5\cos\theta+12\sin\theta=13$ ⇒ $(\frac{5}{13})\cos\theta+(\frac{12}{13})\sin\theta=1$ Comparing it with $\cos^2\theta+\sin^2\theta=1$ we get, $\cos\theta=\frac{5}{13}$ and $\sin\theta=\frac{12}{13}$ Hence, the correct answer is $\frac{12}{13}$.
Question : What is the ratio between the HCF and LCM of the numbers whose LCM is 48 and the product of the numbers is 384?
Option 1: 1 : 4
Option 2: 1 : 6
Option 3: 1 : 3
Option 4: 2 : 5
Correct Answer: 1 : 6
Solution : Let the HCF be a. We know that, HCF × LCM = Product of two numbers ⇒ 48 × a = 384 $\therefore$ a = 8 $\therefore$ HCF : LCM = 8 : 48 = 1 : 6 Hence, the correct answer is
Question : Directions: Select the option in which the following figure is embedded. (Rotation is NOT allowed)
Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer:
Solution : Since the rotation is restricted. So, the embedded figure will have the same orientation as the main figure. By comparison of all the option figures, the given question figure is embedded only in the first option figure.
Hence, the first option is correct.
Question : A six-digit number is divisible by 33. If 54 is added to the number, then the new number formed will also be divisible by:
Option 1: 3
Option 2: 2
Option 3: 5
Option 4: 7
Correct Answer: 3
Solution : A six-digit number is divisible by 33. Let the number be $x$. So, that digit is also divided by the factor of 33, which is 3, 11. If 54 is added then the number will be $x + 54$. Factors of 54 are 2, 3,
Question : If $x+\frac{1}{x}=9$, find $x^4+\frac{1}{x^4}$.
Option 1: 5431
Option 2: 6561
Option 3: 6156
Option 4: 6239
Correct Answer: 6239
Solution : Given: $x+\frac{1}{x} = 9$ Squaring both sides of the given equation, we get, $⇒(x+\frac{1}{x})^2 = 9^2$ $⇒(x^2+\frac{1}{x^2}+2) =81$ $⇒(x^2+\frac{1}{x^2}) = 79$ Again, squaring both sides of the above equation, we get, $⇒(x^2+\frac{1}{x^2})^2 = (79)^2$ $⇒(x^4+\frac{1}{x^4}+2) = 6241$ $⇒(x^4+\frac{1}{x^4}) = 6239$ Hence, the correct answer is
Question : Simplify $\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
Option 1: $\frac{3 \sqrt{2}+\sqrt{6}}{8}$
Option 2: $\frac{\sqrt{3}}{2 \sqrt{2}-2 \sqrt{6}}$
Option 3: $\frac{3 \sqrt{2}-\sqrt{6}}{8}$
Option 4: $\frac{\sqrt{3}}{2 \sqrt{6}-2 \sqrt{2}}$
Correct Answer: $\frac{3 \sqrt{2}-\sqrt{6}}{8}$
Solution : $\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$ = $\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}$ = $\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}$ = $\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}\times\frac{\sqrt{2}(\sqrt{3}-1)}{\sqrt{2}(\sqrt{3}-1)}$ = $\frac{3 \sqrt{2}-\sqrt{6}}{8}$ Hence, the correct answer is $\frac{3 \sqrt{2}-\sqrt{6}}{8}$.
Question : 25 men and 45 women can complete a piece of work in 15 days, while 15 men and 60 women can complete it in 20 days. In how many days can 69 men and 67 women complete the work?
Option 1: 10
Option 2: 5
Option 3: 8
Option 4: 6
Correct Answer: 6
Solution : According to the question, they complete the same work ⇒ M1D1 = M2D2 ⇒ (25M + 45W) × 15 = (15M + 60W) × 20 ⇒ (25M + 45W) × 3 = (15M + 60W) × 4 ⇒ 75M
Question : Directions: In a certain code language, ARTIST is written as ASTITT, and ADVICE is written as AEVIDE. How will ASPECT be written in that language?
Option 1: ATEPDT
Option 2: ATQFDT
Option 3: ATPEDT
Option 4: ATPECT
Correct Answer: ATPEDT
Solution : Given: ARTIST is written as ASTITT, and ADVICE is written as AEVIDE.
Add 1 to the place value of the second and fifth letter, and all the remaining letters will be the same, to obtain the required code –
Thus, ARTIST is coded as ASTITT.
Question : Directions: Select the set in which the numbers are related in the same way as the numbers of the given sets. (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.) (4, 6, 8) (6, 8, 10)
Option 1: (2, 12, 8)
Option 2: (8, 12, 16)
Option 3: (5, 30, 10)
Option 4: (18, 18, 16)
Correct Answer: (8, 12, 16)
Solution : Given: (4, 6, 8); (6, 8, 10)
In the given sets, add the first and third numbers and divide the resultant by 2 to get the second number. (4, 6, 8)→4 + 8 = 12; 12 ÷ 2 = 6 (6, 8, 10)→6
Question : If $\sec \theta+\frac{1}{\cos \theta}=2$, find the value of $\sec ^{55} \theta+\frac{1}{\sec ^{55} \theta}$.
Option 1: 2
Option 2: 0
Option 3: 1
Option 4: 55
Correct Answer: 2
Solution : $\sec \theta+\frac{1}{\cos \theta}=2$ ⇒ $\sec\theta \cos \theta + 1 = 2 \cos\theta$ ⇒ $1 + 1 = 2 \cos\theta$ ⇒ $\cos\theta = 1$ ⇒ $\sec\theta = 1$ ⇒ $\sec ^{55} \theta+\frac{1}{\sec ^{55} \theta}= 1 + 1 = 2$ Hence, the correct answer is 2.
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