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Question : If $\theta$ is positive acute angle and $7\cos^{2}\theta+3\sin^{2}\theta =4$, then the value of $\theta$ is:
Option 1: $60^{\circ}$
Option 2: $30^{\circ}$
Option 3: $45^{\circ}$
Option 4: $90^{\circ}$
Correct Answer: $60^{\circ}$
Solution : $7\cos^{2}\theta +3\sin^{2}\theta = 4$ $⇒7\cos^{2}\theta +3(1-\cos^{2}\theta) = 4$ ........[we know that $\sin^{2}\theta + \cos^{2}\theta=1$] $⇒4\cos^{2}\theta +3 = 4$ $⇒\cos^{2}\theta = \frac{1}{4}$ $⇒\cos^{2}\theta - \frac{1}{4}$ = 0 $⇒(\cos\theta + \frac{1}{2})(\cos\theta - \frac{1}{2})$ = 0 $⇒\cos\theta =\frac{1}{2}\ \text{or,} -\frac{1}{2}$ $⇒\cos\theta =\cos 60^{\circ}\ \text{or,}\cos 120^{\circ}$ Since $\theta$ is
Question : Simplify the given expression. $\frac{x^3+y^3+z^3-3 x y z}{(x-y)^2+(y-z)^2+(z-x)^2}$
Option 1: $\frac{1}{3}(x+y+z)$
Option 2: $(x+y+z)$
Option 3: $\frac{1}{4}(x+y+z)$
Option 4: $\frac{1}{2}(x+y+z)$
Correct Answer: $\frac{1}{2}(x+y+z)$
Solution : By polynomial identity $x^3+y^3+z^3−3xyz = (x+y+z)(x^2+y^2+z^2−xy−yz−zx)$ = $\frac{1}{2}(x+y+z)(2x^2+2y^2+2z^2−2xy−2yz−2zx)$ = $\frac{1}{2}(x+y+z)(x^2−2xy+y^2+y^2−2yz+z^2+z^2−2xz+x^2)$ = $\frac{1}{2}[(x+y+z)(x−y)^2+(y−z)^2+(z−x)^2]$ $\therefore \frac{x^3+y^3+z^3-3 x y z}{(x-y)^2+(y-z)^2+(z-x)^2}$ = $\frac{\frac{1}{2}[(x+y+z)(x−y)^2+(y−z)^2+(z−x)^2]}{(x-y)^2+(y-z)^2+(z-x)^2}$ = $\frac{1}{2}(x+y+z)$ Hence, the correct answer is $\frac{1}{2}(x+y+z)$.
Question : Directions: In the following question, a sentence is given with a blank that is to be filled in with an appropriate word. Four alternatives are suggested; choose the correct alternative out of them as your answer.
Many important projects have reached the final stage of _________.
Option 1: Accomplishment
Option 2: Initiation
Option 3: Resolution
Option 4: Implementation
Correct Answer: Implementation
Solution : To complete the sentence correctly, we need a word that conveys that these projects have reached the final stage where conclusions or solutions are being determined. The word "resolution" fits here. In the context of projects, it refers to reaching a final decision or finding
Question : Directions: In the following question, some parts of the sentence may have 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'.
Acceptance is not about allowing everything to occur or to go on, (1) / it is neither related to passivity and weakness, (2) / nor is it about confirmation or mediocrity. (3) / No Error (4)
Option 1: 1
Option 2: 2
Option 3: 3
Option 4: 4
Correct Answer: 3
Solution : The error lies in the third part of the sentence because 'is it' should be replaced with 'it is' in order to follow parallelism in case of correlative conjunction neither......nor.
Therefore, the correct sentence should be "Acceptance is not about allowing everything to occur or
Question : Comprehension: In the following passage, some words have been deleted. Fill in the blanks with the help of the alternatives given. Select the most appropriate option for each number. I was away in a (1)_____ part of Japan waiting for a (2)______ train at a small (3)______ station. A young girl was also waiting (4)______ the train. I watched her (5)______ out candy from her bag and eat it. Afterward, she walked (6)______ to a nearby waste bin provided on the (7)_____ to deposit the wrapper. Such is the level of (8)_____to keep surroundings clean from a young age. It is the (9)______ of cleanliness and tidiness which is (10)______ from an early age. Question: Select the most appropriate option to fill in the blank No. 6.
Option 1: briskly
Option 2: briefly
Option 3: wearily
Option 4: loudly
Correct Answer: briskly
Solution : The most appropriate option is the first option.
Explanation: In the context of the passage, the sentence describes the girl's actions after eating the candy. Walking briskly to a nearby waste bin indicates a quick and energetic movement, which is appropriate in the given situation
Question : Directions: Select the correct mirror image of the given figure when the mirror is placed at MN as shown below.
Option 1:
Option 2:
Option 3:
Option 4:
Correct Answer:
Solution : As per the mirror image properties, closer things appear close to the mirror in the reflection. Here, according to the information provided, the mirror is placed on the right side of the figure (on line MN). So, the left side of the reflected image will appear
Question : Directions: Select the option that is related to the third letter in the same way as the second letter is related to the first letter: A : Z :: I : ?
Option 1: V
Option 2: S
Option 3: R
Option 4: T
Correct Answer: R
Solution : Given: A : Z :: I : ?
Like, A : Z→Z is the opposite letter of A. Similarly, follow the same pattern for I : ?→R is the opposite letter of I.
So, the required missing letter is R. Hence, the third option is
Question : The average of the set of scores 512, 528, 556, 584, and 615 is:
Option 1: 586
Option 2: 559
Option 3: 530
Option 4: 600
Correct Answer: 559
Solution : Given: The set of scores 512, 528, 556, 584 and 615. $\text{Average}=\frac{\text{Sum of obseravtions}}{\text{The total number of obseravations}}$ The average of the set of scores 512, 528, 556, 584, and 615 $=\frac{512+528+556+584+615}{5}=\frac{2795}{5}=559$ Hence, the correct answer is 559.
Question : Comprehension: In the following passage, some words have been deleted. Read the passage carefully and select the most appropriate option to fill in each blank. The entertainment (1)_______ built by Walt Disney is sometimes called the ‘House of Mouse’ in acknowledgement of the leading role played by Disney’s most famous creation, Mickey Mouse. Mickey’s round mouse ears, either in silhouette or adorning a hat, have become one of the most (2)_______ corporate logos in the world. But if a few business decisions in the 1920s had gone differently, they could have been rabbit ears. In the mid-1920s Disney was a young (3)________ producing a series called the Alice Comedies, short films that combined live-action footage with animated images so that human actors appeared to interact with cartoon characters. But he had grown tired of the series and wanted to move on to making completely animated films. In 1927 he got his wish, (4)_______ a deal with Universal to produce a new series based on the adventures of a character called Oswald the Lucky Rabbit. Both the name and the species of the character had been picked by Universal—'Oswald’ was (5)________ picked out of a hat of names, and the character was made a rabbit because there were already too many cartoon cats on the market. Question: Select the most appropriate option to fill in the blank number 2.
Option 1: visible
Option 2: measurable
Option 3: notifiable
Option 4: recognisable
Correct Answer: recognisable
Solution : The fourth option is the correct answer.
The term "recognisable" best captures the idea that Mickey Mouse's ears are widely identified and distinguished.
The meanings of the other options are as follows:
Visible means able to be seen.
Measurable means able to be measured.
Notifiable
Question : The capital of Rwanda is:
Option 1: Libreville
Option 2: Bogota
Option 3: Kigali
Option 4: Copenhagen
Correct Answer: Kigali
Solution : The correct answer is Kigali.
Rwanda is also known as the land of a thousand hills. Kigali is the capital city of Rwanda. It is the largest and most populous city in Rwanda. Kigali is also famous for the Kigali Amendment to the Montreal
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