3rd aug25

Maths 1

Question 3

Topic: Set Theory & Relations

Consider the following relations defined on the set of integers:

$R_{1} = {(x, y) : x, y \in Z and ∣ x - y ∣ \leq 5}$ .
$R_{2} = {(x, y) : x, y \in Z and 5 divides x - y}$ .

Choose the correct option:

$R_{1}$ is reflexive and transitive.
$R_{2}$ is reflexive and not transitive.
$R_{2}$ is symmetric but not transitive.
$R_{2}$ is an equivalence relation but $R_{1}$ is not an equivalence relation.

Answer: Option 4. $R_{2}$ is an equivalence relation but $R_{1}$ is not an equivalence relation.

Question 4

Topic: Algebra (Quadratic Equations)

Aman and Prakash want to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in getting roots as 3 and 4. Prakash made a mistake in writing down the coefficient of x and got the roots as 2 and 3. Consider the leading coefficient to be 1 in all cases. The correct roots of the quadratic equation are:

1 and 5
2 and 6
1 and 6
2 and 5

Answer: Option 3. 1 and 6

Question 5

Topic: Coordinate Geometry

Which of the following options is/are true?

If $m$ and $n$ are $x -$ intercept and $y -$ intercept of the line $2 x + 3 y = 6$ , respectively, then $m + n = 5$ .
The $x -$ intercept of a line is the perpendicular distance to the line from the origin.
The lines $2 x + 3 y = 6$ and $3 x - 2 y = 6$ are parallel to each other.
The distance between two parallel lines $2 x + 3 y = 6$ and $4 x + 6 y = 12$ is 2.

Answer: Option 1. If $m$ and $n$ are $x -$ intercept and $y -$ intercept of the line $2 x + 3 y = 6$ , respectively, then $m + n = 5$ .

Question 6

Topic: Coordinate Geometry

Let $l$ be the equation of line which passes through the point (1,6) and parallel to $y = 7 x + 6$ . Then which of the following are correct.

The slope of $l$ is 8.
The y-intercept of $l$ is 1.
The equation of line $l$ is $y = 7 x - 1$ .
The equation of line $l$ is $y = - x + 7$ .

Answer: Option 3. The equation of line $l$ is $y = 7 x - 1$ .

Question 7

Topic: Algebra (Polynomials)

Consider the three polynomials

$p (x) = 4 x^{5} + 9 x^{4} + b_{1} x^{2} + c_{1}$ .
$q (x) = - 5 x^{4} + 8 x^{2} + b_{2} x + c_{2}$ .
$s (x) = - x^{7} + 72 x^{5} + b_{3} x^{3} + c_{2} x^{2} + d_{3} x + e_{3}$ .

Which of the following options is/are true?

$q (x) \to - \infty$ as $x \to \infty$ .
$p (x) \to \infty$ as $x \to \infty$ .
$q (x) \to \infty$ as $x \to \infty$ .
$s (x) \to - \infty$ as $x \to \infty$ .

Answer: Options 1, 2, and 4.

Question 8

Topic: Algebra (Polynomial Graph Analysis)

Consider the following polynomial $p (x)$ whose graph is given below:-

Which of the following options is/are correct?

Multiplicity of root -1 and 1 may be the same.
Multiplicity of root 2 may be 3.
The number of turning points is 5.
$p (x) \to \infty$ as $x \to \infty$ .

Answer: Options 1, 2, and 4.

Question 9

Topic: Set Theory

The Cartesian product $A \times A$ has 9 elements. Two of the elements of the Cartesian product are (3, 2) and (2, 7). Find the sum of all the elements in set A.

Answer: 12

Question 10

Topic: Set Theory (Venn Diagrams)

In a survey of 500 people, it was found that 49% liked to watch comedy movies, 53% liked thriller movies, and 62% liked romantic movies. Also, 27% liked to watch both comedy and thriller, 29% liked to watch both thriller and romantic, and 28% liked watching both comedy and romantic. 5% liked none of the above genres. How many people like all the three movies?

Answer: 75

Question 11

Topic: Functions & Optimization

A function $f (x)$ , fit for the data given in the Table-1 recorded by a student, is $y = f (x) = (x - 2) (x - 4) (x - 6) + k$ The data is:

x	1	3	5	6
y	-11	7	1	4

What will be the value of $k$ , so that SSE (Sum Squared Error) will be minimum?

Answer: 4

Comprehension for Questions 12 & 13

Topic: Algebra (Polynomials & Quadratics)

The polynomial $p (x) = a (x - 4) (x - 6) (x - 8) (x - 10)$ passes through the vertex of the quadratic function $q (x) = - (x - 7)^{2} - 9$ . Use this information to answer the given subquestions.

Question 12

Topic: Algebra (Quadratic Functions)

Enter the x-coordinate of the vertex of $q (x)$ .

Answer: 7

Question 13

Topic: Algebra (Polynomials)

Enter the value of $a$ .

Answer: -1

Comprehension for Questions 14 & 15

Topic: Algebra (Polynomial Graphs)

Consider the polynomials $p_{1} (x) = (x^{2} - 4 x + 3) (x^{2} - b x + c)$ and $p_{2} (x) = (x^{2} - 4 x + 3) (x^{2} + b x + c)$ where $b, c \in R$ . Suppose $p_{2} (- 1) = 0$ and Y-intercept of $p_{1} (x)$ is (0, 3). Consider the following figures.

Question 14

Topic: Algebra (Polynomial Graphs)

Which of following graphs can represent the graph of $p_{1} (x)$ ?

Figure 1
Figure 2
Figure 3
Figure 4

Answer: Option 4. Figure 4

Question 15

Topic: Algebra (Polynomial Graphs)

Which of following graphs can represent the graph of $p_{2} (x)$ ?

Figure 1
Figure 2
Figure 3
Figure 4

Answer: Option 1. Figure 1

English 1

Comprehension for Questions 17 to 26

Topic: Reading Comprehension

Read the passage and answer the given subquestions.

I spotted it in a junk shop in Bridport, a roll-top desk. The man said it was early nineteenth century, and oak. I had wanted one, but they were far too expensive. This one was in a bad condition, the roll-top in several pieces, one leg clumsily mended, scorch marks all down one side. It was going for very little money. I thought I could restore it. It would be a risk, a challenge, but I had to have it. I paid the man and brought it back to my workroom at the back of the garage. I began work on it on Christmas Eve.

I removed the roll-top completely and pulled out the drawers. The veneer had lifted almost everywhere — it looked like water damage to me. Both fire and water had clearly taken their toll on this desk. The last drawer was stuck fast. I tried all I could to ease it out gently. In the end I used brute force. I struck it sharply with the side of my fist and the drawer flew open to reveal a shallow space underneath, a secret drawer. There was something in there. I reached in and took out a small black tin box. Sello-taped to the top of it was a piece of lined notepaper, and written on it in shaky handwriting: “Jim’s last letter, received January 25, 1915. To be buried with me when the time comes.” I knew as I did it that it was wrong of me to open the box, but curiosity got the better of my scruples. It usually does.

Inside the box there was an envelope. The address read: “Mrs Jim Macpherson, 12 Copper Beeches, Bridport, Dorset.” I took out the letter and unfolded it. It was written in pencil and dated at the top — “December 26, 1914”.

(Source: The Best Christmas Present In The World - Michael Morpurgo)

Question 17

Topic: Vocabulary

What is the meaning of scorch marks?

Burn marks
Birthmarks
Watermarks
Telemarks

Answer: Option 1. Burn marks

Question 18

Topic: Vocabulary (Phrases)

“Both fire and water had clearly taken their toll on this desk.” What does the phrase “take their toll on” mean?

To achieve a very good result
To ignore
To cause damage
To go crazy

Answer: Option 3. To cause damage

Question 19

Topic: Reading Comprehension

The desk is made of __________.

Teak wood
Oak wood
Pine wood
Ash wood

Answer: Option 2. Oak wood

Question 20

Topic: Vocabulary

What is the meaning of scruples?

To be fearful or frightened
Feelings that make you hesitate to do something wrong
A feeling of pensive sadness
A feeling of intense excitement and happiness

Answer: Option 2. Feelings that make you hesitate to do something wrong

Question 21

Topic: Reading Comprehension

What did the author find in a junk shop?

A desk
A lamp
A bed
A mirror

Answer: Option 1. A desk

Question 22

Topic: Reading Comprehension

Fill in the blank with the correct option. A piece of lined notepaper, and written on it in shaky handwriting: “Jim’s last letter, received _________ To be buried with me when the time comes.”

December 26, 1914
December 26, 1915
January 25, 1915
January 25, 1914

Answer: Option 3. January 25, 1915

Question 23

Topic: Vocabulary (Phrases)

“The last drawer was stuck fast.” What is the meaning of “stuck fast”?

To avoid close contact
To stay calm
To have work to do
Shut tight

Answer: Option 4. Shut tight

Question 24

Topic: Reading Comprehension

Choose the correct option. What did the author find in a secret drawer?

A brown tin box
A yellow tin box
A pink tin box
A black tin box

Answer: Option 4. A black tin box

Question 25

Topic: Reading Comprehension

Choose the correct option. When did the author begin work on the desk?

Christmas Eve
New Year’s day
Valentine’s day
Mother’s day

Answer: Option 1. Christmas Eve

Question 26

Topic: Vocabulary

Choose the correct option for the following statement: Veneer is a thin layer of decorative wood that is glued to a base of inferior wood.

TRUE
FALSE

Answer: Option 1. TRUE

Question 27

Topic: Phonetics (Consonant Clusters)

The word scamper has ____ consonant clusters in it. The word begins with a ____ cluster.

2; CCC
1; CC
2; CC
1; CCC

Answer: Option 3. 2; CC

Question 28

Topic: Grammar (Nouns)

Which one of the following words is an abstract noun?

Shoes
Cake
Jealousy
Teacher

Answer: Option 3. Jealousy

Question 29

Topic: Grammar (Pronouns)

“Whose brilliant idea was this?” What type of pronoun is “whose”?

Possessive
Interrogative
Demonstrative
Proper

Answer: Option 2. Interrogative

Question 30

Topic: Grammar (Prepositions)

“He’s been living in Mumbai ________ ten years” Choose the correct preposition.

Since
For
During
While

Answer: Option 2. For

Question 31

Topic: Grammar (Parts of Speech)

Choose which part of speech the underlined part belongs to: He loves his sister dearly.

Noun
Pronoun
Verb
Adverb

Answer: Option 4. Adverb

Question 32

Topic: Grammar (Adverbs)

Identify the adverb in the following sentence. Never underestimate the power of a woman.

Never
underestimate
power
woman

Answer: Option 1. Never

Question 33

Topic: Grammar (Articles)

Fill in the blank with the correct option. Generally, ____boys are physically stronger than girls.

A
The
No article
An

Answer: Option 3. No article

Question 34

Topic: Grammar (Articles)

Fill in the blank with the correct option. I left it at ________ office.

A
The
No article
An

Answer: Option 2. The

Question 35

Topic: Grammar (Sentence Structure)

Which of the following sentences is correct?

He is fastest runner in the team.
He is a fastest runner in the team.
He is the fastest runner in the team.
He is an fastest runner in the team.

Answer: Option 3. He is the fastest runner in the team.

Question 36

Topic: Grammar (Interjections)

Identify the interjection expressing disgust:

Bravo!
Eek!
Yuck!
Whoa!

Answer: Option 3. Yuck!

Question 37

Topic: Phonetics (Vowel Sounds)

The vowel sound between b and n in the word bun is the same as:

That which occurs between w and n in the word wan
That which occurs between r and m in the word crumb
That which occurs between b and n in the word bean
That which occurs between b and n in the word bin

Answer: Option 2. That which occurs between r and m in the word crumb

Question 38

Topic: Phonetics (Consonant Sounds)

Choose the word with a “ng” sound from the following.

Song
Hang
Feign
Both Song and Hang

Answer: Option 4. Both Song and Hang

Question 39

Topic: Grammar (Nouns)

Which one of the following words is a noun?

Book
Quietly
Frequently
Repeatedly

Answer: Option 1. Book

Question 40

Topic: Grammar (Parts of Speech)

The underlined word is a/an ______. As soon as he entered the room, he went and sat beside his best friend.

Preposition
Adjective
Adverb
Article

Answer: Option 1. Preposition

Question 41

Topic: Grammar (Articles)

Choose the correct option: There is ___ cup of coffee on the table.

A
An
The
None of these

Answer: Option 1. A

Question 42

Topic: Vocabulary (Idioms)

Fill in the blank with the most appropriate idiom from the given options. Rahul tore up his classmate’s notebook and the teacher_______.

Broke a leg
Cut corners
Missed the boat
Read him the riot act

Answer: Option 4. Read him the riot act

Question 43

Topic: Vocabulary (Idioms)

Identify what the idiom in bold means. ‘I manage to travel for pleasure once in a blue moon.’

Once a month, when there is full moon
Twice a month, when there is no moon and when there is a full moon
Very rarely
Never

Answer: Option 3. Very rarely

Question 44

Topic: Vocabulary (Phrasal Verbs)

Fill in the blank with the appropriate option. She takes after her mother. The meaning of the phrasal verb ‘take after’ is __________.

Begin a new hobby
Resemble somebody in appearance or character
Accept the job
Write on a paper

Answer: Option 2. Resemble somebody in appearance or character

Question 45

Topic: Vocabulary (Phrasal Verbs)

Fill in the blank with the appropriate option. The kids were well __________.

Got off
Gave up
Brought up
Pick up

Answer: Option 3. Brought up

Question 46

Topic: Grammar (Modal Verbs)

Fill in the blank with the most suitable modal verb. Stealing __________ get us into no end of trouble. Hint: Future possibility about which we are not sure.

Wouldn’t
Must
Shan’t
Might

Answer: Option 4. Might