PYQs

Maths 1 - Previous Year Questions (PYQs)

This file contains a comprehensive “Problem Bank” of Previous Year Questions for Quiz 1, Quiz 2, and Quiz 3 / Qualifier exams. It includes questions, options, and accepted answers, along with “Mental Algorithms” for key patterns.

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Quiz 1: Foundation (Sets, Relations, Graphics)

October 2025 Version

Question 1

Topic: Real Number System & Rationalization Which of the following numbers are rational?

1. $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$
2. $(\sqrt{8}-\sqrt{2})(\sqrt{18}+\sqrt{2})$
3. $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$
4. \dfrac{\sqrt{8}+& \sqrt{2}}{\sqrt{8}-\sqrt{2}}

Accepted Answers:

• Options 1 and 3.

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Question 2

Topic: Functions & Domain Let $f(x)=\frac{1}{\sqrt{x^2-5x+6}}$. Find the domain of $f$.

• $(-\infty ,2)\cup (3,\infty )$
• $[2,3]$
• $(2,3)$
• $\mathbb{R}\setminus \{2,3\}$

Accepted Answers:

• $(-\infty ,2)\cup (3,\infty )$

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Question 3

Topic: Quadratic Functions (Vertex & Intersections) Consider the two quadratic functions $y_1=x^2-4x+1$ and $y_2=x^2-6x-3$. A quadratic function $f(x)=a{x}^2+bx+c$ has its vertex at the intersection point of $y_1$ and $y_2$, its $y$-intercept is 2, and its axis of symmetry is parallel to the $y$-axis. What is the equation of $f(x)$?

• $y=\frac{-11}{4}(x+2{)}^2+13$
• $y=\frac{-11}{4}(x-2{)}^2+13$
• $y=\frac{11}{4}(x+2{)}^2+13$
• $y=\frac{-11}{4}(x+2{)}^2-13$

Accepted Answers:

• $y=\frac{-11}{4}(x+2{)}^2+13$

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Question 5

Topic: Relations (Equivalence) Let $M$ be the set of all online degree courses.

• $R=\{(A,B)\mid A\text{ and }B\text{ are courses in the same level}\}$
• $S=\{(A,B)\mid A\text{ is a prerequisite for }B\}$ Which of the following statements is/are correct?
• $R$ is an equivalence relation.
• $S$ is an equivalence relation.
• $R$ is not transitive.
• $S$ is not symmetric.

Accepted Answers:

• $R$ is an equivalence relation.
• $S$ is not symmetric.

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Question 6

Topic: Coordinate Geometry (Parabolas & Lines) Which of the following statements is(are) true?

• $y-6=3(x-10{)}^2$ is an equation of a parabola whose vertex is at $(10,6)$.
• $y-6=3(x-10{)}^2$ is an equation of a parabola whose vertex is at $(6,10)$.
• $-5x+4y-1=0$ and $\frac{x}{4}-\frac{y}{5}=1$ are perpendicular.
• $-5x+4y-1=0$ and $\frac{x}{4}-\frac{y}{5}=1$ are parallel.

Accepted Answers:

• $y-6=3(x-10{)}^2$ is an equation of a parabola whose vertex is at $(10,6)$.
• $-5x+4y-1=0$ and $\frac{x}{4}-\frac{y}{5}=1$ are parallel to each other.

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Question 8

Topic: Quadratic Formulas (Incorrect Recording Error) A quadratic equation of the form $a{x}^2+bx+c=0$ has its vertex at $(3,-2)$ and its $y$-intercept at 4. Due to a recording error, the coefficient $b$ was written as $5$ instead of its correct value. Find the absolute difference between the sum of the roots of the incorrectly recorded equation and that of the correct equation.

Accepted Answers:

• 13.5 (Accepted range: 13 to 14)

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Question 9

Topic: Set Operations (Cardinality) In an online degree program, 72 students took Mathematics, 85 took Statistics, and 30 took both. How many students took only one of the two subjects?

Accepted Answers:

• 97

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August 2025 Version (Highlights)

Question 1

Topic: Sets & Survey Problems In a survey of 100 people, it was found that 60 like Coffee, 50 like Tea, and 20 like neither. How many like both?

Mental Algorithm:

1. Union: $100-20=80$.
2. Both: $60+50-80=30$.

Accepted Answer: 30

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Question 3

Topic: Reflection Across X-axis A ray of light passing through $A(1,2)$ reflects at $B$ on the X-axis and passes through $(5,3)$. Find the equation of line segment $AB$.

Mental Algorithm:

1. Reflect $A$ to $A^{\prime }(1,-2)$.
2. Find line $A^{\prime }B$ connecting $(1,-2)$ and $(5,3)$.
3. Intersection with X-axis is $B$.
4. Line through $A$ and $B$.

Accepted Answer: $5x+4y=13$

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Quiz 2: Development (Functions, Limits, Derivatives)

Question 1

Topic: Function Matching Consider the function $f(x)=3\ln (x)-2$. Match the function type and its graph:

• Type: Logarithmic function
• Graph: Passes through $(e^{2/3},0)$.

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Question 2

Topic: Rational Limits at Infinity Find $\underset{n\to \infty }{\lim }\left(\frac{12n^2}{3n+6}-\frac{4n^2-6}{n+15}\right)$.

Accepted Answer: 52

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Question 5

Topic: Tangent Line Application The tangent line to $f$ at $(3,0)$ passes through $(5,4)$. What is $f^{\prime }(3)$?

Mental Algorithm: Slope $m=\frac{4-0}{5-3}=2$.

Accepted Answer: 2

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Quiz 3: Mastery (Graphs, Calculus, Multivariable)

Question 1

Topic: Graph Theory (Independent Sets) Find the cardinality of the maximum independent set for the given adjacency matrix. (Matrix omitted for brevity, see source)

Accepted Answer: 5

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Question 7

Topic: Multivariable Calculus (Critical Points) How many critical points does $f(x,y)=x{y}^2-y^2-x+3$ have?

Accepted Answer: 2 (Both are saddle points)

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Professional Patterns & Mental Algorithms 🧠

TIP

Use these “Mental Algorithms” to solve any variation of these common exam patterns.

Pattern	Goal	Step-by-Step Logic
Parallelogram Vertex	Find $D$ given $A,B,C$	Use $D=A+C-B$ (Midpoint Hack).
Reflect Point	$P^{\prime }$ mirror of $P$ across line $L$	$m_{\perp }=-1/m$. Line through $P\to$ intersection $M\to$ $M$ is midpoint.
Optimization	Max/Min of $a{x}^2+bx+c$	Vertex occurs at $x=-b/2a$.
Domain of Composite	Domain of $f(g(x))$	Intersect [Domain of $g$] and [Values of $g$ staying in Domain of $f$].
Limit of Ratio	${\lim }_{n\to \infty }P(n)/Q(n)$	If degrees equal, ratio of leading coefficients.
Tangent Equation	Line at $(x_0,y_0)$	Point $(x_0,f(x_0))$; Slope $m=f^{\prime }(x_0)$; Eq: $y-y_0=m(x-x_0)$.