Assignments

Maths 1 - Weekly Assignments

This file contains all graded assignments from Week 1 to Week 12.

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Week 01 - Assignments

Question 1

Which of the following are irrational numbers?

• $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$
• $(\sqrt{8}-\sqrt{2})(\sqrt{18}+\sqrt{2})$
• $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$
• $\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}$

Accepted Answers:

• $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$
• $\frac{\sqrt{6}+\sqrt{3}}{\sqrt{6}-\sqrt{3}}$

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Question 2

Suppose $f:D⟶\mathbb{R}$ is a function defined by $f(x)=\frac{\sqrt{x^2-16}}{x+4}$, where $D\subset \mathbb{Z}$. Let $A$ be the set of integers which are not in the domain of $f$, then find the cardinality of the set $A$.

Accepted Answers:

• 8

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Question 3

Consider the set $S=\{a\mid a\in \mathbb{N},a\le 33\}$. Let $R_1\text{and}{R}_2$ be relations from $S$ to $S$ defined as $R_1=\{(x,y)\mid x,y\in S,y=3x\}$ and $R_2=\{(x,y)\mid x,y\in S,y=x^2\}$. Find the cardinality of the set $R_1\setminus (R_1\cap R_2)$.

Accepted Answers:

• 10.0

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Question 4

In a Zoo, there are 6 Bengal white tigers and $6$ Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers in the Zoo.

Accepted Answers:

• 2

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Question 5

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers $(F)$, 245 ponds are polluted by pesticides $(P)$ and 257 ponds are polluted by pharmaceutical products $(Ph)$. 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.

Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?

• Relation is reflexive.
• Relation is transitive.
• Relation is symmetric.
• This is an equivalence relation.

Accepted Answers:

• Relation is transitive.
• Relation is symmetric.

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Question 6

Consider the following table of a few materials and their dielectric constant values.

Material	Dielectric constant
Air	1
Vaccum	2
Paper	3
Glass	8
Nerve membrane	7
Silicon	13

We can think of this as a function $f$ from the set of materials to the set of dielectric constant values consisting of the elements $\{1,2,3,8,7,13\}$. Now pick out the correct statement from the following.

• $f$ is neither one to one nor onto.
• $f$ is one to one but not onto.
• $f$ is onto but not one to one.
• $f$ is bijective.

Accepted Answers:

• $f$ is bijective.

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Question 7

Consider the following sets.

• $A=\{x\in \mathbb{N}\mid x\text{ mod }2=0\text{ and }1\le x\le 10\}$
• $B=\{x\in \mathbb{N}\mid x\text{ mod }5=0\text{ and }6\le x\le 25\}$
• $C=\{x\in \mathbb{N}\mid x\text{ mod }7=0\text{ and }7\le x\le 29\}$

What is the cardinality of $(A\setminus (B\cup C))\cup (B\setminus (C\cup A))\cup (C\setminus (B\cup A))$

Accepted Answers:

• 11

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Question 8

Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh. This family tree is shown in the figure below. Let us define two relations, $R$ and $S$, on the set $M$, which is the collection of all family members, as follows,

$R:=\{(A,B)|A\text{ and }B\text{ are cousins, i.e. their parents are siblings}\}$.

$S:=\{(A,B)|A\text{ is son of }B\}$.

If $m$ is the cardinality of the set $R$ and $n$ is the cardinality of the set $S$, then find the value of $m+n$.

Accepted Answers:

• 16

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Question 9

Define a function $f:\mathbb{Q}\to \mathbb{Z}$, such that $f(p/q)=p-q$, where $gcd(p,q)=1$. Which of the following option(s) is(are) true?

• $f$ is one to one but not onto
• $f$ is neither one to one nor onto.
• $f$ is onto but not one to one.
• $f$ is a bijective function.

Accepted Answers:

• $f$ is onto but not one to one.

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Question 10

Suppose $f_1(x)$ and $f_2(x)$ are functions defined on domains $D_1\subset \mathbb{R}$ and $D_2\subset \mathbb{R}$, respectively and codomains are subset of $\mathbb{R}$ . What will be the domain of the function $f_1(x)+f_2(x)$?

• $D_1\cup D_2$
• $D_1\setminus D_2$
• $D_2\setminus D_1$
• $D_1\cap D_2$

Accepted Answers:

• $D_1\cap D_2$

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Week 02 - Assignments

Question 1

A bird is flying along the straight line $2y-6x=6$. In the same plane, an aeroplane starts to fly in a straight line and passes through the point $(4,12)$. Consider the point where aeroplane starts to fly as origin. If the bird and plane collides then enter the answer as 1 and if not then 0. Note: Bird and aeroplane can be considered to be of negligible size.

Accepted Answers:

• 0

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Question 2

A rock is thrown in a pond, and creates circular ripples whose radius increases at a rate of 0.2 meter per second. What will be the value of $\frac{A}{\pi }$, where $A$ is the area (in square meter) of the circle after 5 seconds?
Hint: The area of a circle = $\pi r^2$, where $r$ is the radius of the circle.

Accepted Answers:

• 1.0

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Question 3

A ray of light passing through the point $A(1,2)$ is reflected at a point $B$ on $X$-axis and then passes through the point $(5,3)$. Then the equation of straight line $AB$ is

• $5x+4y=13$
• $5x-4y=-3$
• $4x+5y=14$
• $4x-5y=-6$

Accepted Answers:

• $5x+4y=13$

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Question 4

Let $ABCD$ be a parallelogram with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$. Which of the following always denotes the coordinate of the fourth vertex $D$?

• $(x_1+x_2+x_3,y_1+y_2+y_3)$
• $(x_1-x_2+x_3,y_1-y_2+y_3)$
• $(x_1+x_2-x_3,y_1+y_2-y_3)$
• $(x_1-x_2-x_3,y_1-y_2-y_3)$

Accepted Answers:

• $(x_1-x_2+x_3,y_1-y_2+y_3)$

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Question 5

Find the y - coordinate of the point of intersection of straight lines represented by (1) and (2), given the following equations:

$ax+by+c=E$ ---- (1)
$bx+cy+d^2=F$ ---- (2)

Given that:

• $E=F=0$
• Arithmetic mean of $a$ and $b$ is $c$.
• Geometric mean of $a$ and $b$ is $d$.

Note: Arithmetic mean of $m,n$ is $\frac{m+n}{2}$. Geometric mean is $\sqrt{mn}$.

• $\left(\frac{2a^{2}b-ab-b^2}{2b^2-a^2-ab}\right)$
• $\left(\frac{a^2}{a-b}-1\right)$
• $\left(\frac{2b^{2}b-ab-b^2}{2b^2-b^2-ab}\right)$
• $\left(\frac{b^2}{a-b}-1\right)$

Accepted Answers:

• $\left(\frac{2a^{2}b-ab-b^2}{2b^2-a^2-ab}\right)$

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Question 6

A mobile company wants to launch its new model in collaboration with a network provider named Astron to attract more customers.

Option 1 - Mobile and 1-year Astron Network costs 34000 rupees (Network offers unlimited calls for one year)
Option 2 - Only Mobile costs 22000 rupees
Lalith needs only 200 minutes per month.

How much will he save per year if he chooses the best option to buy the mobile compared to the collaborated offer given by the company?

Accepted Answers:

• 8400

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Question 7

State Government wants to connect the state road to the national highway from a town. There are 3 possible locations in the town A, B and C to connect to the National Highway whose locations are given by coordinates $(3,8)$, $(5,7)$, $(6,9)$. The National Highway connects the 2 points $(2,1)$, $(10,7)$. Always select the shortest path.
Note: 1 unit = 100 meter
What point will you select to build the road?

• A
• B
• C
• None

Accepted Answers:

• B

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Question 8

(Same context as Q7) What is the minimum length of road in meter required to construct to connect to the National Highway?

Accepted Answers:

• 300

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Question 9

A fitness trainer came up with an equation $W=-8t+98$, where W = Weight in Kg, t= time in months.

Equation is said to be well fitted to data if the SSE is less than 5. Is this equation well fitted?

• True
• False

Accepted Answers:

• True

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Question 10

(Same context as Q9) How many days are required for you to loss weight from 100 kg to 72 kg? Note: 1 month has 30 days.

Accepted Answers:

• 105

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Question 11

A function $f(x)$ which is the best fit for the data given in Table 1 is $f(x)=-(x-1{)}^2(x-3)(x-5)(x-7)+c$. What will be the value of $c$, so that SSE (Sum Squared Error) will be minimum?

Accepted Answers:

• 3.4

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Question 12

A bird is flying along the straight line $2y-6x=6$. After some time an aeroplane also follows the straight line path with a slope of 2 and passes through the point (4, 8). Let $(\alpha ,\beta )$ be the point where the bird and airplane can collide. Then find the value of $\alpha +\beta$.

Accepted Answers:

• -9

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Question 13

Consider a triangle $∆ABC$, whose co-ordinates are $A(-3,3),B(1,7)$ and $C(2,-2)$. Let point $M$ divides the line $AB$ in $1:3$, point $N$ divides the line $AC$ in $2:3$ and the point $O$ is the mid-point of $BC$. Find out the area of triangle $∆MNO$ (in sq. unit).

Accepted Answers:

• 4.5

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Question 14

Choose the point where $L_1$ and $L_2$ intersect. (Referring to internal data: $L_1$ is a line with x-intercept 10 and y-intercept -3, $L_2$ is a line with x-intercept -1 and y-intercept 2).

• (10, 18)
• (5, 8)
• (−10, −18)
• (6, 6)

Accepted Answers:

• (−10, −18)

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Question 15

If θ is the angle between $L_1$ and $L_2$, then $\tan \theta$ is equal to:

• $\frac{1}{8}$
• $\frac{1}{6}$
• $\frac{3}{8}$
• $\frac{1}{4}$

Accepted Answers:

• $\frac{1}{8}$

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Question 16

Consider triangle ABC and PAB. Coordinates: $A(4,3),B(2,2),C(8,3)$ and $P(t,t^2)$. The area of triangle ABC is 4 times the area of triangle PAB. What is the area of triangle ABC?

Accepted Answers:

• 2

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Question 17

(Same context as Q16) Choose all the possible options for P.

• (0, 0)
• (2, 4)
• (−2, 4)
• (−1, 1)

Accepted Answers:

• (−1, 1)

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Question 18

Radhika fitted a best-fit line to her data $y=4x+2$. What is the value of SSE?

Amount spent (y)	6	14	24	29	39	45
Number of outings (x)	1	3	5	7	9	11

Accepted Answers:

• 7

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Week 03 - Assignments

Question 1

If the slope of parabola $y=a{x}^2+bx+c$, where $a,b,c\in \mathbb{R}\setminus \{0\}$ at points $(3,2)$ and $(2,3)$ are $32$ and $2$ respectively, then find the value of $a$.

Accepted Answers:

• 15.0

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Question 2

A class of $140$ students are arranged in rows such that the number of students in a row is one less than thrice the number of rows. Find the number of students in each row.

Accepted Answers:

• 20

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Question 3

The product of two consecutive odd natural numbers is $143$. Find the largest number among them.

Accepted Answers:

• 13

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Question 4

The slope of a parabola $y=3x^2-11x+10$ at a point $P$ is 1. Find the $y-$ coordinate of the point $P$.

Accepted Answers:

• 0.0

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Question 5

Two parabolas $y=x^2+3x+2$ and $y=-x^2-5x-4$ are intersecting at two points $A$ (point $A$ is not on the $X-$ axis) and $B$. Suppose a straight line ${\ell }_1$ passes through the point $A$ with slope equal to the slope of the parabola $y=-x^2-5x-4$ at point $A$ and two straight lines ${\ell }_2$ and ${\ell }_3$ pass through the point $B$ with slopes equal to the slopes of the parabolas $y=x^2+3x+2$ and $y=-x^2-5x-4$ at point $B$, respectively. Which of the following is/are true?

• ${\ell }_1$ and ${\ell }_2$ are parallel.
• ${\ell }_1$ and ${\ell }_3$ are parallel.
• ${\ell }_1$ and ${\ell }_3$ are intersecting at point (-2, 3).
• ${\ell }_2$ and ${\ell }_3$ are intersecting at point (-1, 0).
• ${\ell }_2$ and ${\ell }_3$ are parallel.

Accepted Answers:

• ${\ell }_1$ and ${\ell }_2$ are parallel.
• ${\ell }_1$ and ${\ell }_3$ are intersecting at point (-2, 3).
• ${\ell }_2$ and ${\ell }_3$ are intersecting at point (-1, 0).

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Question 6

In order to cover a fixed distance of $48$ km, two vehicles start from the same place. The faster one takes $2$ hrs less and has a speed $4$ km/hr more than the slower one. Using the given information, What is the time (in hrs) taken by the faster one?

Accepted Answers:

• 4

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Question 7

The maximum value of a quadratic function $f$ is $-3$, its axis of symmetry is $x=2$ and the value of the quadratic function at $x=0$ is $-9$. What will be the coefficient of $x^2$ in the expression of $f$?

• $-1$
• 1
• $-1.5$
• $-0.5$

Accepted Answers:

• $-1.5$

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Question 8

A water fountain is designed to shoot a stream of water in the shape of a parabolic arc. The equation of the parabola is given by $h(t)=-0.5t^2+4t+1$, where $h(t)$ represents the height of the water stream in meters and $t$ represents the time in seconds since the water was shot. Answer the following questions.
Determine the time (in seconds) it takes for the water stream to reach its maximum height.

Accepted Answers:

• 4

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Question 9

Find out the points where the curve $y=4x^2+x+6$ and the straight line passing through the points $(1,6)$ and $(4,5)$ intersect.

• $(\frac{-1}{2},\frac{13}{2})$
• $(\frac{1}{6},\frac{113}{18})$
• $(0,6)$
• The curve and the straight line do not intersect.

Accepted Answers:

• $(\frac{-1}{2},\frac{13}{2})$
• $(\frac{1}{6},\frac{113}{18})$

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Question 10

If the slope of parabola $y=A{x}^2+Bx+C$, where $A,B,C\in \mathbb{R}$ at points (3, 2) and (2, 3) are 16 and 12 respectively.
Calculate the value of $A$.

Accepted Answers:

• 2

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Question 11

Context for Q11-Q13: A ballistic missile is launched from a fighter jet flying at a height of 40 m from the ground. The missile hits the tank which is present on the ground, as shown in Figure M1G3T5-2. The function $h(t)=-8t^2+32t+40$ represents the height (in meters) of the missile after $t$ seconds. Assume the dimensions of the tank and the fighter jet are negligible.

Find out the maximum height (in meters) attained by the missile.

Accepted Answers:

• 72

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Question 12

(Same context as Q11) Find out the time (in seconds) when the missile hits the tank.

Accepted Answers:

• 5

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Question 13

(Same context as Q11) Suppose an air defense system is present at the origin, and it follows the straight line path $h(t)=10t$, find the height from the ground at which the air defense missile will destroy the ballistic missile in the air.

• 40 m
• 12.5 m
• 4 m
• 1.25 m

Accepted Answers:

• 40 m

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Question 14

The polynomial $p(x)=a(x-4)(x-6)(x-8)(x-10)$ passes through the vertex of the quadratic function $q(x)=-(x-7{)}^2-9$. Calculate the value of $a$.

Accepted Answers:

• -1

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Week 04 - Assignments

Question 1

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be two functions, defined as $f(x)=x^3-8x^2+7$ and $g(x)=-2f(x)$ respectively. Choose the correct option(s) from the following.

• $f$ has two turning points and there are no turning points with negative $y-$ coordinate.
• $f$ is strictly increasing in $[10,\infty )$.
• $g$ has two turning points and $y-$coordinate of only one turning point is negative.
• $g$ has two turning points and there are no turning points with positive $y-$ coordinate.

Accepted Answers:

• $f$ is strictly increasing in $[10,\infty )$.
• $g$ has two turning points and $y-$coordinate of only one turning point is negative.

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Question 2

Which among the following function first increases and then decreases in all the intervals $(-4,-3)\text{and}(-1,2)\text{and}(5,6)$?

• $\frac{1}{10000}{\left(x+1\right)}^2\left(x-2\right){\left(x+3\right)}^2\left(x+4\right)\left(x-5\right){\left(x-6\right)}^2$
• $\frac{1}{10000}{\left(x+1\right)}^2\left(x-2\right)\left(x+3\right)\left(x+4\right){\left(x-5\right)}^2{\left(x-6\right)}^2$
• $\frac{-1}{10000}{\left(x+1\right)}^2\left(x-2\right){\left(x+3\right)}^2{\left(x+4\right)}^2\left(5-x\right){\left(x-6\right)}^2$
• $\frac{1}{10000}{\left(x+1\right)}^2\left(x-2\right)\left(x+3\right)\left(x+4\right){\left(x-5\right)}^2{\left(x-6\right)}^2\left(x+7\right)$

Accepted Answers:

• $\frac{1}{10000}{\left(x+1\right)}^2\left(x-2\right){\left(x+3\right)}^2\left(x+4\right)\left(x-5\right){\left(x-6\right)}^2$
• $\frac{-1}{10000}{\left(x+1\right)}^2\left(x-2\right){\left(x+3\right)}^2{\left(x+4\right)}^2\left(5-x\right){\left(x-6\right)}^2$

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Question 3

Consider a polynomial function $p(x)=-\left(x^2-16\right){\left(x-3\right)}^2{\left(2-x\right)}^2\left(x+9\right)$. Choose the set of correct options.

• $p(x)$ is strictly increasing when $x\in (-\infty ,-9)$
• Total number of turning points of $p(x)$ are 6.
• $p(x)$ first increases then decreases in the interval $(2,3)$
• Total number of turning points of $p(x)$ are 7.

Accepted Answers:

• Total number of turning points of $p(x)$ are 6.
• $p(x)$ first increases then decreases in the interval $(2,3)$

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Question 4

An ant named $B$, wants to climb an uneven cliff and reach its anthill (i.e., home of ant). On its way home, $B$ makes sure that it collects some food. A group of ants have reached the food locations which are at $x-$intercepts of the function $f(x)=\left(x^2-23\right)\left({\left(x-10\right)}^3-1\right)$. As ants secrete pheromones (a form of signals which other ants can detect and reach the food location), $B$ gets to know the food location. Then the sum of the $x$-coordinates of all the food locations is:

Accepted Answers:

• 11

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Question 5

The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads $r_1$ and $r_2$. Two roads are connected if they intersect. The shape of the two roads $r_1$ and $r_2$ follows polynomial curve $f(x)=\left(x-9\right){\left(x-7\right)}^2$ and $g(x)=-\left(x-9\right)\left(x-7\right)$ respectively. What will be the $x-$coordinate of the third aspirational district, if the first two are at $x-$intercepts of $f(x)$ and $g(x)$.

Accepted Answers:

• 6

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Question 6

Consider a polynomial function $f(x)$ of degree 4 which intersects the X-axis at $x=2,x=-3$ and $x=-4$. Moreover, $f(x)<0$ when $x\in (1,2)$, and $f(x)>0$ when $x\in (-1,1)$. Find out the equation of the polynomial

• $a(x-2{)}^2(x^2+7x+12),a>0$
• $a(x^4+4x^3-7x^2-22x+24),a>0$
• $a(x-2{)}^2(x^2+2x-8),a>0$
• $a(x^4-5x^3-7x^2-50x-24),a>0$

Accepted Answers:

• $a(x^4+4x^3-7x^2-22x+24),a>0$

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Question 7

Consider a polynomial function $P(x)=(x^4+4x^3+x+10)$ and $Q(x)=(x^3+2x^2-6)$. If $M(x)$ is the equation of the straight line passing through $(2,Q(2))$ and having slope 3, then find out the equation of $P(x)+M(x)Q(x)$.

Accepted Answers:

• $4x^4+14x^3+8x^2-17x-14$

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Question 8

Consider two polynomials $p(x)=-x^5+5x^4-7x-2$ and $q(x)=-x^5+5x^4-x^2-2$. Which of the following options is/are true?

• $q(x)⟶\infty$ as $x⟶\infty$.
• $p(x)⟶-\infty$ as $x⟶\infty$.
• $p(x)$ has at most 4 turning points.
• The quotient obtained while dividing $q(x)$ by $p(x)$ is a constant.

Accepted Answers:

• $p(x)⟶-\infty$ as $x⟶\infty$.
• $p(x)$ has at most 4 turning points.
• The quotient obtained while dividing $q(x)$ by $p(x)$ is a constant.

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Question 9

Ritwik wrote 12 mock tests. His score in each mock test $M(n)$ is represented as $M(n)=-(\frac{n^2}{1000})\left(n^3-15n^2+50n\right)+40,$ where $n$ represents the mock test number i.e., $n\in \{1,2,...,11,12\}$. He should score 40 or above to pass the assignment. In total, how many mock tests did Ritwik pass?

Accepted Answers:

• 6

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Question 10

The height of a roller coaster at a given time is modeled by the polynomial function $h(t)=(-0.01t^3+0.35t^2-3.5t+10)(t+5{)}^2(t-5)(t+1)(2-t{)}^3$, where t represents time in seconds.

• The roller coaster will first go up and then go down in the interval $(-5,-1)$.
• The roller coaster will first go down and then go up in the interval $(10,20)$.
• The roller coaster will first go up and then go down in the interval $(-1,-2)$.
• The roller coaster will first go up and then go down in the interval $(2,5)$.

Accepted Answers:

• The roller coaster will first go up and then go down in the interval $(-5,-1)$.
• The roller coaster will first go down and then go up in the interval $(10,20)$.
• The roller coaster will first go up and then go down in the interval $(2,5)$.

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Week 05 - Assignments

Question 1

A graph is shown in the below figure, $\circ$ symbol signifies that the straight line does not touch the point and the $\bullet$ symbol signifies that the line touches the point. Choose the correct option(s)

• The graph can be of a function, because it passes the vertical line test.
• The graph represents the graph of neither even function nor odd function.
• The graph represents the graph of either even function or odd function.
• The graph cannot be of a function, because it passes the vertical line test but fails the horizontal line test.
• The graph fails the horizontal line test thus it can be an injective function.

Accepted Answers:

• The graph can be of a function, because it passes the vertical line test.
• The graph represents the graph of neither even function nor odd function.

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Question 2

For $y=x^n$,where $n$ is a positive integer and $x\in \mathbb{R}$, which of the following statements are true?

• For all values of $n$, $y$ is not a one-to-one function.
• For all values of $n$, $y$ is an injective function.
• $y$ is not a function.
• If $n$ is an even number, then $y$ is not an injective function.
• If $n$ is an odd number, then $y$ is an injective function.

Accepted Answers:

• If $n$ is an even number, then $y$ is not an injective function.
• If $n$ is an odd number, then $y$ is an injective function.

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Question 3

If $4m-n=$ -2, then the value of $(\frac{16^m}{2^n}+\frac{27^n}{9^{6m}})$ is:

Accepted Answers:

• (Type: Range) 656.33, 802.18

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Question 4

Half-life of an element is the time required for half of a given sample of radioactive element to change to another element.The rate of change of concentration is calculated by the formula $A(t)=A_o(\frac{1}{2}{)}^{(\frac{t}{\gamma })}$ where $\gamma$ is the half-life of the material, $A_o$ is the initial concentration of the radioactive element in the given sample, $A(t)$ is the concentration of the radioactive element in the sample after time $t$. If Radium has a half-life of 1000 years and the initial concentration of Radium in a sample was 100%, then calculate the percentage of Radium in that sample after 2000 years.

Accepted Answers:

• 25

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Question 5

If $f(x)=(1-x{)}^{\frac{1}{2}}$ and $g(x)=(1-x^2)$, then find the domain of the composite function $g\circ f$.

• $\mathbb{R}$
• $((-\infty ,1]\cap [-2,\infty ))\cup (-\infty ,-2)$
• $[1,\infty )$
• $\mathbb{R}\setminus (1,\infty )$

Accepted Answers:

• $((-\infty ,1]\cap [-2,\infty ))\cup (-\infty ,-2)$
• $\mathbb{R}\setminus (1,\infty )$

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Question 6

Find the domain of the inverse function of $y=x^3+1$.

• $\mathbb{R}$
• $\mathbb{R}\setminus \{1\}$
• $[1,\infty )$
• $\mathbb{R}\setminus [1,\infty )$

Accepted Answers:

• $\mathbb{R}$

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Question 7

If $f(x)=x^3$, then choose the points where the graphs of the functions $f(x)$ and $f^{-1}(x)$ intersect each other?

• (-1,-1)
• (1,-1)
• (0,0)
• (-2,-8)
• (1,1)
• (2,8)

Accepted Answers:

• (-1,-1)
• (0,0)
• (1,1)

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Question 8

In a survey, the population growth in an area can be predicted according to the equation $\alpha (T)={\alpha }_o(1+\frac{d}{100}{)}^T$ where $d$ is the percentage growth rate of population per year and $T$ is the time since the initial population count ${\alpha }_o$ was taken. If in 2016, the population of Adyar was 44000.0 and the population growth rate is 3$\%$ per year, then what will be the approximate population of Adyar in 2021?

Accepted Answers:

• (Type: Range) 45907.25, 56108.87

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Question 9

An ant moves along the curve whose equation is $f(x)=x^2+1$ in the restricted domain $[0,\infty )$. Let a mirror be placed along the line $y=x$. If the reflection of the ant with respect to the mirror moves along the curve $g(x)$, then which of the following options is(are) correct?

• $g(x)=f^{-1}(x)$
• $g(x)=f(x)$
• $g(x)=\sqrt[2]{(x-1)}$
• $g(x)=\sqrt[2]{(x+1)}$

Accepted Answers:

• $g(x)=f^{-1}(x)$
• $g(x)=\sqrt[2]{(x-1)}$

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Question 10

Suppose a textile shop provides two different types of offers during a festival season. The first offer($D_1$) is “shop for more than ₹14,999 and pay only ₹9,999”. The second offer($D_2$) is “avail 30% discount on the total payable amount”. If Shalini wants to buy two dresses each of which costs more than ₹8,000 and she is given the choice to avail both offers simultaneously, then which of the following options is(are) correct?

• The minimum amount she should pay after applying two offers cannot be determined because the exact values of the dresses she wanted to buy are unknown.
• The minimum amount she should pay after applying the two offers simultaneously is approximately ₹6,999.
• The amount she is supposed to pay after applying $D_2$ only is approximately ₹11,199.
• The amount she is supposed to pay after applying $D_1$ only is approximately ₹9,999.
• Suppose the total payable amount is ₹17,999 for the two dresses. In order to pay minimum amount Shalini should avail offer $D_1$ first and offer $D_2$ next.
• Suppose the total payable amount is ₹17,999 for the two dresses. If Shalini avails offer $D_2$ first, then she cannot avail offer $D_1$.
• Suppose the total payable amount is ₹17,999 for the two dresses. In order to pay minimum amount Shalini should avail offer $D_2$ first and offer $D_1$ next.

Accepted Answers:

• The minimum amount she should pay after applying the two offers simultaneously is approximately ₹6,999.
• The amount she is supposed to pay after applying $D_1$ only is approximately ₹9,999.
• Suppose the total payable amount is ₹17,999 for the two dresses. In order to pay minimum amount Shalini should avail offer $D_1$ first and offer $D_2$ next.
• Suppose the total payable amount is ₹17,999 for the two dresses. If Shalini avails offer $D_2$ first, then she cannot avail offer $D_1$.

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Question 11

If $f(x)=x^2$ and $h(x)=x-1$, then which of the following options is(are) incorrect?

• $f\circ h$ is not an injective function.
• $f(f(h(x)))\times h(x)=(x-1{)}^5$
• $h\circ f$ is not an injective function.
• There are two distinct solution for $h(h(f(x)))=0$.
• $f\circ h$ is an injective function.
• $f(f(h(x)))\times h(x)=(x-1{)}^4$.
• $h\circ f$ is an injective function.

Accepted Answers:

• $f\circ h$ is an injective function.
• $f(f(h(x)))\times h(x)=(x-1{)}^4$.
• $h\circ f$ is an injective function.

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Question 12

Let $f(x)$, $g(x)$, $p(x)$ and $q(x)$ be the functions defined on $R$. Refer Figure 3 (A and B) and choose the correct option(s) from the following.

• $g(x)$ may be the inverse of $f(x)$.
• $p(x)$ and $q(x)$ are even functions but $f(x)$ and $g(x)$ are neither even functions nor odd functions.
• $q(x)$ could not be the inverse function of $p(x)$.
• $p(x)$, $q(x)$ can be an even degree polynomial functions and $f(x)$ can be an odd degree polynomial functions.

Accepted Answers:

• $g(x)$ may be the inverse of $f(x)$.
• $p(x)$ and $q(x)$ are even functions but $f(x)$ and $g(x)$ are neither even functions nor odd functions.
• $q(x)$ could not be the inverse function of $p(x)$.
• $p(x)$, $q(x)$ can be an even degree polynomial functions and $f(x)$ can be an odd degree polynomial functions.

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Week 06 - Assignments

Question 1

If $18^x-12^x-(2\times 8^x)=0$, then the value of $x$ is:

• $\frac{\ln 2}{\ln 3-\ln 2}$
• $\frac{\ln 18}{\ln 12-\ln 8}$
• $\ln 2$
• $\ln 18$

Accepted Answers:

• $\frac{\ln 2}{\ln 3-\ln 2}$

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Question 2

Suppose three distinct persons $A$, $B$ and $C$ are standing on the $X$- axis of the $XY$- plane and the distance between $B$ and $A$ is same as the distance between $C$ and $B$. The coordinates of $A$, $B$ and $C$ are $({\log }_{5}3,0)$, $({\log }_5(3^x-\frac{9}{2}),0)$ and $({\log }_5(3^x-\frac{9}{4}),0)$ respectively. What is the distance between $C$ and $B$?

Accepted Answers:

• (Type: Range) 0.24, 0.26

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Question 3

In a city, a rumour is spreading about the safety of corona vaccination. Suppose $N$ number of people live in the city and $f(t)$ is the number of people who have not yet heard about the rumour after $t$ days. Suppose $f(t)$ is given by $f(t)=N{e}^{-kt}$, where $k$ is a constant. If the population of the city is $1000$, and suppose $40$ have heard the rumor after the first day. After how many days (approximately) half of the population would have heard the rumor?
Note: Enter the nearest integer value.

Accepted Answers:

• 17

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Question 4

Which of the following is true about $f$?

• $f$ is not a one to one function.
• $f$ is a one to one function.
• Range of $f$ is $\mathbb{R}$.
• $f$ is a bijective function.

Accepted Answers:

• $f$ is a one to one function.

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Question 5

The inverse of $f$ would be:

• $\ln (\frac{2x}{2-3x})$
• $\ln (\frac{x}{2-3x})$
• $\ln (\frac{x}{2-x})$

Accepted Answers:

• $\ln (\frac{x}{2-3x})$

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Question 6

If $m>n>9$, then choose the correct option(s).

• $f(m)>f(n)$
• $f(m)<f(n)$
• $f(m)=f(n)$
• $f(m)\le f(n)$

Accepted Answers:

• $f(m)>f(n)$

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Question 7

Choose the correct option(s).

• The jeweler sold $540$ kg gold in $2019$.
• The jeweler sold at least $730$ kg gold in $2019$.
• The jeweler sold at least $2$ kg gold daily throughout the year $2019$.
• The jeweler sold at least $10$ kg gold daily throughout the year $2019$.

Accepted Answers:

• The jeweler sold at least $730$ kg gold in $2019$.
• The jeweler sold at least $2$ kg gold daily throughout the year $2019$.

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Question 8

The stock market chart of a tourism company $(A)$ is shown roughly in the Figure below. This company was listed in February $(x=2)$ and experiences a logarithmic fall after the COVID$-19$ outbreak which is given by $y=-alog(x-h)+a$. $x$ represents the number of months since the beginning of the year and $y$ represents the stock price in ₹(1000). During the $10^{th}$ month the pharmacy company announced that the vaccine is made for the COVID$-19$. Thereafter, the stock price of the company $(A)$ is raised exponentially $y=10^{\frac{x}{b}}-b$. Choose the correct set of options.

• For logarithmic fall the value of $a=1.5$ and $h=2$.
• For exponential rise passing through $(10,0)$ the value of $b=10$.
• The stock price in $12^{th}$ month is ₹4000.
• If the vaccine was not made and the stock price just followed the same logarithmic function through out, then the investor would have lost his/her entire investment on the $12^{th}$ month.

Accepted Answers:

• For logarithmic fall the value of $a=1.5$ and $h=2$.
• For exponential rise passing through $(10,0)$ the value of $b=10$.
• If the vaccine was not made and the stock price just followed the same logarithmic function through out, then the investor would have lost his/her entire investment on the $12^{th}$ month.

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Question 9

If $m^{{\log }_{3}2}+2^{{\log }_{3}m}=$ 4, then what is the value of $m$ ?

Accepted Answers:

• 3

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Question 10

Choose the correct options with respect to the graph of a function $f(x)$ shown below.

• The given function is not defined in the restricted domain $(-2,-1)\cup (3,4)\cup (5,6)$.
• The given function is invertible in the restricted domain $(-\infty ,-2)\cup (1,3)\cup (4,5)\cup (6,\infty ]$
• The given graph is a graph of a polynomial.
• The range of the given function could be $(-\infty ,\infty )$.
• The graph of $f(x)$ could be a graph of ${\log }_{10}\left(1+\left(x+2\right)\left(x+1\right)\left(x-3\right)\left(x-4\right)\left(x-5\right)\left(x-6\right)\right)$
• The function is invertible in restricted domain $[5,\infty )$

Accepted Answers:

• The given function is not defined in the restricted domain $(-2,-1)\cup (3,4)\cup (5,6)$.
• The range of the given function could be $(-\infty ,\infty )$.
• The graph of $f(x)$ could be a graph of ${\log }_{10}\left(1+\left(x+2\right)\left(x+1\right)\left(x-3\right)\left(x-4\right)\left(x-5\right)\left(x-6\right)\right)$

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Question 11

Choose the set of correct options.

• ${\log }_{5}2$ is a rational number
• If 0 < $b$ < 1 and 0 < $x$ < 1 then ${\log }_{b}x$ > 0
• If ${\log }_3({\log }_{5}x)$ = 1 then $x$ = 625
• If 0 < $b$ < 1 and 0 < $x$ < $y$ then ${\log }_{b}x$ > ${\log }_{b}y$

Accepted Answers:

• If 0 < $b$ < 1 and 0 < $x$ < 1 then ${\log }_{b}x$ > 0
• If 0 < $b$ < 1 and 0 < $x$ < $y$ then ${\log }_{b}x$ > ${\log }_{b}y$

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Question 12

Find the maximum value of $g(x)$.

Accepted Answers:

• 81

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Question 13

Find the maximum value of $h(x)$.

Accepted Answers:

• 1

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Question 14

Find the number solution(s) of the equation $\ln (7)+\ln (2-4x^2)=\ln (14)$.

Accepted Answers:

• 1

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Question 15

Consider the function $f(x)=|\log (x+1)|$. Choose the correct option(s) from the following.

• The domain of $f$ is $(-1,\infty )$.
• The domain of $f$ is $(-\infty ,1)$.
• $f(x)$ is not a one-one function when $x\in (-1,1)$.
• $f(x)$ is a one-one function when $x\in (-1,1)$.

Accepted Answers:

• The domain of $f$ is $(-1,\infty )$.
• $f(x)$ is not a one-one function when $x\in (-1,1)$.

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Question 16

Suppose three distinct persons A, B and C are standing on the X-axis of the XY-plane and the distance between B and A is same as the distance between C and B. The coordinates of A, B and C are $(lo{g}_{5}3,0),(lo{g}_5(3^x-\frac{9}{2}),0)$ and $(lo{g}_5(3^x-\frac{9}{4}),0)$ respectively. What is the distance between C and B?

Accepted Answers:

• (Type: Range) 0.24, 0.26

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Week 07 - Assignments

Question 1

Table: M2W1G1 gives functions in Column A with their types in column B and their graphs in Column C.

Choose the options which represent the correct matching of a given function with its type and its graph.

• i) $\to$ d) $\to$ 2)
• i) $\to$ a) $\to$ 2)
• ii) $\to$ c) $\to$ 1)
• ii) $\to$ c) $\to$ 4)
• iii) $\to$ d) $\to$ 2)
• iii) $\to$ a) $\to$ 3)
• iv) $\to$ b) $\to$ 1)
• iv) $\to$ b) $\to$ 3)

Accepted Answers:

• i) $\to$ d) $\to$ 2)
• ii) $\to$ c) $\to$ 4)
• iii) $\to$ a) $\to$ 3)
• iv) $\to$ b) $\to$ 1)

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Question 2

Suppose $f(x)$ is a strictly increasing function and $g(x)$ is a strictly decreasing function. If the curves represented by $f(x)$ and $g(x)$ intersect at $x=x_0$, then choose the set of correct options.

• $f(x)\ge g(x)$ for all $x\ge x_0$.
• There exists a point $x_1>x_0$ such that $f(x_1)=g(x_1)$.
• $g(x)\ge f(x)$ for all $x\ge x_0$.
• $g(x)\ge f(x)$ for all $x_0\ge x$.
• $f(x)\ge g(x)$ for all $x_0\ge x$.

Accepted Answers:

• $f(x)\ge g(x)$ for all $x\ge x_0$.
• $g(x)\ge f(x)$ for all $x_0\ge x$.

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Question 3

Suppose $a_n=\frac{12n^2}{3n+6}-\frac{4n^2-6}{n+15}$. Find the value of $\underset{n\to \infty }{\lim }{a}_n$.

Accepted Answers:

• 52

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Question 4

In the graphs given below, how many of the curves have a (unique) tangent at the origin (i.e., (0, 0))?

Accepted Answers:

• 2

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Question 5

Limits of some standard functions are given below:

• $\underset{x\to 0}{\lim }\frac{\log (1+x)}{x}=1$
• $\underset{x\to 0}{\lim }\frac{e^x-1}{x}=1$
• $\underset{n\to \infty }{\lim }n(\frac{\sqrt{2\pi n}}{n!}{)}^{\frac{1}{n}}=e$
• $\underset{n\to \infty }{\lim }\frac{n}{\sqrt[n]{n!}}=e$

Using the given information, find the value of $\underset{n\to \infty }{\lim }e\sqrt[n]{n!}\left[\log \right(1+\frac{24}{n})-\frac{(e^{(\frac{5}{n})}-1)}{(\sqrt{2\pi n}{)}^{\frac{1}{n}}}]$.

Accepted Answers:

• 19

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Question 6

Find the limit of the sequence given by $a_n=\frac{12+20+28+...+(8n-4)}{n^2}$, (where $n\in \mathbb{N}\setminus \{0\}$).

Accepted Answers:

• 4

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Question 7

Find the value of $5\underset{x\to 13^{+}}{\lim }\lfloor x\rfloor -3\underset{x\to 3^{-}}{\lim }\lfloor x\rfloor$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Accepted Answers:

• 59

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Question 8

Suppose a company runs three algorithms to estimate its future growth. Suppose the error in the estimation depends on the available number $(n)$ of data as follows:

• Algorithm 1: $a_n=\frac{n^2+5n}{6n^2+1}$
• Algorithm 2: $b_n=\frac{1+(-1{)}^n}{n}$
• Algorithm 3: $c_n=\frac{8^n+4}{7e^n}$ (assume $n\to \infty$). Which of the following statements is (are) correct?
• Error in estimation by Algorithm 2 will be 0.500.
• Error in estimation by Algorithm 2 will give the minimum error.
• Error in estimation by Algorithm 2 will give the maximum error.
• Both Algorithm 1 and Algorithm 2 will give the same error and that will be the maximum.
• Error in estimation by Algorithm 1 will be 0.166 approximately.

Accepted Answers:

• Error in estimation by Algorithm 2 will give the minimum error.
• Error in estimation by Algorithm 1 will be 0.166 approximately.

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Question 9

Suppose a new algorithm is designed and the error in estimation is given by $a_n-b_n$. Choose the set of correct options.

• The error in estimation using the new algorithm is less than the error in estimation using Algorithm 1.
• The error in estimation using Algorithm 2 is less than the error in estimation using the new algorithm.
• The error in estimation using the new algorithm is less than the error in estimation using Algorithm 3.
• The error in estimation using the new algorithm cannot be compared with the error in estimation using Algorithm 3.

Accepted Answers:

• The error in estimation using the new algorithm is less than the error in estimation using Algorithm 1.
• The error in estimation using the new algorithm is less than the error in estimation using Algorithm 3.

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Question 10

Suppose the company modified the error in estimation by Algorithm 3 as $c_n^{{}^{\prime }}=n{e}^{\left(\frac{1}{15n}\right)}-n$. What will be the new error in estimation by Algorithm 3? (Correct upto 3 decimal places)

Accepted Answers:

• (Type: Range) 0.060, 0.070

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Week 08 - Assignments

Question 1

Match the given functions in Column A with the equations of their tangents at the origin $(0,0)$ in column B and the plotted graphs in Column C.

• ii) $\to$ a) $\to$ 1.
• i) $\to$ b) $\to$ 3.
• iii) $\to$ b) $\to$ 1.
• iii) $\to$ c) $\to$ 2.
• i) $\to$ a) $\to$ 1.

Accepted Answers:

• ii) $\to$ a) $\to$ 1.
• i) $\to$ b) $\to$ 3.
• iii) $\to$ c) $\to$ 2.

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Question 2

Consider the following two functions $f(x)$ and $g(x)$. $f(x)=\{\begin{array}{ll}\frac{x^3-9x}{x(x-3)}& \text{if }x\ne 0,3\\ 3& \text{if }x=0\\ 0& \text{if }x=3\end{array}$ $g(x)=\{\begin{array}{ll}|x|& \text{if }x\le 2\\ \lfloor x\rfloor & \text{if }x>2\end{array}$ Choose the set of correct options.

• $f(x)$ is discontinuous at both $x=0$ and $x=3$.
• $f(x)$ is discontinuous only at $x=0$.
• $f(x)$ is discontinuous only at $x=3$.
• $g(x)$ is discontinuous at $x=2$.
• $g(x)$ is discontinuous at $x=3$.

Accepted Answers:

• $f(x)isdiscontinuousonlyatx=3$.
• $g(x)isdiscontinuousatx=3$.

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Question 3

Consider the graphs given below:

Choose the set of correct options from the below.

• Curve 1 is both continuous and differentiable at the origin.
• Curve 2 is continuous but not differentiable at the origin.
• Curve 2 has derivative 0 at $x=0$.
• Curve 3 is continuous but not differentiable at the origin.
• Curve 4 is not differentiable anywhere.
• Curve 4 has derivative 0 at $x=0$.

Accepted Answers:

• Curve 1 is both continuous and differentiable at the origin.
• Curve 2 has derivative 0 at $x=0$.
• Curve 3 is continuous but not differentiable at the origin.

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Question 4

Choose the set of correct options considering the function given below: $f(x)=\{\begin{array}{ll}\frac{\sin x}{x}& \text{if }x\ne 0,\\ 1& \text{if }x=0\end{array}$

• $f(x)$ is not continuous at $x=0$.
• $f(x)$ is continuous at $x=0$.
• $f(x)$ is not differentiable at $x=0$.
• $f(x)$ is differentiable at $x=0$.
• The derivative of $f(x)$ at $x=0$ (if exists) is $0$.
• The derivative of $f(x)$ at $x=0$ (if exists) is $1$.

Accepted Answers:

• $f(x)$ is continuous at $x=0$.
• $f(x)$ is differentiable at $x=0$.
• The derivative of f(x) at x=0 (if exists) is 0.

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Question 5

Let $f$ be a polynomial of degree 5: $f(x)=a_5{x}^5+a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x+a_0.$ Let $f^{\prime }(b)$ denote the derivative. Choose correct options.

• $a_1=f^{\prime }(0)$
• $5a_5+3a_3=\frac{1}{2}(f^{\prime }(1)+f^{\prime }(-1)-2f^{\prime }(0))$
• $4a_4+2a_2=\frac{1}{2}(f^{\prime }(1)-f^{\prime }(-1))$
• None of the above.

Accepted Answers:

• $a_1=f^{\prime }(0)$
• $5a_5+3a_3=\frac{1}{2}(f^{\prime }(1)+f^{\prime }(-1)-2f^{\prime }(0))$
• $4a_4+2a_2=\frac{1}{2}(f^{\prime }(1)-f^{\prime }(-1))$

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Question 6

Let $f$ be differentiable at $x=3$. The tangent line to the graph of $f$ at $(3,0)$ passes through $(5,4)$. Find $f^{\prime }(3)$.

Accepted Answers:

• 2

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Question 7

Let $f$ and $g$ be two differentiable functions. $f(x)=g(x^2+5x)$ and $f^{\prime }(0)=10$. Find $g^{\prime }(0)$.

Accepted Answers:

• 2

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Question 8

Consider $f(x)=\{\begin{array}{ll}\frac{\sin 12x+A\sin x}{15x^3}& \text{if }x\ne 0,\\ B& \text{if }x=0.\end{array}$ If $f(x)$ is continuous at $x=0$, find $90B-A$.

Accepted Answers:

• -1704

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Question 9

Distance car travels: $d(t)=g(7t^3+3t^2+8t+7)$. Find instantaneous speed after 5 min if $g^{\prime }(997)=2$.

Accepted Answers:

• 1126

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Question 10

Find the number of correct statements.

• Statement P: $p(t)$ and $q(t)$ are continuous.
• Statement Q: $p(t)$ and $q(t)$ are not differentiable.
• Statement R: $p(t)$ is continuous, $q(t)$ is differentiable.
• Statement S: $q(t)$ is continuous, $p(t)$ is not differentiable.
• Statement T: Neither $p(t)$ nor $q(t)$ is continuous.

Accepted Answers:

• 2

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Question 11

If $L_p(t)=At+B$ denotes the best linear approximation of $p(t)$ at $t=1$, find $\frac{-8}{e^{-4}-1}(A+B)$.

Accepted Answers:

• 4

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Question 12

Consider $q(t)=|t(t-2)(t-8)|$. If $m$ is the slope of tangent at $t=3/2$, find $m-27/4$.

Accepted Answers:

• -14

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Question 16

Consider $f(x)=\{\begin{array}{ll}\frac{x}{(x+1)(x+2)}& x\ge 1\\ \frac{1}{x-5}& x<0\end{array}$. Which is correct?

• $li{m}_{x\to -2^{+}}f(x)=\infty$
• The function $f$ is continuous.
• $li{m}_{x\to 5^{+}}f(x)=li{m}_{x\to 5^{-}}f(x)=\frac{5}{42}$
• At $x=1$, the function $f$ is discontinuous.

Accepted Answers:

• $li{m}_{x\to 5^{+}}f(x)=li{m}_{x\to 5^{-}}f(x)=\frac{5}{42}$
• At $x=1$, the function $f$ is discontinuous.

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Question 19

$f(x)=\{\begin{array}{ll}3mx+n& x<1,\\ 11& x=1,\\ 5mx+2n& x>1.\end{array}$. If $f$ is continuous at $x=1$, find $m+n$.

Accepted Answers:

• -11

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Question 20

LEDs Price: $f(x)=1000-x$. Cost: $g(x)=30000+300x$ (if $x\le 400$) or $100x+110000$ (if $400<x\le 800$). Max production 400. Find $x$ for max profit.

Accepted Answers:

• 350

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Week 09 - Assignments

Question 1

Match the functions in Column A with the corresponding (signed) area between its graph and the interval $[-1,1]$ on the X-axis in column B and the images in Column C.

• i) $\to$ b) $\to$ 1), iii) $\to$ a) $\to$ 2).
• i) $\to$ b) $\to$ 3), ii) $\to$ c) $\to$ 1).
• ii) $\to$ c) $\to$ 1), iii) $\to$ a) $\to$ 2).
• i) $\to$ b) $\to$ 1), ii) $\to$ c) $\to$ 3), iii) $\to$ a) → 2).

Accepted Answers:

• i) $\to$ b) $\to$ 3), ii) $\to$ c) $\to$ 1).
• ii) $\to$ c) $\to$ 1), iii) $\to$ a) $\to$ 2).

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Question 2

A cylinder of radius $x$ and height $2h$ is to be inscribed in a sphere of radius $R$ centered at O. The volume $V=2\pi x^{2}h$ and the surface area $S=4\pi xh$. Choose the set of correct options.

• The cylinder has maximum volume when $h=R$.
• The cylinder has maximum volume when $h=\sqrt{3}R$.
• The cylinder has maximum volume when $h=\frac{R}{\sqrt{3}}$.
• The cylinder has maximum surface area of its curved surface when $h=2R$.
• The cylinder has maximum surface area of its curved surface when $h=\frac{R}{\sqrt{2}}$.
• The cylinder has maximum surface area of its curved surface when $h=\sqrt{2}$.

Accepted Answers:

• The cylinder has maximum volume amongst all cylinders which can be inscribed when $h=\frac{R}{\sqrt{3}}$.
• The cylinder has maximum surface area of its curved surface, amongst all cylinders which can be inscribed, when $h=\frac{R}{\sqrt{2}}$.

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Question 3

Which of the curves in the following figures enclose a negative area on the $X$ axis in the interval $[0,1]$?

• Curve 1
• Curve 2
• Curve 3
• Curve 4

Accepted Answers:

• Curve 2
• Curve 4

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Question 4

What will the absolute difference between the minimum values of $f_2$ and $g_2$ in the interval $[0,1]$ be?

Accepted Answers:

• 1

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Question 5

Suppose the area of the region bounded by two curves in the interval $[0,1]$ is defined to be the error in prediction.

• The error in prediction for company A is $\frac{5}{12}$.
• The error in prediction for company A is $\frac{11}{12}$.
• The error in prediction for company A is more than that for company B.
• The error in prediction for company B is more than that for company A.
• The error in prediction for Company A and Company B cannot be compared.

Accepted Answers:

• The error in prediction for company A is $\frac{5}{12}$.
• The error in prediction for company B is more than that for company A.

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Question 6

Let $f(x)=x^3-3x+25$. What is the local minimum value of $f$ attained at a critical point?

Accepted Answers:

• 23

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Question 7

Let $f(x)=10x^2+\frac{40}{6}$, $0\le x\le 6$. The estimated area obtained by dividing the interval into 3 sub-intervals of equal length and the left end points of the sub-intervals for height of the rectangles is:

Accepted Answers:

• 440

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Question 8

Let $f(x)=\{\begin{array}{ll}-5x+1& 0\le x\le 10\\ x^2& 10<x\le 20\end{array}$. What is the global minimum of $f$ on $[0,20]$?

Accepted Answers:

• -49

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Question 9

If $x-y=34$, find the least value of $2xy$.

Accepted Answers:

• -578

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Week 10 - Assignments

Question 1

The maximum number of non-zero entries in an adjacency matrix of a simple graph having $n$ vertices can be:

• $n^2$
• $\frac{n(n-1)}{2}$
• $\frac{n(n-1)}{4}$
• $n(n-1)$

Accepted Answers:

• $n(n-1)$

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Question 2

We have a graph $G$ with 6 vertices. Which of the following is a possible listing of the degrees in descending order?

• $6,5,4,3,2,1$
• $5,5,2,2,1,1$
• $5,3,3,2,2,1$
• $2,1,1,1,1,1$

Accepted Answers:

• $5,3,3,2,2,1$

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Question 3

We start at the entrance of a maze. If we reach a dead end, we come back to the most recent intersection where we still have an unexplored direction. What is a good data structure to keep track?

• List
• Stack
• Queue
• Array

Accepted Answers:

• Stack

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Question 4

Suppose we obtain the following BFS tree rooted at node 1 for an undirected graph. Which of the following cannot be an edge in the original graph?

• $(8,11)$
• $(3,10)$
• $(4,5)$
• $(6,9)$

Accepted Answers:

• $(8,11)$

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Question 5

Which of the following graphs satisfies: $|VC(G)|=3$, $|PM(G)|=3$, and is a 3-colouring?

• 
• 
• 
• 

Accepted Answers:

• Image 6.3
• Image 6.4

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Question 6

Which of the following statements is(are) true?

• BFS can be used to identify the vertex farthest from $v$.
• BFS and DFS identifies all vertices reachable from $v.$
• BFS cannot check for cycles while DFS can.
• DFS identifies shortest distance in terms of number of edges.

Accepted Answers:

• BFS can be used to identify the vertex which is at the farthest distance from $v$ in terms of number of edges.
• BFS and DFS identifies all the vertices reachable from the starting vertex $v.$

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Question 7

If $A=\left[\begin{array}{ccccccc}0& 1& 0& 1& 0& 0& 0\\ 1& 0& 1& 1& 0& 1& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 1& 1& 1\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\end{array}\right]$, find the cardinality of the maximum independent set.

Accepted Answers:

• 5

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Question 8

Chemicals $x_1$ to $x_{10}$ are stored. Incompatible chemicals (shown in graph) must be in different compartments. What is the least number of compartments required?

Accepted Answers:

• 3

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Question 9

Find the number of maximum edges that can be added such that coloring is retained and the graph is planar.

Accepted Answers:

• 6

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Week 11 - Assignments

Question 1

An undirected graph G has 38 vertices and the degree of each vertex is at least 7. What is the minimum number of edges?

Accepted Answers:

• 133.0

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Question 2

If G is connected, degree at most 7, and shortest path length at most 2, what is the maximum number of vertices?

Accepted Answers:

• 50

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Question 3

Suppose $A^2=\left[\begin{array}{cccccc}1& 1& 1& 0& 1& 1\\ 1& 1& 1& 1& 0& 1\\ 1& 1& 1& 1& 1& 0\\ 0& 1& 1& 1& 1& 1\\ 1& 0& 1& 1& 1& 1\\ 1& 1& 0& 1& 1& 1\end{array}\right]$. Which represents graph G?

Accepted Answers:

• Image W103.3.

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Question 4

Adjacency matrix $A=\left[\begin{array}{cccccc}0& 0& 1& 0& 0& 1\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$. Which is true?

• G is a directed acyclic graph.
• From vertex 4, every other vertex is reachable.
• The longest path has length 4.
• The longest path is $4⟶0⟶2⟶3$.

Accepted Answers:

• The graph $G$ is a directed acyclic graph.
• From vertex 4, every other vertex is reachable.
• The longest path in the graph is $4⟶0⟶2⟶3$.

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Question 5

Order in which Shreya can perform tasks (referring to topological sort table)?

• $A,C,B,D,E,I,F,H,G,J$
• $A,D,C,B,E,I,F,H,G,J$
• $C,A,D,E,B,I,F,G,H,J$
• $D,C,A,B,E,I,F,H,G,J$

Accepted Answers:

• $A,C,B,D,E,I,F,H,G,J$
• $A,D,C,B,E,I,F,H,G,J$
• $D,C,A,B,E,I,F,H,G,J$

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Question 6

If each task takes 5 mins and independent tasks are done simultaneously, time taken is:

Accepted Answers:

• 30

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Question 7

Positive integer $x$ such that unique shortest path from $a$ to $j$ contains $(b,e)$ using Dijkstra?

• $\{x|x\in \mathbb{Z},0<x<8\}$
• $\{x|x\in \mathbb{Z},0<x<7\}$
• $\{x|x\in \mathbb{Z},0<x<6\}$
• $\{x|x\in \mathbb{Z},0<x<9\}$

Accepted Answers:

• $\{x|x\in \mathbb{Z},0<x<7\}$

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Question 8

Shortest path $v_0$ to $v_4$. Correct statements?

• Dijkstra can be used.
• Bellman-Ford can be used because of negative edges.
• Weight of shortest path is 1.
• Bellman-Ford cannot be used because there is a negative cycle.

Accepted Answers:

• Bellman-Ford algorithm cannot be used to find the shortest path from $v_0$ to $v_4$ because there is a negative cycle in the given graph.

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Question 9

Which is INCORRECT?

• Unique MCST if all edge weights are different.
• Bellman-Ford fails if there is a weight 0 cycle.
• Connected if acyclic and $n-1$ edges.
• Min weight edge is always in MCST.

Accepted Answers:

• If there is a cycle of weight $0$ in a directed graph $G$, then we cannot find the shortest path using Bellman-Ford algorithm.
• In a graph $G$, every edge with the minimum weight will be in the minimum cost spanning tree.

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Question 11

Route for visiting each branch once with minimum fare?

• $C_2⟶C_3⟶C_1⟶C_0⟶C_4⟶C_5$
• $C_2⟶C_1⟶C_3⟶C_4⟶C_0⟶C_5$
• $C_2⟶C_3⟶C_1⟶C_4⟶C_0⟶C_5$
• Such a route is not possible.

Accepted Answers:

• $C_2⟶C_1⟶C_3⟶C_4⟶C_0⟶C_5$

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Question 14

Prim’s algorithm vertex addition order starting from A?

• $A,C,F,G,B,D,E$
• $A,B,D,E,G,C,F$
• $A,B,G,C,F,D,E$
• $A,C,F,G,E,D,B$

Accepted Answers:

• $A,B,G,C,F,D,E$

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Question 15

Kruskal’s algorithm - which edges are NOT added?

• $(A,E)$
• $(B,D)$
• $(G,F)$
• $(A,G)$

Accepted Answers:

• $(A,E)$
• $(G,F)$

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Week 12 - Assignments (Qualifier Section)

Question 1

Consider the function $f(x,y)=x{y}^2-y^2-x+3$. How many critical points does $f$ have?

• 0
• 1
• 2
• Infinitely many

Accepted Answers:

• 2

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Question 8

Using the Hessian test for the function $f(x,y)$ in Q1, what can we conclude?

• $f$ does not have critical points.
• The Hessian test is inconclusive for all points.
• All critical points are local minima.
• All critical points are saddle points.

Accepted Answers:

• All critical points of $f$ are saddle points.

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