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CMI Data Science 2019 Question Paper with Solutions: 5 Questions, Answer Key & Section-wise Analysis

CMI Data Science 2019 previous year paper: 5 questions with answer key and detailed solutions, section-wise breakdown and free sample questions.

5 Qs

Total Questions

15 Marks

Total Marks

0.525 Mins

Duration

+3 / -1 / 0

Marking Scheme

Section-wise Paper Structure

School Level Mathematics

5 Qs

100% of total marks

Free Solved Questions with Step-by-Step Solutions

Authentic examination problems with detailed derivations and answer keys.

Question 1

2019 PYQ

Level 3: Exam Standard

Let θ = lo g_{e} (2) . For - θ \leq x \leq θ, let f (x) = \frac{e x p ( x )}{1 + e x p ( x )} . Compute α and β where α = - θ \leq x \leq θ max f (x) and β = - θ \leq x \leq θ min f (x) . Justify your answers.

Question 2

2019 PYQ

Level 3: Exam Standard

Ani is training for the olympics with Usain Bolt. After a few days of training Usain challenges Ani to catch him. Usain sets off running very slowly with a view to encourage Ani. He covers 70m the first minute, 100m the next minute, then 130m the minute afterwards and so on. Ani is told to start 3 minutes later. Having trained hard he runs 100m the first minute, 150 m the second minute, then 200m and so on. Ani catches Usain at an integral multiple of a minute. How many minutes did Ani run before catching up with Usain. What were their respective speeds?

Question 3

2019 PYQ

Level 3: Exam Standard

A small circular fire is spreading with its radius increasing at the rate of 1.5 metres per minute. When the radius of the fire is 5 metres, how fast is the burned area growing?

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CMI Data Science 2019 Question Paper with Solutions: 5 Questions, Answer Key & Section-wise Analysis

CMI Data Science 2019 previous year paper: 5 questions with answer key and detailed solutions, section-wise breakdown and free sample questions.

Paper breakdown

5 questions · 15 marks · 0.525 minutes. School Level Mathematics: 5

Free sample questions from CMI Data Science 2019 Question Paper

Question 1 · School Level Mathematics · 2019 SUB

Let $θ = lo g_{e} (2)$ . For $- θ \leq x \leq θ$ , let $f (x) = \frac{e x p ( x )}{1 + e x p ( x )}$ . Compute $α$ and $β$ where

$α = - θ \leq x \leq θ max f (x) and β = - θ \leq x \leq θ min f (x) .$

Justify your answers.

Correct Answer:

0.667

Step-by-Step Solution

Key idea: Find the global maximum and minimum of $f (x) = \frac{e ^{x}}{1 + e ^{x}}$ on the closed interval $[- θ, θ]$ where $θ = ln 2$ .

Step 1: Analyze monotonicity of $f (x)$ .

$f (x) = \frac{e ^{x}}{1 + e ^{x}}$ .

Differentiate using quotient rule:

$f^{'} (x) = \frac{e ^{x} ( 1 + e ^{x} ) - e ^{x} ( e ^{x} )}{( 1 + e ^{x} ) ^{2}} = \frac{e ^{x}}{( 1 + e ^{x} ) ^{2}}$ .

Since $e^{x} > 0$ for all real $x$ , $f^{'} (x) > 0$ for all $x$ .

Therefore, $f (x)$ is strictly increasing on $R$ .

Step 2: Evaluate at endpoints.

Since $f$ is strictly increasing on $[- θ, θ]$ :

Minimum $β$ occurs at $x = - θ$ .

Maximum $α$ occurs at $x = θ$ .

Step 3: Calculate values.

Given $θ = ln 2$ .

Calculate $α = f (θ) = f (ln 2)$ :

$f (ln 2) = \frac{e ^{l n 2}}{1 + e ^{l n 2}} = \frac{2}{1 + 2} = \frac{2}{3}$ .

Calculate $β = f (- θ) = f (- ln 2)$ :

Note $e^{- l n 2} = \frac{1}{e ^{l n 2}} = \frac{1}{2}$ .

$f (- ln 2) = \frac{1/2}{1 + 1/2} = \frac{1/2}{3/2} = \frac{1}{3}$ .

Answer: 0.667

Question 2 · School Level Mathematics · 2019 SUB

Ani is training for the olympics with Usain Bolt. After a few days of training Usain challenges Ani to catch him. Usain sets off running very slowly with a view to encourage Ani. He covers 70m the first minute, 100m the next minute, then 130m the minute afterwards and so on. Ani is told to start 3 minutes later. Having trained hard he runs 100m the first minute, 150 m the second minute, then 200m and so on. Ani catches Usain at an integral multiple of a minute. How many minutes did Ani run before catching up with Usain. What were their respective speeds?

Correct Answer:

Step-by-Step Solution

Key idea: Arithmetic Progression (AP) in motion problems.

This is recognisable because distances covered in successive equal time intervals (each minute) increase by a constant amount, requiring the sum of an AP formula to find total distance.

Step 1: Formulate Usain's distance.

Usain's minute-by-minute distances form an AP: $70, 100, 130, \dots$

First term $a_{U} = 70$ , common difference $d_{U} = 30$ .

Distance covered by Usain in $n$ minutes: $S_{U} (n) = \frac{n}{2} [2 (70) + (n - 1) 30] = 15 n^{2} + 55 n$ .

Step 2: Formulate Ani's distance.

Ani's distances form an AP: $100, 150, 200, \dots$

First term $a_{A} = 100$ , common difference $d_{A} = 50$ .

Distance covered by Ani in $k$ minutes: $S_{A} (k) = \frac{k}{2} [2 (100) + (k - 1) 50] = 25 k^{2} + 75 k$ .

Step 3: Account for the time offset.

Ani starts 3 minutes later. When Ani has run for $k$ minutes, Usain has run for $k + 3$ minutes.

They meet when their total distances are equal: $S_{U} (k + 3) = S_{A} (k)$ .

Step 4: Solve for $k$ .

$15 (k + 3)^{2} + 55 (k + 3) = 25 k^{2} + 75 k$

$15 (k^{2} + 6 k + 9) + 55 k + 165 = 25 k^{2} + 75 k$

$15 k^{2} + 90 k + 135 + 55 k + 165 = 25 k^{2} + 75 k$

$15 k^{2} + 145 k + 300 = 25 k^{2} + 75 k$

$10 k^{2} - 70 k - 300 = 0$

Divide by 10: $k^{2} - 7 k - 30 = 0$ .

Factor: $(k - 10) (k + 3) = 0$ .

Since $k > 0$ , $k = 10$ .

Answer: 10

Question 3 · School Level Mathematics · 2019 SUB

A small circular fire is spreading with its radius increasing at the rate of 1.5 metres per minute. When the radius of the fire is 5 metres, how fast is the burned area growing?

Correct Answer:

Step-by-Step Solution

Key idea: This is a Related Rates problem involving the area of a circle, recognisable because we are given the rate of change of the radius and asked for the rate of change of the area at a specific instant.

Step 1: Identify the geometric formula.

For a circle of radius $r$ , the area $A$ is:

$A = π r^{2}$

Step 2: Differentiate both sides with respect to time $t$ using the Chain Rule.

Since $r$ changes with time,

$\frac{d A}{d t} = \frac{d}{d t} (π r^{2}) = 2 π r \frac{d r}{d t}$

Step 3: Substitute the known values at the specific instant.

Given $r = 5$ m and $\frac{d r}{d t} = 1.5$ m/min.

$\frac{d A}{d t} = 2 π (5) (1.5)$

$\frac{d A}{d t} = 15 π m^{2} / min$

Step 4: Interpret the NAT answer format.

The mathematical rate is $15 π$ square metres per minute. Since the expected NAT answer is the coefficient of $π$ , the answer is $15$ .

Common trap: Forgetting the factor of $2$ when differentiating $r^{2}$ , or including $π$ in the final numeric answer when the NAT format expects only the coefficient.

Answer: 15

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CMI Data Science 2019 Question Paper with Solutions: 5 Questions, Answer Key & Section-wise Analysis

Paper breakdown

Free sample questions from CMI Data Science 2019 Question Paper

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Other CMI Data Science papers