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Jan 2026 - Mathematics I - MOCK 3week1-4

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MOCK 3(week1-4)

Course: Jan 2026 - Mathematics I
MOCK 3(week1-4)

Introduction

Instructions:
  • There are some questions which have functions with discrete valued domains (such as day, month, year etc). For simplicity, we treat them as continuous functions.
  • For NAT type question, enter only one right answer even if you get multiple answers for that particular question. 
  • Notations:
 ∙\bullet R\mathbb{R}= Set of real numbers  ∙\bullet Q\mathbb{Q}= Set of rational numbers  ∙\bullet Z\mathbb{Z}= Set of integers  ∙\bullet N\mathbb{N}= Set of natural numbers
  •  The set of natural numbers includes 0.
Based on this information answer the following questions (Q3, Q4) For the purposes of a research, a survey of 1000 students is conducted in a IITM BSc Degree Program. The results show that 52% liked CT, 45% liked Statistics and 60% liked Mathematics. In addition, 25% liked both CT and Statistics, 28% liked both Mathematics and Statistics and 30% liked both CT and Mathematics. 6% liked none of these subjects. Based on this information answer the following questions
Based on this information answer the following questions (Q5 and Q6) Consider a triangle ABCABC (See Figure 1), the line segments ABAB, BCBC, and CACA represent the sides of the triangle ABCABC. It is given that angle ∠ABC=∠BCA=α\angle ABC = \angle BCA =\alpha, where α\alpha is an acute angle.
     
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Based on this information answer the following questions
Based on this information answer the following questions (Q7 and Q8) Rizwan wants to cross the river represented by r(x)=0.05(x−1)(x−5)(x−10)+k,r\left(x\right)=0.05\left(x-1\right)\left(x-5\right)\left(x-10\right)+k, where kk is an integer constant (Refer Figure 2). He chooses to cross the river r(x)r(x) using the bridge. In order to cross the river he has to identify the coordinates of the point AA. He realises that the coordinates of the point AA can be identified in such a way that the curve r(x)r(x) can be best fitted according to the data given in the table (refer Table 1). Consider the river and the bridge to be of negligible thickness. Based on these information answer the following questions    
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Based on this information answer the following questions (Q10 and Q11) A total of ₹₹300 is raised by a group AA, by collecting equal amounts from a certain number of people. Another group BB, contains 5 more people and each person contributes ₹₹ 10 less to raise the same amount as that of group AA. Based on this information answer the following questions
Based on this information answer the following questions (Q12) Suppose the function f(x)=−x2+4x−kf(x)=-x^2+4x-k and g(x)=x2−kx+4g(x)= x^2-kx+4 intersects at most at one point, where k∈Zk\in \mathbb{Z}. Based on this information answer the following questions
Based on this information answer the following questions (Q13 and Q14): Consider two polynomial function p(x)=0.1(x−1)(x−15)(x−20)p(x)=0.1\left(x-1\right)\left(x-15\right)\left(x-20\right) and q(x)=0.4(x−1)(x−20)q(x)=0.4\left(x-1\right)\left(x-20\right) defined in the interval (1,20)(1,20). A line l(x)l(x) passes through x−x-intercept of p(x)p(x) and the intersection point of p(x)p(x) and q(x)q(x).

Question 1

Let r(x)r(x) be a polynomial function which is obtained as the quotient after dividing the polynomial p(x)=−(x+5)(x−3)(x2−16)p(x)=-\left(x+5\right)\left(x-3\right)\left(x^{2}-16\right) by the polynomial q(x)=−(x−4)(x+4)q(x)=-\left(x-4\right)\left(x+4\right). Choose the correct option(s) which represent(s) the polynomial r(x)r(x) most appropriately.
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Feedback/Explanation:
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Accepted Answers:
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Question 2

Let RR= {(a,c),(d,b),(b,d),(b,c),(c,a)(a, c),(d, b),(b, d),(b, c),(c, a)}, be a relation on the set   A ={a,b,c,da, b, c, d}. The relation RR is
  • a function
  • reflexive
  • transitive
  • not symmetric
Feedback/Explanation: not symmetric
Accepted Answers:
not symmetric

Question 3

How many students like all the three subjects?
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 200
Accepted Answers:
(Type: Numeric) 200

Question 4

Find the number of students who like only one of the three subjects.
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 510
Accepted Answers:
(Type: Numeric) 510

Question 5

Which of the following options are correct w.r.t triangle ABCABC
  • Slope of the line segment ABAB is 12\frac{1}{2}
  • Slope of the line segment BCBC is −12-\frac{1}{2}
  • tan⁡α=−12\tan \alpha = -\frac{1}{2}
  • tan⁡α=34\tan \alpha = \frac{3}{4}
Feedback/Explanation: Slope of the line segment ABAB is 12\frac{1}{2}
tan⁡α=34\tan \alpha = \frac{3}{4}
Accepted Answers:
Slope of the line segment ABAB is 12\frac{1}{2}
tan⁡α=34\tan \alpha = \frac{3}{4}

Question 6

Which of the following options represents the equation of the line ACAC
  • 2y+11x−59=0{2y+11x-59=0}
  • 6y=2x+2{6y=2x+2}
  • 3y−x−1=0{3y-x-1=0}
  • y=−5.5x+29.5{y=-5.5x+29.5}
Feedback/Explanation: 2y+11x−59=0{2y+11x-59=0}
y=−5.5x+29.5{y=-5.5x+29.5}
Accepted Answers:
2y+11x−59=0{2y+11x-59=0}
y=−5.5x+29.5{y=-5.5x+29.5}

Question 7

Using the above information, find the value of kk.
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 15
Accepted Answers:
(Type: Numeric) 15

Question 8

Using the above information, what will be the y−y-coordinate of the point AA?
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 12.5
Accepted Answers:
(Type: Numeric) 12.5

Question 9

Choose the most appropriate option for the statement given below: "The equation of the line joining the point (−2,0)(-2, 0) to the point of intersection of the lines y+4x−2=0y+4x-2=0 and y−3x−2=0y-3x-2=0 is equidistant from the points (0,0)(0, 0) and (2,2)(2, 2).
  • True
  • False
  • Data insufficient
  • None of the above
Feedback/Explanation: True
Accepted Answers:
True

Question 10

How many people actually contributed in group A ?
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 10
Accepted Answers:
(Type: Numeric) 10

Question 11

What is the contribution (₹₹) made by each person in group BB?
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) 20
Accepted Answers:
(Type: Numeric) 20

Question 12

Which of the following could be the value of kk.
  • 4
  • -5
  • 10
  • -10
Feedback/Explanation: 4
Accepted Answers:
4

Question 13

Which of the following are correct.
  • The x−x-intercept of p(x)p(x) in the given domain is 15.
  • In the given domain p(x)p(x) has 2 turning points.
  • The x−x-coordinate of the vertex of q(x)q(x) is 5.5.
  • The x−x-coordinate of the intersection point of p(x)p(x) and q(x)q(x) is 19.
Feedback/Explanation: The x−x-intercept of p(x)p(x) in the given domain is 15.
In the given domain p(x)p(x) has 2 turning points.
The x−x-coordinate of the intersection point of p(x)p(x) and q(x)q(x) is 19.
Accepted Answers:
The x−x-intercept of p(x)p(x) in the given domain is 15.
In the given domain p(x)p(x) has 2 turning points.
The x−x-coordinate of the intersection point of p(x)p(x) and q(x)q(x) is 19.

Question 14

What is the slope of the line l(x)l(x)?
Your Answer: (Not answered)
Feedback/Explanation: (Type: Numeric) -1.8
Accepted Answers:
(Type: Numeric) -1.8
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