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CMI Entrance Exam

Mock test #3 ‘22: Full-syllabus test

Timings: 14:00-17:00 Hrs    Date: 8 April 2022

Instructions

  • You are responsible for keeping time. Email all your solutions by 17:05 Hrs IST.
  • Write your answers with a dark pen on white paper.
  • Find an email from me with the subject line ‘CMI Tomato: Third mock test of 2022’. Send your solutions (images) as replies to this email.
  • Adjust/Reduce the resolution of the camera so that each image is less than 500 KB in size.
  • As per the rules of CMI entrance exam, a simple calculator may be used.
  • Total marks: 100 (10x4=40 for Part A + 6x10=60 for Part B)

For students who miss the live test
Self-administer the mock test and email your solutions before 9 April, 23:59 Hrs IST. Your solutions will be evaluated but marks won’t be counted for official use in the future. Solutions submitted after 9 April, 23:59 Hrs will not be evaluated.

Part A: Short-answer type questions

Submission file: Write answers to all the ten questions on a single sheet of paper. Email a picture of your answer sheet. Name the file as PartA.jpg or PartA.png.

For this part, answers must be written without any explanation.

  1. How many factors of \(2^{5} 3^{6} 5^{2}\) are perfect squares?
    A) 24
    B) 20
    C) 30
    D) 36

  2. Find the value of expression:
    \[ \sum_{n=1}^{\infty} \frac{\sin n}{n} \]

  3. Let \(\omega \neq 1\) be a complex cube root of unity. If \[\left(3-3 \omega+2 \omega^{2}\right)^{4 n+3}+\left(2+3 \omega-3 \omega^{2}\right)^{4 n+3}+\left(-3+2 \omega+3 \omega^{2}\right)^{4 n+3}=0\] then the possible value(s) of \(n\) is (are)
    A) 1
    B) 2
    C) 4
    D) 5

  4. Let \(a, b \in R\) and \(f: R \rightarrow R\) be defined by \(f(x)=a \cos \left(\left|x^{3}-x\right|\right)+b|x| \sin \left(\left|x^{3}+x\right|\right)\). Then \(f\) is
    A) differentiable at \(x=0\) if \(a=0\) and \(b=1\)
    B) differentiable at \(x=1\) if \(a=1\) and \(b=0\)
    C) NOT differentiable at \(x=0\) if \(a=1\) and \(b=0\)
    D) NOT differentiable at \(x=1\) if \(a=1\) and \(b=1\)

  5. What is value of the series given below: \[(20)^2+ \dfrac{(21)^2}{1!} + \dfrac{(22)^2}{2!} + \dfrac{(23)^2}{3!} + \cdots \]

  6. Let \(g\) be a continuous function which is not differentiable at 0 and let \(g(0)=8\). If \(f(x)=x g(x)\), then \(f^{\prime}(0)=\)
    A) 0
    B) 4
    C) 2
    D) 8

  7. If the circles \(x^{2}+y^{2}=1\) and \((x-a)^{2}+(y-b)^{2}=1\) have exactly one point in common, then the point \((a, b)\) lies on
    A) \(x^{2}+y^{2}=1\)
    B) \((x-a)^{2}+(y-b)^{2}=1\)
    C) \(x^{2}+y^{2}=2\)
    D) \(x^{2}+y^{2}=4\).

  8. If \(x\) and \(y\) are non-zero real numbers, then \(x^{2}+x y+y^{2}\)
    A) is always negative
    B) takes the value zero for some \(x, y\)
    C) is always positive
    D) takes both positive and negative values.

  9. The set of all real numbers \(x\) such that || \(3-x|-| x+2||=5\) is
    A) \([3, \infty)\)
    В) \((-\infty,-2] \cup[3, \infty)\)
    C) \((-\infty,-2]\)
    D) \((-\infty,-3] \cup[2, \infty)\).

  10. If \(f(x)=nx , g(x) = e^{2x},\) and \(h(x)=g(f(x))\). If \(h'(0)=100\), find \(n\).

Part B: Subjective questions

Submission files: Each question in this part must be answered on a page of its own. Name the files as B1.jpg, B2.jpg, etc. In case you have multiple files for the same question, say B4, name the corresponding files as B4-1.jpg, B4-2.jpg, etc.

Clearly explain your entire reasoning. No credit will be given without reasoning. Partial solutions may get partial credit.

B1. Five people each choose an integer between \(1\) and \(3\), inclusive. What is the probability that all three numbers get chosen?

B2. In a convex quadrilateral \(ABCD\) ,\(\angle ABC\) and \(\angle BCD\) are \(\geq 120^\circ\). Prove that \(|AC|\) + \(|BD| \geq |AB|+|BC|+|CD|\). Notation. \(|XY|\) denotes the length of the segment \(XY\).

B3. Consider a \(3 \times 3\) grid with the first \(9\) positive integers placed in the grid. Take the greatest integer in each row and let \(r\) be the smallest of those numbers. Take the smallest integer in each column and let \(c\) be the greatest of those numbers.
(a) True or false: Regardless of the arrangement of the numbers, \(r \leq c\) always. Give explanation if true or a counterexample, if false.
(b) How many arrangements are there such that \(r \leq c \leq 4\)?

B4. Three squares have average area \(\bar{A}=100 \mathrm{~m}^{2}\). The average of the lengths of their sides is \(\bar{l}=10 \mathrm{~m}\). Find all possible values of the size of the largest of the three squares. The answer must be given along with reasoning.

B5. Let \(t\) be a randomly chosen positive divisor of 20!. What is the probability that \(t\) can be written as \(a^{2}+b^{2}\) for some integers \(a\) and \(b\)?

B6. Compute \[ \int_{\frac{1}{2}}^{2} \frac{\tan ^{-1} x}{x^{2}-x+1} dx \]