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CMI Entrance Exam

Mock test #4 ‘22: Full-syllabus test

Timings: 14:00-17:00 Hrs    Date: 29 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: Fourth 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 30 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 30 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. Let \(I \subseteq R\) be an interval and \(f : I \rightarrow \mathbb{R}\) be a differentiable function. Let \[J = \Bigg\{ \dfrac{f(b) - f(a)}{b-a} : a, b \in I, a< b \Bigg\} \] Which of the following are true?
    (a) \(J\) is an interval.
    (b) \(J \subseteq f'(I)\)
    (c) \(f'(I) - J\) contains at most two elements.
    (d) \(f'(I) - J\) can contain infinite number of elements.

  2. The number of subsets of the set \(\{1,2, \cdots, 10\}\) containing at least one odd integer is
    (a) \(2^{10}\)
    (b) \(2^{5}\)
    (c) \({ }^{10} C_{5}\)
    (d) \(2^{10}-2^{5}\)

  3. The number of negative solutions of the equation \(e^{x}-\sin x=0\) is
    (a) 1
    (b) 2
    (c) 0
    (d) infinite.

  4. The last two digits of \(17^{400}\) are
    (a) 17
    (b) 09
    (c) 01
    (d) 89

  5. Let \(a_{1}=1, a_{n+1}=\left(\frac{1+n}{n}\right) a_{n}\) for \(n \geq 1\). Then the sequence \(\left\{a_{n}\right\}\) is
    (a) divergent
    (b) decreasing
    (c) convergent
    (d) bounded.

  6. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \[ f(x+h)-f(x)=h f^{\prime}\left(x+\frac{1}{2} h\right) \] for all real \(x\) and \(h\).
    Which of the following is/are true?
    (a) \(f\) is a polynomial of degree at most 1.
    (b) \(f\) is a polynomial of degree at most 2.
    (c) \(f\) is a polynomial of degree at most 3.
    (d) \(f\) need not be a polynomial.

  7. Let \(S=\{a, b, c\}, T=\{1,2\}\). If \(m\) denotes the number of one-one functions and \(n\) denotes the number of onto functions from \(S\) to \(T\), then the values of \(m\) and \(n\) respectively are
    (a) 6,0
    (b) 0,6
    (c) 5,6
    (d) 0,8

  8. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be differentiable functions such that \(f^{\prime}(x)>g^{\prime}(x)\) for every \(x\). Then the graphs \(y=f(x)\) and \(y=g(x)\)
    (a) intersect exactly once.
    (b) intersect at most once.
    (c) do not intersect.
    (d) could intersect more than once.

  9. The equation \(x^{4}+x^{2}-1=0\) has
    (a) two positive and two negative roots
    (b) one positive, one negative and two non-real roots
    (c) all positive roots
    (d) no real root

  10. The value of \(\lim_{x \rightarrow 1} \frac{\int_{1}^{x} e^{t^{2}} d t}{x^{2}-1}\) is
    (a) 0
    (b) 1
    (c) \(e\)
    (d) \(e/2\)

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. Let \(f : (0,\infty) \implies \mathbb{R}\) be a continuous function satisfying \(f(1)=5\) and \(f\left( \dfrac{x}{x+1} \right) = f(x) + 2\) for all positive reals \(x\).
(a) Find \(\lim_{x \rightarrow \infty} f(x)\).
(b) Show that \(\lim_{x \rightarrow 0^+} f(x) = \infty\)
(c) Find one example of such a function.

B2. If \(n\) is a natural number, prove that
(a) \(\log_{10}(n+1)>\frac{3}{10 n}+\log_{10} n\)
(b) \(\log n !>\frac{3 n}{10}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-1\right)\).

B3. Show that any integer greater than \(10\) whose digits are all members of \(\{1, 3, 7, 9\}\) has a prime factor \(\geq 11\).

B4. Let \(f(x)=\dfrac{a_i}{x+a_i} + \dfrac{a_2}{x+a_2} + \cdots + \dfrac{a_n}{x+a_n}\) where \(a_i\) are unequal positive reals. Find the sum of the lengths of the intervals in which \(f(x) \geq 1\).

B5. Let ABCD be a cyclic quadrilateral. Suppose that there exists a circle with center in AB that is tangent to the other sides of the quadrilateral.
(a) Prove that \(AB = AD+BC\).
(b) Determine, in terms of \(x = AB\) and \(y =CD\), the maximum possible area of the quadrilateral.

B6. Let \(o(n)\) be the number of \(2n-\)tuples \((a_1, a_2, \cdots , a_n, b_1, b_2, \cdots , b_n)\) such that each \(a_i, b_j = 0\) or \(1\) and \(a_1b_1 + a_2b_2 + \cdots + a_nb_n\) is odd. Similarly, let \(e(n)\) be the number for which the sum is even. Find the value of the ratio \(o(n)/e(n)\) in terms of \(n\).