Part I Problem 1 Let \(XZ\) be a diameter of circle \(\omega\). Let Y be a point on \(XZ\) such that \(XY=7\) and \(YZ=1\). Let W be a point on \(\omega\) such that \(WY\) is perpendicular to \(XZ\). What is the square of the length of the line segment \(WY\) ? (a) 7(b) 8(c) 10(d) […]
PART I Problem 1 Answer: A Problem 2 Answer: D Problem 3 Answer: D Problem 4 Answer: A Problem 5 Answer: D Problem 6 Answer: D Problem 7 Answer: D Problem 8 Answer: C Problem 9 Answer: B Problem 10 For positive real numbers a and b, the minimum value of\( \left18 a+\frac{1}{3 b}\right\left3 b+\frac{1}{8 […]
Problem 1 Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$. Find the largest positive integer $m$ such that $m$ divides $a b c d$ for all $(a, b, c, d) \in S$. Solution Notice that $(2, 2, 1, 3)\in S\Rightarrow m$ is a divisor of $12 […]
Try this ISI BStat - BMath Entrance 2023, Problem no. 2 with hints and final solution.
In the math world, a unique challenge emerges, combining algebra and number theory. The goal: show that specific equation solutions aren't simple fractions. We reveal a key insight about b² - 4ac: it's always a distinct odd perfect square. Using "parity check," No matter the numbers, the left side remains even, while the right side stays odd. The result: these solutions differ from simple fractions, highlighting math's power in problem-solving.
In 2023, the following Cheenta students are successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 100 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 times every week. CMI B.Sc. Math Entrance I.S.I. […]
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Please use the following form to contribute problems of CMI BSc Math Entrance 2023. We will work on the solutions. Also come back to this page to see the updates on Problems and Solutions of CMI BSc Math Entrance 2023. B1. Let \( n \) be an odd positive number greater than 1. We have […]
Questions, Solutions and discussions on ISI BStat-BMath Entrance 2023.
Question 1 What is \[ \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? \] (a) \(-1\) (b) \(\frac{5}{36}\) (c) \(\frac{7}{12}\) (d) \(\frac{49}{20}\) (e) \(\frac{43}{3}\) Question 2 Mr.\ Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr.\ Green's steps is 2 feet long. Mr.\ Green expects half a […]
Question 1 Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? (a) 33 (b) 35 (c) 37 (d) 39 (e) 41 Question 2 What […]
Question 1 What is the value of $$ \frac{2 a^{-1}+\frac{a^{-1}}{2}}{a} $$ when $a=\frac{1}{2}$ ? (a) 1 (b) 2 (c) $\frac{5}{2}$ (d) 10 (e) 20 Question 2 If $n \circlearrowleft m=n^{3} m^{2}$, what is $\frac{294}{49^{2}}$ ? (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) 1 (d) 2 (e) 4 Question 3 Let $x=-2016$. What is the value of $|||x|-x|-|x||-x$ […]
Question 1 What is the value of \(2-(-2)^{-2}\)? (a) \(-2\) (b) \(\frac{1}{16}\) (c) \(\frac{7}{4}\) (d) \(\frac{9}{4}\) (e) \(6\) Question 2 Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task? […]
Question 1 Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. What was Mary's number? (a) 11 (b) 12 (c) 13 (d) 14 (e) 15 Question 2 Sofia ran 5 laps around the […]
Question 1 Kate bakes 20 -inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? (a) 90 (b) 100 (c) 180 (d) 200 (e) 360 Question 2 Sam drove 96 miles in 90 minutes. His average […]
Question 1 What is the value of $$ 1-(-2)-3-(-4)-5-(-6) ? $$ (a) -20 (b) -3 (c) 3 (d) 5 (e) 21 Question 2 Carl has 5 cubes each having side length 1 , and Kate has 5 cubes each having side length 2 . What is the total volume of these 10 cubes? (a) 24 […]
Question 1 How many integer values of $x$ satisfy $|x|<3 \pi$ ? (a) 9 (b) 10 (c) 18 (d) 19 (e) 20 Question 2 What is the value of $\sqrt{(3-2 \sqrt{3})^{2}}+\sqrt{(3+2 \sqrt{3})^{2}}$ ? (a) 0 (b) $4 \sqrt{3}-6$ (c) 6 (d) $4 \sqrt{3}$ (e) $4 \sqrt{3}+6$ Question 3 In an after-school program for juniors and […]
Question 1 What is the value of $1234+2341+3412+4123$ ? (a) 10,000 (b) 10,010 (c) 10,110 (d) 11,000 (e) 11,110 Question 2 What is the area of the shaded figure shown below? (a) 4 (b) 6 (c) 8 (d) 10 (e) 12 Question 3 The expression $\frac{2021}{2020}-\frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ […]
Question 1 Define $x \diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of $$ (1 \diamond(2 \diamond 3))-((1 \diamond 2) \diamond 3) ? $$ (a) -2 (b) -1 (c) 0 (d) 1 (e) 2 Question 2 In rhombus $A B C D$, point $P$ lies on segment […]
Question 1 A target of ${ }^{7} \mathrm{Li}$ is bombarded with a proton beam of current $10^{-4}$ ampere for 1 hour to produce ${ }^{7} \mathrm{Be}$ of activity $1.8 \times 10^{8}$ disintegrations per second. Assuming that bombarding of 1000 protons produces one ${ }^{7} \mathrm{Be}$ radioactive nucleus, the half-life of ${ }^{7} \mathrm{Be}$ is estimated […]
Question 1 An air filled parallel plate capacitor, with plate area $A$ and plate separation $d$, is connected to a battery of emf $V$ volt having negligible internal resistance. One of the plates of the capacitor vibrates with amplitude ' $a$ ' ( $a \ll d$ ) and angular frequency $\omega$. If the instantaneous current […]
Question 1 Rod AB of radius 2 r is joined with rod BC of radius r . They are of same material and are of same length. The combination carries a current I. Choose the correct statement. (a) $V_{A B}=4 V_{B C}$ (b) Current per unit in AB and BC are equal (c) Resistance of […]
Question 1 Gravitational collapse is the contraction of an astronomical object under its own gravity. This draws the matter inwards towards the centre of gravity. A neutron star is an example of the collapsed core of a giant star. A certain neutron star of radius 10 km is of mass $1.5 M_{\odot}$. The acceleration due […]
Question 1 The axes of a coordinate system $\mathrm{S}_{2}$ are inclined at an angle $\theta$ to those of another coordinate system $\mathrm{S}_{1}$. The origins of both the systems are coinciding. A particle $\mathrm{P}_{1}$ at rest in system $\mathrm{S}_{1}$, starts from point ($-2,0$) and travels along positive direction of $\mathrm{X}_{1}$ axis with uniform acceleration of $1.25 […]
Question 1 A student, intending to catch a bus on the bus stand, finds that the bus starts and accelerates away from him/her with uniform acceleration $a=2 m s^{-2}$ when he is 16 m far from the bus. He runs fast enough with a uniform speed so as just to catch the bus. The minimum […]
Question 1 A particle of mass 0.3 kg starts moving from rest, in one dimension, under a force that delivers constant power $\mathrm{P}=1.5$ watt. The kinetic energy of the particle will be $\mathrm{KE}=15$ Joule after a time of (a) 5 S (b) 10 S (c) 12 S (d) 15 S Question 2 A trolley of […]
Question 1 Select the correct order of dielectric constant, refractive index and intermolecular forces for water $\left(\mathrm{H}_{2} \mathrm{O}\right)$ and heavy water $\left(\mathrm{D}_{2} \mathrm{O}\right)$ at 293 K respectively among those given below (i) Dielectric constant $-\mathrm{H}_{2} \mathrm{O}>\mathrm{D}_{2} \mathrm{O}$ (ii) Dielectric constant $-\mathrm{D}_{2} \mathrm{O}>\mathrm{H}_{2} \mathrm{O}$ (iii) Refractive index $-\mathrm{H}_{2} \mathrm{O}>\mathrm{D}_{2} \mathrm{O}$ (iv) Refractive index $-\mathrm{D}_{2} \mathrm{O}>\mathrm{H}_{2} \mathrm{O}$ […]
Question 1 The magnitude of electrostatic force between two tiny spherical balls carrying charge $\mathrm{q}_{1}$ and $\mathrm{q}_{2}$ separated by a distance r in free space is given by $\mathrm{F}=\mathrm{K} \frac{\mathrm{q}_{1} \mathrm{q}_{2}}{\mathrm{r}^{2}}$ where the constant $\mathrm{K}=9 \times 10^{9}$ in SI units Two tiny spherical balls of carbon $\left({ }_{6}^{12} \mathrm{C}\right)$ weighing 1 g each are kept […]
Question 1 Which of the following graphs is correct for a particle moving in a circle of radius $r$ at a speed of v (where ' $a$ ' is magnitude of acceleration) ? Question 2 Electronic configuration of $\mathrm{Na}^{+}$ is $(2,8)$ and that of sodium element is $(2,8,1)$. Choose the correct statements. (i) $\quad \mathrm{Na}^{+}{ […]