What this CMI Entrance quiz checks
The CMI Entrance paper assumes a body of school mathematics is already fluent, and tests whether you can reason with it rather than whether you have met it. This quiz covers that assumed ground: algebra, functions, coordinate geometry, trigonometry, sequences, combinatorics and elementary calculus.
Use it as a diagnostic before you begin preparation proper. A question you cannot start is a prerequisite gap, and closing it now is far cheaper than discovering it in the middle of a past paper. A question you can solve slowly is a fluency problem, which practice fixes.
Anyone sitting CMI Entrance should treat past papers as the primary resource, attempting each one under time before reading any solution. Because CMI Entrance and the ISI paper test overlapping skills, work done for one is rarely wasted on the other, and a realistic plan gives the final third of preparation almost entirely to written proof.
ISI/CMI Entrance Prerequisite Mathematics Quiz
A foundational assessment of 40 questions covering high-school level mathematics required for ISI and CMI entrance exams. Focuses on Algebra, Calculus, Combinatorics, and Geometry.
How this quiz works:
- Select one option per question. Your choice is final — no re-selection after clicking.
- The correct answer and a full explanation appear immediately after you choose.
- Your running score updates in the panel above.
- Use the Reset Quiz button to start over.
Question 1
Solve for $x$: $\,x^3 - 6x^2 + 11x - 6 = 0$
Answer: A. Factorising: $x^3-6x^2+11x-6 = (x-1)(x-2)(x-3)=0$, so $x=1,2,3$. Check by evaluating: $f(1)=1-6+11-6=0$, then divide to get the quadratic $(x-2)(x-3)$.
Question 2
If $\alpha, \beta$ are roots of $x^2 - 5x + 7 = 0$, find $\alpha^3 + \beta^3$.
Answer: B. $\alpha+\beta=5$, $\alpha\beta=7$. Using $\alpha^3+\beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta) = 125 - 3(7)(5) = 125 - 105 = 20$.
Question 3
Find the number of real solutions of $|x^2 - 1| = x + 1$.
Answer: C. Case 1: $x^2\geq 1$ gives $x^2-1=x+1 \Rightarrow x^2-x-2=0 \Rightarrow x=2$ or $x=-1$. Both satisfy $x^2\geq 1$. Case 2: $x^2<1$ gives $1-x^2=x+1 \Rightarrow x^2+x=0 \Rightarrow x=0$ or $x=-1$. Only $x=0$ satisfies $x^2<1$ and $x+1\geq 0$. Solutions: $x=-1,0,2$. Total = 3.
Question 4
If $a + b + c = 0$, evaluate $\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}$.
Answer: D. Combine over common denominator $abc$: $\frac{a^3+b^3+c^3}{abc}$. When $a+b+c=0$, we have $a^3+b^3+c^3 = 3abc$. Hence the expression equals $\frac{3abc}{abc} = 3$.
Question 5
Solve: $\log_2(x-1) + \log_2(x+3) = 3$.
Answer: A. $\log_2[(x-1)(x+3)] = 3 \Rightarrow (x-1)(x+3)=8 \Rightarrow x^2+2x-11=0 \Rightarrow x = -1\pm 2\sqrt{3}$. Since we need $x-1>0$, only $x=-1+2\sqrt{3}\approx 2.46$ is valid.
Question 6
Find the minimum value of $f(x) = x^2 + \dfrac{1}{x^2}$ for $x > 0$.
Answer: C. By AM-GM: $x^2 + \frac{1}{x^2} \geq 2\sqrt{x^2 \cdot \frac{1}{x^2}} = 2$, with equality when $x^2 = \frac{1}{x^2}$, i.e., $x=1$.
Question 7
If $z = 1 + i$, find $|z^{2024}|$.
Answer: A. $|z|=\sqrt{1^2+1^2}=\sqrt{2}$. Then $|z^{2024}|=|z|^{2024}=(\sqrt{2})^{2024}=2^{1012}$.
Question 8
Find the number of distinct real roots of $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$.
Answer: B. $f(x) = (x-1)^4$. Expanding confirms this is $(x-1)^4$, so $x=1$ is the only distinct real root (of multiplicity 4).
Question 9
The sum of an infinite geometric series is $6$ and its first term is $2$. Find the common ratio.
Answer: B. $S = \frac{a}{1-r} = \frac{2}{1-r} = 6$, so $1-r = \frac{1}{3}$, giving $r = \frac{2}{3}$. Check: $|r|<1$, valid.
Question 10
Let $f(x) = x^3 + ax^2 + bx + c$. If $f(1) = f(2) = f(3) = 0$, find $a + b + c$.
Answer: A. $f(x)=(x-1)(x-2)(x-3) = x^3-6x^2+11x-6$. So $a=-6, b=11, c=-6$. Thus $a+b+c = -6+11-6 = -1$. Wait — but the question asks $f(0) = c = -6$? Expanding, $f(x)=(x-1)(x-2)(x-3)=x^3-6x^2+11x-6$, so $a=-6$, $b=11$ and $c=-6$. Therefore $f(0)=c=-6$. Answer: A ($-6$).
Question 11
Evaluate $\displaystyle\lim_{x \to 0} \frac{\sin 5x}{3x}$.
Answer: B. $\lim_{x\to 0}\frac{\sin 5x}{3x} = \lim_{x\to 0}\frac{\sin 5x}{5x}\cdot\frac{5x}{3x} = 1\cdot\frac{5}{3} = \frac{5}{3}$.
Question 12
Find $f'(x)$ if $f(x) = x \ln x$.
Answer: B. By the product rule: $f'(x) = 1\cdot\ln x + x\cdot\frac{1}{x} = \ln x + 1$.
Question 13
Evaluate $\displaystyle\int_0^1 x(1-x)^4 \, dx$.
Answer: A. Using the Beta function: $\int_0^1 x^1(1-x)^4\,dx = B(2,5) = \frac{\Gamma(2)\Gamma(5)}{\Gamma(7)} = \frac{1!\cdot 4!}{6!} = \frac{24}{720} = \frac{1}{30}$. Alternatively, expand $(1-x)^4$ and integrate term by term.
Question 14
Find the area bounded by $y = x^2$ and $y = 2x$.
Answer: B. Intersection: $x^2=2x \Rightarrow x=0,2$. Area $= \int_0^2(2x-x^2)\,dx = \left[x^2-\frac{x^3}{3}\right]_0^2 = 4-\frac{8}{3} = \frac{4}{3}$.
Question 15
Evaluate $\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$.
Answer: B. By L'Hôpital (twice): $\lim\frac{e^x-1}{2x}=\lim\frac{e^x}{2}=\frac{1}{2}$. Or from the Taylor series $e^x=1+x+\frac{x^2}{2}+\cdots$, so $\frac{e^x-1-x}{x^2}\to\frac{1}{2}$.
Question 16
Find the derivative of $\sin^{-1}(2x\sqrt{1-x^2})$ for $|x| < \dfrac{1}{\sqrt{2}}$.
Answer: A. Let $x=\sin\theta$. Then $2x\sqrt{1-x^2}=2\sin\theta\cos\theta=\sin 2\theta$. For $|x|<\frac{1}{\sqrt{2}}$, we have $2\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$, so $\sin^{-1}(\sin 2\theta)=2\theta$. Thus $f(x)=2\sin^{-1}x$ and $f'(x)=\frac{2}{\sqrt{1-x^2}}$.
Question 17
Evaluate $\displaystyle\int \frac{dx}{x^2 + a^2}$ where $a \neq 0$.
Answer: A. This is a standard integral. Let $x = a\tan\theta$, then $dx = a\sec^2\theta\,d\theta$ and $\int\frac{a\sec^2\theta}{a^2\sec^2\theta}\,d\theta = \frac{1}{a}\theta + C = \frac{1}{a}\tan^{-1}\frac{x}{a}+C$.
Question 18
Determine the intervals where $f(x) = x^3 - 3x + 2$ is strictly increasing.
Answer: A. $f'(x)=3x^2-3=3(x-1)(x+1)$. $f'(x)>0$ when $x<-1$ or $x>1$. So $f$ is strictly increasing on $(-\infty,-1)\cup(1,\infty)$.
Question 19
Evaluate $\displaystyle\lim_{x \to 0} \frac{\tan x - x}{x^3}$.
Answer: C. Using Taylor: $\tan x = x + \frac{x^3}{3}+\cdots$, so $\frac{\tan x - x}{x^3}\to\frac{1}{3}$. By L'Hôpital: $\lim\frac{\sec^2 x - 1}{3x^2} = \lim\frac{\tan^2 x}{3x^2}=\frac{1}{3}$.
Question 20
The value of $\displaystyle\int_{-\pi/2}^{\pi/2} \sin^3 x \, dx$ is:
Answer: A. $\sin^3 x$ is an odd function: $\sin^3(-x)=-\sin^3 x$. The integral of an odd function over a symmetric interval $[-a,a]$ is $0$.
Question 21
A committee of 3 is to be formed from 5 men and 4 women. In how many ways can this be done if at least one woman must be included?
Answer: B. Total ways $= \binom{9}{3}=84$. Ways with no women $= \binom{5}{3}=10$. At least one woman $= 84-10 = 74$.
Question 22
Find the coefficient of $x^5$ in the expansion of $(1+x)^{10}$.
Answer: C. By the Binomial Theorem, the coefficient of $x^5$ is $\binom{10}{5}=\frac{10!}{5!\cdot 5!}=252$.
Question 23
How many positive integers less than 1000 are divisible by 3 or 5?
Answer: A. By inclusion-exclusion: $\lfloor 999/3\rfloor + \lfloor 999/5\rfloor - \lfloor 999/15\rfloor = 333 + 199 - 66 = 466$.
Question 24
In how many ways can the letters of "MISSISSIPPI" be arranged?
Answer: A. 11 letters with M:1, I:4, S:4, P:2. Number of arrangements $= \frac{11!}{1!\cdot 4!\cdot 4!\cdot 2!} = \frac{11!}{4!\cdot 4!\cdot 2!} = 34650$.
Question 25
A box contains 4 red and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?
Answer: A. $P = \frac{\binom{4}{2}}{\binom{10}{2}} = \frac{6}{45} = \frac{2}{15}$. Alternatively: $\frac{4}{10}\cdot\frac{3}{9}=\frac{2}{15}$.
Question 26
How many integers from 1 to 1000 are NOT divisible by 2, 3, or 5?
Answer: A. By inclusion-exclusion, divisible by 2, 3, or 5: $500+333+200-166-100-66+33=734$. Not divisible: $1000-734=266$.
Question 27
Find the number of surjections (onto functions) from a set of 4 elements to a set of 3 elements.
Answer: B. By inclusion-exclusion: $3^4 - \binom{3}{1}\cdot 2^4 + \binom{3}{2}\cdot 1^4 = 81 - 48 + 3 = 36$.
Question 28
How many 5-digit numbers have all digits distinct?
Answer: A. First digit (1-9): 9 choices. Second: 9 (0-9 minus first). Third: 8. Fourth: 7. Fifth: 6. Total $= 9\times 9\times 8\times 7\times 6 = 27216$.
Question 29
From a group of 7 people, how many ways to choose a president, vice-president, and secretary (all different people)?
Answer: C. This is a permutation: $P(7,3) = 7\times 6\times 5 = 210$.
Question 30
What is the 10th term of the arithmetic progression $3, 7, 11, 15, \ldots$?
Answer: A. $a=3$, $d=4$. $a_{10} = a + 9d = 3 + 36 = 39$.
Question 31
In a triangle with sides 5, 12, and 13, find the area.
Answer: A. $5^2+12^2=25+144=169=13^2$, so this is a right triangle. Area $= \frac{1}{2}\times 5\times 12 = 30$.
Question 32
Find the distance between the points $(1, 2)$ and $(4, 6)$.
Answer: A. Distance $= \sqrt{(4-1)^2+(6-2)^2} = \sqrt{9+16} = \sqrt{25} = 5$.
Question 33
The line $3x + 4y = 12$ intersects the coordinate axes at $A$ and $B$. Find the area of triangle $OAB$ where $O$ is the origin.
Answer: A. $x$-intercept: set $y=0$, get $A=(4,0)$. $y$-intercept: set $x=0$, get $B=(0,3)$. Area $= \frac{1}{2}\times 4\times 3 = 6$.
Question 34
Find the eccentricity of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$.
Answer: A. $a^2=9$, $b^2=4$. $c^2=a^2-b^2=9-4=5$, so $c=\sqrt{5}$. Eccentricity $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$.
Question 35
Two circles of radii 3 and 4 touch each other externally. Find the distance between their centers.
Answer: C. For external tangency, the distance between centers equals the sum of the radii: $3+4=7$.
Question 36
In $\triangle ABC$, $\angle A = 60°$, $AB = 5$, $AC = 8$. Find $BC$.
Answer: A. By the cosine rule: $BC^2 = AB^2+AC^2-2(AB)(AC)\cos A = 25+64-2(5)(8)\cos 60° = 89-40 = 49$. So $BC=7$.
Question 37
Find the equation of the circle with center $(3, -2)$ and radius $4$.
Answer: A. The standard form is $(x-h)^2+(y-k)^2=r^2$ where $(h,k)=(3,-2)$ and $r=4$. Thus $(x-3)^2+(y+2)^2=16$.
Question 38
The angle of elevation of the top of a tower from a point on the ground 30 m away is $30°$. Find the height of the tower.
Answer: A. $\tan 30° = \frac{h}{30}$, so $h = 30\tan 30° = 30\cdot\frac{1}{\sqrt{3}} = 10\sqrt{3}$ m.
Question 39
Find the reflection of the point $(2, 3)$ about the line $x + y = 0$.
Answer: B. Reflection about $y=-x$ (i.e., $x+y=0$) maps $(x,y)$ to $(-y,-x)$. So $(2,3)\to(-3,-2)$. Derivation: the foot of perpendicular from $(2,3)$ to $x+y=0$ is $\left(-\frac{1}{2},\frac{1}{2}\right)$, and reflecting gives $(-3,-2)$.
Question 40
Find the area of the region enclosed by the parabola $y = x^2$ and the line $y = 4$.
Answer: B. Intersection: $x^2=4 \Rightarrow x=\pm 2$. Area $= \int_{-2}^{2}(4-x^2)\,dx = \left[4x-\frac{x^3}{3}\right]_{-2}^{2} = (8-\frac{8}{3})-(-8+\frac{8}{3}) = 16-\frac{16}{3} = \frac{32}{3}$.
Candidates preparing for CMI Entrance often ask how much of the paper is proof-based. The answer is: most of what separates the top band from the rest.
The CMI Entrance paper rewards a clearly written argument over a correct answer with no working, which is the habit that takes longest to build.
Anyone sitting CMI Entrance should treat past papers as the primary resource, attempting each one under time before reading any solution.
Because CMI Entrance and the ISI paper test overlapping skills, work done for one is rarely wasted on the other.
A realistic CMI Entrance plan allocates the final third of preparation almost entirely to written proof.
The practical points about the CMI Entrance are gathered here rather than scattered across the guide.
