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isi, cmi practice problem: Let $f:\mathbb{N} \to \mathbb{N}$ satisfying the conditions: (i) $x - f(x) = 19 \left[ \frac{x}{19} \right] - 90 \left[ \frac{f(x)}{90} \right]$ (ii) $1900 < f(1900) < 2000$ where $[x]$ denotes the ...

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isi, cmi practice problem: Let $f(x) = \left[ \frac{1}{\cos\{x\}} \right]$, where $[\,\cdot\,]$ and $\{\,\cdot\,\}$ denote the greatest integer and fractional part functions respectively. Then find the range of the function $f$...

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isi, cmi practice problem: Let $A$ be a non-empty set of real numbers and $f: A \to A$ such that $f(f(x)) = x$ for all $x$ in $A$. Then which of the followings are true. (a) a bijection ...

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isi, cmi practice problem: Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1$. What is the remainder when $g(x^{12})$ is divided by the polynomial $g(x)$ ? (a) 6 (b) $5 - x$ (c) $4 - x + x^2$ (d) $3 - x + x^2 - x^3$

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isi, cmi practice problem: If $P(x)$ is a polynomial satisfying $P(x^2) = x^2 P(x)$ and $P(0) = -2, P'\left(\frac{3}{2}\right) = 0, P(1) = 0$. Find the maximum value of $P(x)$ is...

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isi, cmi practice problem: $y = ax^2 + 2bx + c \ (a, b, c \in R, abc \neq 0)$ never meets x-axis then $a, b, c$ can be in (a) A.P. (b) G.P. (c) H.P. (d) None of these

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isi, cmi practice problem: Find the number of ordered triples $(x, y, z)$, $x, y, z > 0$ which satisfies following inequalities \[ x(1 - y) > \frac{1}{4} \quad,\quad y(1 - z) > \frac{1}{4} \quad,\quad z(1 - x) > \frac{1}{4} ...

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isi, cmi practice problem: If ‘$a$’ and ‘$b$’ are distinct positive integers and the quadratic equations $(a - 1)x^2 - (a^2 + 2)x + (a^2 + 2a) = 0$ and $(b - 1)x^2 - (b^2 + 2)x + (b^2 + 2b) = 0$ have a common root...

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isi, cmi practice problem: Show that, if $p$ is a prime then $P \mid \binom{p}{r}$ for $0 < r < p$.

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isi, cmi practice problem\text{Find the value of} \cot \left\{ \sum_{n=1}^{23} \cot^{-1} \left( 1 + \sum_{k=1}^{n} 2k \right) \right\} \begin{matrix} \text{(a) } ...

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ISI / CMI PRACTICE PROBLEM AND SOLUTION: \lim_{n \to \infty} \frac{\{x\} + \{2x\} + \{3x\} + \cdots + \{nx\}}{n^2} = \text{[where } \{x\} = x - [x] ...

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Isi practice problem from sequence and series : The sequence $\{x_k\}$ is defined by $x_{k+1} = x_k^2 + x_k$ and $x_1 = \frac{1}{2}$...

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