SRMJEEE Aptitude Syllabus

Define mean proportional and third proportional.

Mean proportional between $a$ and $b$ is $\sqrt{ab}$. The third proportional to $a$ and $b$ is $\frac{b^{2}}{a}$ (where $a:b = b:x$).

What is the formula for the average (arithmetic mean) of $n$ quantities?

$$\text{Average} = \frac{\text{Sum of all quantities}}{\text{Number of quantities}} = \frac{1}{n}\sum_{i=1}^{n} x_{i}$$

What is the average of the first $n$ natural numbers and of $n$ consecutive numbers?

Average of the first $n$ natural numbers is $\frac{n+1}{2}$. The average of any set of consecutive numbers equals the average of the first and last terms.

State the rule of alligation for mixing two ingredients.

$$\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{\text{Price of dearer} - \text{Mean price}}{\text{Mean price} - \text{Price of cheaper}}$$

A container has $x$ litres of pure liquid; $y$ litres are removed and replaced with water, repeated $n$ times. How much pure liquid remains?

$$\text{Pure liquid left} = x \left(1 - \frac{y}{x}\right)^{n}$$

What is the basic relationship between speed, distance and time?

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Distance} = \text{Speed} \times \text{Time}, \quad \text{Time} = \frac{\text{Distance}}{\text{Speed}}$$

How do you convert between km/h and m/s?

$$1 \text{ km/h} = \frac{5}{18} \text{ m/s}, \qquad 1 \text{ m/s} = \frac{18}{5} \text{ km/h}$$

What is the average speed for a journey covering equal distances at speeds $u$ and $v$?

$$\text{Average speed} = \frac{2uv}{u+v} \text{ (the harmonic mean of the two speeds)}$$

How do relative speeds combine for objects moving in the same and opposite directions?

Opposite directions: relative speed $= u + v$. Same direction: relative speed $= |u - v|$.

How long does a train of length $L$ take to cross a stationary pole versus a platform of length $P$?

To cross a pole: $\text{time} = \frac{L}{\text{speed}}$. To cross a platform: $\text{time} = \frac{L + P}{\text{speed}}$.

In boats and streams, what are the downstream and upstream speeds in terms of boat speed $b$ and stream speed $s$?

Downstream speed $= b + s$; upstream speed $= b - s$. Hence $b = \frac{1}{2}(\text{down} + \text{up})$ and $s = \frac{1}{2}(\text{down} - \text{up})$.

If a person can do a piece of work in $n$ days, what fraction of the work is done in one day?

One day's work $= \frac{1}{n}$ of the total work.

If A and B can finish a work in $a$ and $b$ days respectively, how long do they take working together?

$$\text{Time together} = \frac{ab}{a+b} \text{ days, since combined one-day work} = \frac{1}{a} + \frac{1}{b}$$

State the work equivalence (MDH) relationship used in time-and-work problems.

$$\frac{M_{1} D_{1} H_{1}}{W_{1}} = \frac{M_{2} D_{2} H_{2}}{W_{2}}$$ where $M$ = men, $D$ = days, $H$ = hours/day, $W$ = work done.

If A is twice as efficient as B, what is the ratio of times they take and the ratio of work done?

Times taken are in the inverse ratio $1:2$ (A takes half the time); in equal time, work done is in the ratio $2:1$.

What is the simple interest formula?

$$SI = \frac{P \times R \times T}{100}$$ where $P$ = principal, $R$ = rate percent per annum, $T$ = time in years.

What is the compound interest amount formula for annual compounding?

$$A = P\left(1 + \frac{R}{100}\right)^{T}, \qquad CI = A - P$$

How is the compound amount adjusted when interest is compounded $n$ times per year?

$$A = P\left(1 + \frac{R}{100n}\right)^{nT}$$ for compounding $n$ times per year over $T$ years.

For the same principal, rate and time, what is the difference between CI and SI for 2 years?

$$CI - SI = P\left(\frac{R}{100}\right)^{2}$$ for a period of $2$ years compounded annually.

In how many years does a sum double at simple interest with rate $R\%$ per annum?

For doubling, $SI = P$, so $\frac{P \cdot R \cdot T}{100} = P$, giving $T = \frac{100}{R}$ years.

What is an arithmetic progression and the formula for its $n$th term?

An AP has a constant common difference $d$. The $n$th term is $a_{n} = a + (n-1)d$, where $a$ is the first term.

What is a geometric progression and the formula for its $n$th term?

A GP has a constant ratio $r$ between consecutive terms. The $n$th term is $a_{n} = a r^{\,n-1}$, where $a$ is the first term.

Identify the pattern type: $2, 6, 12, 20, 30, \ldots$

The differences are $4, 6, 8, 10, \ldots$ (increasing by $2$). The terms follow $a_{n} = n(n+1)$, so the next term is $42$.

In an alphabet series, what positional approach helps solve letter-series problems?

Assign each letter its position number ($A=1, B=2, \ldots, Z=26$), find the numeric pattern in the gaps, then convert the resulting position back to a letter.

In Coding-Decoding, what is letter-shift (Caesar) coding? Give an example.

Each letter is shifted by a fixed number of places. For a $+1$ shift, $\text{CAT} \to \text{DBU}$. To decode, shift back by the same amount.

What is the EJOTY rule and how is it used in coding problems?

EJOTY gives easy reference positions: $E=5, J=10, O=15, T=20, Y=25$. It helps quickly locate a letter's position when computing shifts in coding-decoding.

In number coding, how can a word be coded using positional sums or products?

Replace each letter with its position value, then apply an operation (sum, product, or pattern). E.g. coding may use the sum of position numbers of the letters as the code for the word.

What is the opposite-letter (mirror) coding relationship in the alphabet?

A letter at position $n$ maps to position $27 - n$ (its mirror): $A \leftrightarrow Z$, $B \leftrightarrow Y$, $C \leftrightarrow X$, etc.

In blood relations, how do you identify maternal versus paternal relatives?

Maternal relatives are connected through the mother's side (e.g. mother's brother = maternal uncle); paternal relatives are connected through the father's side (e.g. father's brother = paternal uncle).

In a direction-sense test, what are the four cardinal and four ordinal directions, and how do left/right turns work facing North?

Cardinals: N, S, E, W; ordinals: NE, NW, SE, SW. Facing North, a right turn faces East and a left turn faces West; a turn behind (about-turn) faces South. Net straight-line distance between two perpendicular legs is found using $\sqrt{x^{2} + y^{2}}$.