MAH CET MCA 2025 Previous Year Questions (PYQs) – Page 3 of 10
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The projections of a line segment on the $X,Y,Z$ axes are $12,4,3$ respectively. The length and direction cosines are
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Solution
Components are $(12,4,3)$. Length $L=\sqrt{12^2+4^2+3^2}=13$.
Direction cosines $=\left(\dfrac{12}{13},\dfrac{4}{13},\dfrac{3}{13}\right)$.
If $\tan^{-1}\left(\dfrac{1-x}{1+x}\right)=\dfrac{1}{2}\tan^{-1}x$, then $x$ is
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Solution
Let $t=\tan\left(\dfrac{1}{2}\tan^{-1}x\right)=\dfrac{1-x}{1+x}$.
Using $\tan(2\theta)=\dfrac{2t}{1-t^2}=x$, we get $x=\dfrac{1-x^2}{2x}\Rightarrow 3x^2=1\Rightarrow x=\pm\dfrac{1}{\sqrt3}$.
From the equation, $x>0$.
Find the total number of ways to put $20$ balls into $5$ boxes so that the first box contains only one ball.
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Solution
Choose which ball goes to Box 1: $20$ ways.
Each of the remaining $19$ balls can go to any of Boxes $2$–$5$: $4^{19}$ ways.
Total $=20\cdot4^{19}$.
Find the number of arrangements of the letters of the word $,\text{INDEPENDENCE},$ such that
(i) the words start with $I$ and end with $P$.
(ii) all the vowels never occur together.
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Solution
Letters in INDEPENDENCE: $E^4,\ N^3,\ D^2,\ I,\ P,\ C$ (total $12$).
(i) Fix $I$ at start and $P$ at end. Arrange the remaining $10$ symbols: $E^4,\ N^3,\ D^2,\ C$.
Number $= \dfrac{10!}{4!,3!,2!}=12600$.
(ii) Total arrangements $= \dfrac{12!}{4!,3!,2!}=1663200$.
“All vowels together”: treat $(\text{vowels}=E^4I)$ as one block. Then items are
${\text{block}, N^3, D^2, C, P}$ → $8$ items with repeats $N^3, D^2$.
Arrangements of items $= \dfrac{8!}{3!,2!}=3360$.
Internal vowel-block arrangements $= \dfrac{5!}{4!}=5$.
Together $= 3360\times5=16800$.
Hence “never together” $=1663200-16800=1646400$.
The number of parallelograms formed by a set of $4$ parallel lines intersecting another set of $3$ parallel lines is
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Solution
Choose any $2$ lines from each set $\Rightarrow \binom{4}{2}\binom{3}{2}=6\times3=18$.
A study finds that employees working fewer hours per week are more productive during their working hours and concludes that reducing working hours can increase productivity and job satisfaction. Which assumption underpins the conclusion?
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Solution
The conclusion requires that higher per-hour productivity offsets the reduced hours, addressing the concern about delays/costs.
A study found that employees who worked fewer hours per week were more productive. Which of the following, if true, weakens the study’s conclusion?
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Solution
The conclusion requires that higher per-hour productivity offsets the reduced hours, addressing the concern about delays/costs.
In a certain code “SAND” is written as “UCPF”. How is “STONE” written in that code?
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Solution
Each letter is shifted $+2$ positions in the alphabet:
$S!\to!U,\ T!\to!V,\ O\to!Q,\ N!\to!P,\ E!\to!G$.
So “STONE” $\to$ “UVQPG”.
Three of the following four words are alike in a certain way and one is different. Select the odd one.
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Solution
The first three are types of eyewear worn on the eyes; an optical reader is a device, not eyewear.
Look at this series: $53,53,40,40,27,27,\ldots$ What number should come next?
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Solution
The numbers decrease by $13$ and each value is repeated:
$53,53,\ 40,40,\ 27,27,\ 14,14,\ldots$
So the next term is $14$.
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