Jamia Millia Islamia MCA 2023 Previous Year Questions (PYQs) – Page 1 of 10
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How many elements does the set
$P\big({\varnothing, a, {a}, {{a}}}\big)$ has;
where $a$ and $b$ are distinct elements and $P$ denotes the power set?
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Solution
The given set ${\varnothing, a, {a}, {{a}}}$ has $4$ elements.
Therefore,
What is the cardinality of these sets in the order of their serial number?
(i) ${a}$
(ii) ${{a}}$
(iii) ${a, {a}}$
(iv) ${a, {a}, {{a}}}$
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Solution
(i) ${a}$ → 1 element
(ii) ${{a}}$ → 1 element
(iii) ${a, {a}}$ → 2 elements
(iv) ${a, {a}, {{a}}}$ → 3 elements
Suppose that $A_i = {1, 2, 3, \ldots, i}$ for $i = 1, 2, 3, \ldots$
Then find $\displaystyle \bigcup_{i=1}^{\infty} A_i = ?$
Here $Z$ denotes the set of integers.
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Solution
$,\text{Find } \displaystyle \bigcup_{i=1}^{\infty} A_i \text{ and } \bigcap_{i=1}^{\infty} A_i,\ \text{for every positive integer } i \text{ where } A_i={-i,i}.$
(Here $\mathbb Z$ denotes the set of integers.)
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Solution
$A_1={-1,1},\ A_2={-2,2},\ldots$ so
$\displaystyle \bigcup_{i=1}^{\infty}A_i={\ldots,-3,-2,-1,1,2,3,\ldots}=\mathbb Z\setminus{0}$,
$\displaystyle \bigcap_{i=1}^{\infty}A_i=\varnothing$.
Which of the following relations are functions?
(i) ${(1,(a,b)),\ (2,(b,c)),\ (3,(c,a)),\ (4,(a,b))}$
(ii) ${(1,(a,b)),\ (2,(b,a)),\ (3,(c,a)),\ (1,(a,c))}$
(iii) ${(1,(a,b)),\ (2,(a,b)),\ (3,(a,b))}$
(iv) ${(1,(a,b)),\ (2,(b,c)),\ (1,(c,a))}$
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Solution
A relation $R$ is a function iff every first component occurs exactly once.
(i) first components $1,2,3,4$ appear once each $\Rightarrow$ function.
(ii) $1$ appears twice $\Rightarrow$ not a function.
(iii) $1,2,3$ appear once each $\Rightarrow$ function.
(iv) $1$ appears twice $\Rightarrow$ not a function.
There is a direct flight from Trichy to New Delhi and 2 direct trains.
There are 6 trains from Trichy to Chennai and 4 trains from Chennai to Delhi.
Also, there are 2 trains from Trichy to Mumbai and 8 flights from Mumbai to New Delhi.
In how many ways can a person travel from Trichy to New Delhi?
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Solution
Routes and counts:
Direct flight: $1$; direct trains: $2$; via Chennai: $6\times4=24$; via Mumbai: $2\times8=16$.
Total $=1+2+24+16=43$.
If $P,Q,R$ have truth values $T,T,F$, then the truth values of
$\Big(P\to(Q\to R)\Big)\to\Big((P\to Q)\to(P\to R)\Big)$ and $P\to(Q\vee R)$ are:
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Solution
$Q\to R = T\to F = F$, so $P\to(Q\to R)=T\to F=F$.
$P\to Q = T\to T = T$, $P\to R = T\to F = F$, hence $(P\to Q)\to(P\to R)=T\to F=F$.
Therefore $(F)\to(F)=T$.
Also $Q\vee R = T\vee F = T$, so $P\to(Q\vee R)=T\to T=T$.
$(A\cap B')\ \cup\ (A'\cap B)\ \cup\ (A'\cap B')$ is equal to
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Solution
$(A'\cap B)\cup(A'\cap B')=A'\cap(B\cup B')=A'$.
Then $A'\cup(A\cap B')=(A'\cup A)\cap(A'\cup B')=\mathbf U\cap(A'\cup B')=A'\cup B'$.
The floor function $[x]$ is
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Solution
The floor function $f(x) = [x]$ maps every real number to the greatest integer less than or equal to $x$.
Different real numbers can have the same floor value, so it is not one-to-one,
but for every integer $n \in \mathbb{Z}$, there exists an $x \in \mathbb{R}$ such that $[x]=n$.
Hence, it is onto but not one-to-one.
The domain of the real-valued function
$f(x) = \sqrt{x - 3} + \sqrt{x - 4}$
is the set of all values of $x$ satisfying
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Solution
For $f(x)$ to be real, both radicals must be defined:
$x - 3 \ge 0 \quad \text{and} \quad x - 4 \ge 0$
$\Rightarrow x \ge 4$
So the domain is $[4, \infty)$.
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