CUET PG MCA 2023 Previous Year Questions (PYQs) – Page 8 of 10
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The simplified form of the Boolean Expression AB+AB' is ______
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Solution
Which of the following is true
A. If $a\cos A=b\cos B$, then the triangle is isosceles or right angled.
B. If in a triangle $ABC$, $\cos A\cos B+\sin A\sin B\sin C=1$, then the triangle is isosceles right angled.
C. If the ex-radii $r_1,r_2,r_3$ of $\triangle ABC$ are in H.P., then its sides are not in A.P.
Choose the correct answer from the options given below:
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Solution
A:
$a\cos A=b\cos B \Rightarrow \cos A\cos A=\cos B\cos B$
This holds when $A=B$ (isosceles) or one angle is $90^\circ$
✔ True
B:
Maximum value of $\cos A\cos B+\sin A\sin B\sin C$ is $1$
This occurs when $A=B=45^\circ,\ C=90^\circ$
✔ True
C:
If ex-radii are in H.P., sides cannot be in A.P.
✔ True
Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A:
If dot product and cross product of vectors $\vec A$ and $\vec B$ are zero, it implies that one of the vectors $\vec A$ or $\vec B$ must be a null vector.
Reason R:
Null vector is a vector with zero magnitude.
Choose the correct answer:
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Solution
$\vec A\times\vec B=0$ ⇒ vectors are parallel
A non-zero vector cannot be both parallel and perpendicular
⇒ one vector must be zero
✔ Assertion A is true
✔ Reason R is true
✔ R explains A
If $A,B,C$ are any three sets, then
(A) $A-(B\cap C)=(A\cap B)-(A\cap C)$
(B) $A-(B\cup C)=(A-B)\cap(A-C)$
(C) $n(A-B)=n(A)-n(A\cap B)$
(D) $A\cap(B-C)=(A\cap B)\cap(A-C)$
Choose the most appropriate answer:
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Solution
A: False (LHS ≠ RHS generally)
B: True (De Morgan’s law)
C: True (basic counting identity)
D: True (set distributive law)
| LIST I | LIST II | ||
|---|---|---|---|
| A. | $|\vec A+\vec B|=|\vec A-\vec B|$ | I. | $45^\circ$ |
| B. | $|\vec A\times\vec B|=\vec A\cdot\vec B$ | II. | $30^\circ$ |
| C. | $|\vec A\cdot\vec B|=\dfrac{AB}{2}$ | III. | $90^\circ$ |
| D. | $|\vec A\times\vec B|=\dfrac{AB}{2}$ | IV. | $60^\circ$ |
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Solution
A:
$|\vec A+\vec B|=|\vec A-\vec B| \Rightarrow \vec A\cdot\vec B=0$
$\Rightarrow$ angle $=90^\circ$ → III
B:
$|\vec A\times\vec B|=\vec A\cdot\vec B$
$AB\sin\theta=AB\cos\theta \Rightarrow \theta=45^\circ$ → I
C:
$|\vec A\cdot\vec B|=AB\cos\theta=\dfrac{AB}{2}$
$\Rightarrow \cos\theta=\dfrac12 \Rightarrow \theta=60^\circ$ → IV
D:
$|\vec A\times\vec B|=AB\sin\theta=\dfrac{AB}{2}$
$\Rightarrow \sin\theta=\dfrac12 \Rightarrow \theta=30^\circ$ → II
Correct Matching:
A-III, B-I, C-IV, D-II
If $x,y,z$ are all distinct and
$\left|
\begin{array}{ccc}
x & x^2 & 1+x^3 \\
y & y^2 & 1+y^3 \\
z & z^2 & 1+z^3
\end{array}
\right|=0$
then the value of $xyz$ is.
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Solution
Expand the determinant by using column operation:
$C_3 \rightarrow C_3 - C_1^3$
Then the determinant becomes a Vandermonde determinant multiplied by $(xyz+1)$.
Since $x,y,z$ are distinct, the Vandermonde determinant is non-zero.
Hence,
$xyz+1=0$
$\Rightarrow xyz=-1$
These are eight members in the family. Bravo and Priya are siblings. Angel is Kajal’s grand daughter, Kajal who
is Priya’s mother-in-law. Ziva is a married woman and is older than Tim. Tim is the son of Sam who is the
brother-in-law of Bravo. Smith is the eldest male in the family. Angel is not Ziva’s daughter. So how is Bravo
related to Ziva?
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Solution
Bravo and Priya are siblings.
Kajal is Priya’s mother-in-law ⇒ Priya is married.
Sam is brother-in-law of Bravo ⇒ Sam is husband of Priya.
Tim is son of Sam ⇒ Tim is Priya’s son.
Ziva is married and older than Tim ⇒ Ziva is Tim’s wife.
Hence, Bravo (brother of Priya) is brother-in-law of Ziva.
Find out the trend and choose the missing character from given alternative.
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Solution
Column-wise differences increase by 6:
Column 1: $2 \to 17 (+15),\ 17 \to 50 (+33)$
Column 2: $5 \to 26 (+21),\ 26 \to 65 (+39)$
Column 3: $10 \to 37 (+27),\ 37 \to 82 (+45)$
So missing number = 26.
The number of possible Boolean functions that can be defined for $n$ Boolean variables over $n$-valued Boolean algebra is ______.
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Solution
There are $2^n$ input combinations and each can take $n$ values.
Total functions $= n^{2^n}$.
CUET PG MCA
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CUET PG MCA
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and More.
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