CUET PG MCA 2023 Previous Year Questions (PYQs) – Page 10 of 10
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If $f:\mathbb{R}\to\mathbb{R}$ is defined as
$f(x)=x^2+1$, then the minimum value of $f(x)$ is
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Solution
Since $x^2\ge0$ for all real $x$,
$f(x)=x^2+1\ge1$
Minimum occurs at $x=0$.
A 32-bit wide main memory with a capacity of $1$ GB is built using $256\text{M}\times4$ bits DRAM chips.
The number of rows per chip is $2^{14}$.
The time taken to perform one refresh operation is $50$ nanoseconds.
The refresh period is $2$ milliseconds.
The percentage (rounded to the nearest integer) of time available for performing memory read/write operations in the main memory unit is ______.
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Solution
Rows to be refreshed $=2^{14}=16384$
Time for one refresh $=50$ ns
Total refresh time:
$16384\times50\text{ ns}=819200\text{ ns}=0.8192\text{ ms}$
Refresh period $=2$ ms
Percentage of time used for refresh:
$\displaystyle \frac{0.8192}{2}\times100\approx41%$
Percentage available for read/write:
$100-41=59%$
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A:
If two circles intersect at two points, then the line joining their centres is perpendicular to the common chord.
Reason R:
The perpendicular bisectors of two chords of a circle intersect at its centre.
In the light of the above statements, choose the correct answer from the options given below:
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Solution
Assertion A is a standard geometric property of intersecting circles ⇒ True.
Reason R is also true (property of chords in a circle), but it does not directly explain why the line joining centres is perpendicular to the common chord of two circles.
If $\sin\beta$ is the G.M. between $\sin\alpha$ and $\cos\alpha$, then $\cos2\beta$ is equal to
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Solution
Since $\sin\beta$ is the G.M. of $\sin\alpha$ and $\cos\alpha$,
$\sin^2\beta=\sin\alpha\cos\alpha=\dfrac12\sin2\alpha$
So,
$\cos2\beta=1-2\sin^2\beta=1-\sin2\alpha$
But
$1-\sin2\alpha=2\sin^2\left(\dfrac{\pi}{4}-\alpha\right)$
If a chord which is normal to the parabola $y^2=4ax$ at one end subtends a right angle at the vertex, then its slope is
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Solution
For parabola $y^2=4ax$, slope of normal at parameter $t$ is $-t$.
Let slope of chord be $m=-t$.
Since the chord subtends a right angle at the vertex,
$m^2=2$
So,
$m=\sqrt2$
If $\hat n_1,\hat n_2$ are two unit vectors and $\theta$ is the angle between them, then $\cos\dfrac{\theta}{2}$ is equal to
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Solution
$|\hat n_1+\hat n_2|^2=2(1+\cos\theta)$
So,
$|\hat n_1+\hat n_2|=2\cos\dfrac{\theta}{2}$
Hence,
$\cos\dfrac{\theta}{2}=\dfrac12|\hat n_1+\hat n_2|$
A 2’s-complement adder–subtracter can add or subtract binary numbers.
Sign-magnitude numbers represent ______ decimal numbers, and 2’s complements stand for ______ decimal numbers.
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Solution
In sign-magnitude representation, a separate sign bit is used, mainly representing positive numbers explicitly.
2’s-complement representation is primarily used to represent negative numbers in binary arithmetic.
If each observation of raw data whose variance is $\sigma^2$ is multiplied by $h$, then the variance of the new set is
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Solution
If all observations are multiplied by a constant $h$, then variance becomes
$h^2$ times the original variance.
So, new variance $=h^2\sigma^2$.
Which of the following functions is differentiable at $x=0$?
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Solution
$|x|$ is not differentiable at $0$.
Near $x=0$, $\sin(|x|)\approx |x|$.
For option (4):
$f(x)=\sin(|x|)-|x|$
Near $0$:
$f(x)\approx |x|-|x|=0$
Both left-hand and right-hand derivatives at $0$ are equal, hence $f(x)$ is differentiable at $0$.
All other options involve $|x|$ in a way that makes the derivative discontinuous at $0$.
Find the missing term in the given number series.
-1,0, 7, 26, 63, ?, 215, 342.......
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Solution
CUET PG MCA
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CUET PG MCA
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and More.
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