Jamia Millia Islamia MCA 2024 Previous Year Questions (PYQs) – Page 6 of 10
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Let $A$ be a non-singular matrix of order $2 \times 2$. Then
$|A^{-1}| =$
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Solution
For any invertible matrix $A$,
$\det(A^{-1}) = \dfrac{1}{\det(A)}$
If $A$ and $B$ are symmetric matrices of the same order, then $(AB - BA)$ is a
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Solution
$(AB - BA)^T = B^TA^T - A^TB^T = BA - AB = -(AB - BA)$
Hence, $(AB - BA)$ is skew-symmetric.
What is $\displaystyle \lim_{x\to 0}\left(\frac{x\tan x}{\cot x}\right)$?
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Solution
$\tan x\sim x,\ \cot x\sim \frac1x \Rightarrow \dfrac{x\tan x}{\cot x}\sim \dfrac{x\cdot x}{1/x}=x^3\to0.$
If $\sec!\left(\dfrac{x-y}{x+y}\right)=a$, then $\dfrac{dy}{dx}$ is
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Solution
Let $u=\dfrac{x-y}{x+y}$. Since $\sec u=a$ (constant), $u' = 0$.
$\displaystyle 0=\frac{d}{dx}!\left(\frac{x-y}{x+y}\right)=\frac{(1-y')(x+y)-(x-y)(1+y')}{(x+y)^2}$
$\Rightarrow 2y-2xy'=0\Rightarrow y'=\dfrac{y}{x}.$
If $x^{,y}y^{,x}=16$, then $\left.\dfrac{dy}{dx}\right|_{(2,2)}$ is
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Solution
$\ln(x^{y}y^{x})=y\ln x+x\ln y=\ln16$
$\Rightarrow y'(\ln x+\tfrac{x}{y})+(\tfrac{y}{x}+\ln y)=0$
At $(2,2)$: $y'=\dfrac{-,(1+\ln2)}{,\ln2+1}=-1.$
The function $f(x)=x+\cos x$ is
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Solution
$f'(x)=1-\sin x\in[0,2]$, so $f'(x)\ge0$ for all $x$ (zero only when $\sin x=1$).
$\displaystyle \int x^{2}\sin(x^{3}),dx =$
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Solution
Let $u=x^{3}\Rightarrow du=3x^{2}dx\Rightarrow x^{2}dx=\tfrac{1}{3}du$.
$\int x^{2}\sin(x^{3})dx=\tfrac13\int\sin u,du=-\tfrac13\cos x+C.$
The objective function of a linear programming problem is
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Solution
In linear programming, the objective function represents the quantity (profit, cost, etc.) to be maximized or minimized, subject to given constraints.
If $|z_1| = 4,\ |z_2| = 3$, then what is the value of $|z_1 + z_2 + 3 + 4i|$ ?
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Solution
Using triangle inequality,
Hence, value is less than 12.
Region represented by $x \ge 0,\ y \ge 0$ is
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Solution
Using triangle inequality,
Hence, value is less than 12.
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