JEE MAIN 2023 Previous Year Questions (PYQs) – Page 16 of 34
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The number of square matrices of order 5 with entries from the set {0,1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is:
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Solution
Key Insight: Each row has exactly one $1$ and rest $0$s. Each column also has exactly one $1$ and rest $0$s.
So no two rows can have $1$ in the same column. This is exactly a Permutation Matrix!
Count choices row by row
Row 1: $5$ columns available $\rightarrow$ 5 choices
Row 2: $4$ columns remaining $\rightarrow$ 4 choices
Row 3: $3$ columns remaining $\rightarrow$ 3 choices
Row 4: $2$ columns remaining $\rightarrow$ 2 choices
Row 5: $1$ column remaining $\rightarrow$ 1 choice
Total number of matrices
$\text{Total} = 5 \times 4 \times 3 \times 2 \times 1 = 5! = 120$
Answer: $\boxed{120}$
So no two rows can have $1$ in the same column. This is exactly a Permutation Matrix!
Count choices row by row
Row 1: $5$ columns available $\rightarrow$ 5 choices
Row 2: $4$ columns remaining $\rightarrow$ 4 choices
Row 3: $3$ columns remaining $\rightarrow$ 3 choices
Row 4: $2$ columns remaining $\rightarrow$ 2 choices
Row 5: $1$ column remaining $\rightarrow$ 1 choice
Total number of matrices
$\text{Total} = 5 \times 4 \times 3 \times 2 \times 1 = 5! = 120$
Answer: $\boxed{120}$
If
\[
\begin{vmatrix}
x+1 & x & x \\
x & x+\lambda & x \\
x & x & x+\lambda^2
\end{vmatrix}
= \dfrac{9}{8}\,(103x+81),
\]
then $\lambda,\ \dfrac{\lambda}{3}$ are the roots of the equation:
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Solution
Let the six numbers $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$ be in A.P. and $a_{1}+a_{3}=10$.
If the mean of these six numbers is $\dfrac{19}{2}$ and their variance is $\sigma^{2}$, then $8\sigma^{2}$ is equal to:
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Solution
Let the mean of 6 observations $1, 2, 4, 5, x, y$ be $5$ and their variance be $10$.
Then their mean deviation about the mean is equal to:
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Solution
The value of
$\left(\dfrac{\,1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}\,}{\,1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}\,}\right)^{3}$ is:
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Solution
If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying
\[
\int_{0}^{\pi/2} f(\sin 2x)\,\sin x\,dx \;+\; \alpha \int_{0}^{\pi/4} f(\cos 2x)\,\cos x\,dx \;=\; 0,
\]
then the value of $\alpha$ is:
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Solution
If $f(x)=x^{3}-x^{2}f'(1)+x f''(2)-f'''(3),\ x\in\mathbb{R}$, then:
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Solution
Let $f$ and $g$ be two functions defined by
\[
f(x)=
\begin{cases}
x+1, & x<0,\\[2pt]
|x-1|, & x\ge 0
\end{cases}
\qquad\text{and}\qquad
g(x)=
\begin{cases}
x+1, & x<0,\\[2pt]
1, & x\ge 0.
\end{cases}
\]
Then $(g\circ f)(x)$ is:
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Solution
$\displaystyle \int_{\tfrac{3\sqrt{2}}{4}}^{\tfrac{3\sqrt{3}}{4}} \dfrac{48}{\sqrt{9-4x^{2}}}\,dx$ is equal to:
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Solution
Let $y=y(x)$ be the solution of the differential equation
\[
\frac{dy}{dx}+\frac{5}{x(x^5+1)}\,y=\frac{(x^5+1)^2}{x^7},\quad x>0.
\]
If $y(1)=2$, then $y(2)$ is equal to:
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Solution
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