JEE MAIN 2023 Previous Year Questions (PYQs) – Page 27 of 34
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The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :
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Solution
Let $\left(a+bx+cx^2\right)^{10}=\displaystyle\sum_{i=0}^{20} p_i x^i,\ a,b,c\in\mathbb{N}.$
If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to:
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Solution
If $\vec a=\hat i+2\hat k$, $\vec b=\hat i+\hat j+\hat k$, $\vec c=7\hat i-3\hat j+4\hat k$,
$\ \ \vec r\times\vec b+\vec b\times\vec c=\vec 0$ and $\vec r\cdot\vec a=0$.
Then $\ \vec r\cdot\vec c$ is equal to:
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Solution
A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:
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Solution
The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :
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Solution
The number of real roots of the equation $x|x|-5|x+2|+6=0$ is:
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Solution
Consider a function $f:\mathbb{N}\to\mathbb{R}$ satisfying
\[
f(1)+2f(2)+3f(3)+\cdots+xf(x)=x(x+1)f(x),\quad x\ge 2,
\]
with $f(1)=1$. Then
\[
\frac{1}{f(2022)}+\frac{1}{f(2028)}
\]
is equal to:
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Solution
If $(\alpha,\beta)$ is the orthocenter of the triangle $ABC$ with vertices $A(3,-7)$, $B(-1,2)$ and $C(4,5)$, then $9\alpha-6\beta+60$ is equal to:
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Solution
Let \( S=\{w_1,w_2,\ldots\} \) be the sample space of a random experiment.
Let the probabilities satisfy
\[
P(w_n)=\frac{P(w_{n-1})}{2},\qquad n\ge 2.
\]
Let
\[
A=\{\,2k+3\ell : k,\ell\in\mathbb{N}\,\},\qquad
B=\{\,w_n : n\in A\,\}.
\]
Then \(P(B)\) is equal to:
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Solution
Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to:
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Solution
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