JEE MAIN 2022 Previous Year Questions (PYQs) – Page 5 of 31
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Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is ${6 \over {11}}$, then n is equal to __________.
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Solution
The value of $\displaystyle \int_{0}^{2}!\left(,|2x^{2}-3x|+\big[x-\tfrac{1}{2}\big]\right),dx$, where $[\cdot]$ is the greatest integer function, is equal to:
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Solution
The number of values of $\alpha$ for which the system of equations :
x + y + z = $\alpha$
$\alpha$x + 2$\alpha$y + 3z = $-$1
x + 3$\alpha$y + 5z = 4
is inconsistent, is
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Solution
Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_{1}$ be twice the area $A_{2}$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point:1
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Solution
If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation 3x2 + $\lambda$x $-$ 1 = 0 is 15, then 6($\alpha$3 + $\beta$3)2 is equal to :
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Solution
The equations of the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the following is NOT true?
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Solution
The set of all values of k for which ${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$, is the interval :
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Solution
Let \( \tan^{-1}x=\theta \).
Then \( \cot^{-1}x=\frac{\pi}{2}-\theta \).
Since \( x\in R \), \( -\frac{\pi}{2}<\theta<\frac{\pi}{2} \).
Given,
\( (\tan^{-1}x)^3+(\cot^{-1}x)^3=k\pi^3 \)
\( \theta^3+\left(\frac{\pi}{2}-\theta\right)^3=k\pi^3 \)
Let \( t=\frac{\theta}{\pi} \).
So, $\dfrac{-1}{2}<t<\dfrac{1}{2}$
Now,
\( k=t^3+\left(\frac{1}{2}-t\right)^3 \)
\( k=t^3+\frac{1}{8}-\frac{3t}{4}+\frac{3t^2}{2}-t^3 \)
\( k=\frac{3t^2}{2}-\frac{3t}{4}+\frac{1}{8} \)
This quadratic is minimum at \( t=\frac{1}{4} \).
\( k_{\min}=\frac{1}{32} \)
As \( t\to-\frac{1}{2} \), \( k\to\frac{7}{8} \), but \( t=-\frac{1}{2} \) is not included.
Therefore,
\( k\in\left[\frac{1}{32},\frac{7}{8}\right) \)
Answer: \( \boxed{(a)\ \left[\frac{1}{32},\frac{7}{8}\right)} \)
Then \( \cot^{-1}x=\frac{\pi}{2}-\theta \).
Since \( x\in R \), \( -\frac{\pi}{2}<\theta<\frac{\pi}{2} \).
Given,
\( (\tan^{-1}x)^3+(\cot^{-1}x)^3=k\pi^3 \)
\( \theta^3+\left(\frac{\pi}{2}-\theta\right)^3=k\pi^3 \)
Let \( t=\frac{\theta}{\pi} \).
So, $\dfrac{-1}{2}<t<\dfrac{1}{2}$
Now,
\( k=t^3+\left(\frac{1}{2}-t\right)^3 \)
\( k=t^3+\frac{1}{8}-\frac{3t}{4}+\frac{3t^2}{2}-t^3 \)
\( k=\frac{3t^2}{2}-\frac{3t}{4}+\frac{1}{8} \)
This quadratic is minimum at \( t=\frac{1}{4} \).
\( k_{\min}=\frac{1}{32} \)
As \( t\to-\frac{1}{2} \), \( k\to\frac{7}{8} \), but \( t=-\frac{1}{2} \) is not included.
Therefore,
\( k\in\left[\frac{1}{32},\frac{7}{8}\right) \)
Answer: \( \boxed{(a)\ \left[\frac{1}{32},\frac{7}{8}\right)} \)
If the length of the perpendicular drawn from the point $P(a,4,2),;a>0$ on the line $\dfrac{x+1}{2}=\dfrac{y-3}{3}=\dfrac{z-1}{-1}$ is $2\sqrt{6}$ units and $Q(\alpha_{1},\alpha_{2},\alpha_{3})$ is the image of the point $P$ in this line, then $a+\sum_{i=1}^{3}\alpha_{i}$ is equal to:
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Solution
For the function $f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$, which one of the following is NOT correct?
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Solution
A six faced die is biased such that
$3 \times P(\text{a prime number}) = 6 \times P(\text{a composite number}) = 2 \times P(1).$
Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is:
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Solution
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