JEE Main 2023 (Online) 31st January Morning Shift

Let the shortest distance between the lines
\(L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0\) and
\(L_{1}: x+1=y-1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\),
then which of the follo...

Let the line \(L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}\) intersect the plane \(2 x+y+3 z=16\) at the point
\(P\). Let the point \(Q\) be the foot of perpendicular from the point \(R(1,-1,-3)\) on the line \(L\). If \(\alpha\) is the ...

Let \(\theta\) be the angle between the planes \(P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9\) and \(P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15\). Let \(\mathrm{L}\) be the line that meets \(P_{2}\) at the point \((4,-2,5)\) a...

A wire of length \(20 \mathrm{~m}\) is to be cut into two pieces. A piece of length \(l_{1}\) is bent to make a square of area \(A_{1}\) and the other piece of length \(l_{2}\) is made into a circle of area \(A_{2}\). If \(2 A_{1}+3 A_{2}\)...

Let for \(x \in \mathbb{R}\),
$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. } $$
Then area bounded by the curve \(y=(f \circ g)(x)\) and the lines $$y=0,2 y-x=1...

The remainder on dividing \(5^{99}\) by 11 is ____________.

Let \(\alpha>0\), be the smallest number such that the expansion of \(\left(x^{\frac{2}{3}}+\frac{2}{x^{3}}\right)^{30}\) has a term \(\beta x^{-\alpha}, \beta \in \mathbb{N}\). Then \(\alpha\) is equal to ___________.

Let a circle \(C_{1}\) be obtained on rolling the circle \(x^{2}+y^{2}-4 x-6 y+11=0\) upwards 4 units on the tangent \(\mathrm{T}\) to it at the point \((3,2)\). Let \(C_{2}\) be the image of \(C_{1}\) in \(\mathrm{T}\). Let \(A\) and \(B\)...

For all \(z \in C\) on the curve \(C_{1}:|z|=4\), let the locus of the point \(z+\frac{1}{z}\) be the curve \(\mathrm{C}_{2}\). Then :

Let \(\alpha \in (0,1)\) and \(\beta = {\log _e}(1 - \alpha )\). Let \({P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)\). Then the integral $$\int\limits_0^\alpha {{{{t^{50}}} \over {1 -...

The value of \(\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x\) is equal to :

Let a differentiable function \(f\) satisfy \(f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3\). Then \(12 f(8)\) is equal to :

Let \(y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)\). Then, at x = 1,

If the maximum distance of normal to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b < 2\), from the origin is 1, then the eccentricity of the ellipse is :

If the domain of the function \(f(x)=\frac{[x]}{1+x^{2}}\), where \([x]\) is greatest integer \(\leq x\), is \([2,6)\), then its range is

If \({\sin ^{ - 1}}{\alpha \over {17}} + {\cos ^{ - 1}}{4 \over 5} - {\tan ^{ - 1}}{{77} \over {36}} = 0,0 < \alpha < 13\), then \({\sin ^{ - 1}}(\sin \alpha ) + {\cos ^{ - 1}}(\cos \alpha )\) is equal to :

\((\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))\) is a tautology
\((\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)\) is a contradiction.
Then

For the system of linear equations
\(x+y+z=6\)
\(\alpha x+\beta y+7 z=3\)
\(x+2 y+3 z=14\)
which of the following is NOT true ?

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 1} \cr 0 & {12} & { - 3} \cr } } \right)\). Then the sum of the diagonal elements of the matrix \({(A + I)^{11}}\) is equal to :

Let \(\mathrm{y}=f(x)\) represent a parabola with focus \(\left(-\frac{1}{2}, 0\right)\) and directrix \(y=-\frac{1}{2}\). Then
\(S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}\) :

Let 5 digit numbers be constructed using the digits \(0,2,3,4,7,9\) with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is __________.

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to ____________.

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is :

The number of real roots of the equation \(\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+6}\), is :

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be in A.P. If \(a_{5}=2 a_{7}\) and \(a_{11}=18\), then
\(12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)\) is equa...

Let \(\mathrm{R}\) be a relation on \(\mathrm{N} \times \mathbb{N}\) defined by \((a, b) ~\mathrm{R}~(c, d)\) if and only if \(a d(b-c)=b c(a-d)\). Then \(\mathrm{R}\) is

If the variance of the frequency distribution

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Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statements:
(A) ...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}\) and \(|\vec{a} \times \vec{b}|=\sqrt{48}\). Then \((\vec{a} \cdot \vec{b})^{2}\) is equal to ___________.