JEE Main 2023 (Online) 1st February Morning Shift

The shortest distance between the lines \({{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}\) and \({{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}\) is :

Let the image of the point \(P(2,-1,3)\) in the plane \(x+2 y-z=0\) be \(Q\). Then the distance of the plane \(3 x+2 y+z+29=0\) from the point \(Q\) is :

Let \(A\) be the area bounded by the curve \(y=x|x-3|\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=2\). Then \(12 A\) is equal to ____________.

The remainder, when \(19^{200}+23^{200}\) is divided by 49 , is ___________.

If the center and radius of the circle \(\left| {{{z - 2} \over {z - 3}}} \right| = 2\) are respectively \((\alpha,\beta)\) and \(\gamma\), then \(3(\alpha+\beta+\gamma)\) is equal to :

\(\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]\) is equal to

If \(\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}\) where \(l, m, n \in \mathbb{N}, m\) and \(n\) are coprime then \(l+m+n\) is equal to ____________.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t\). If \(f(0)=e^{-2}\), then \(2 f(0)-f(2)\) is equal to ____________.

The area enclosed by the closed curve \(\mathrm{C}\) given by the differential equation \(\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0\) is \(4 \pi\).
Let \(P\) and \(Q\) be the points of intersection of the curve \(\mathrm{C}\) and the \(y\)-...

If \(y=y(x)\) is the solution curve of the differential equation \(\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1\), then \(y\left(\frac{\pi}{6}\right)\) is equal to

Let \(f(x) = 2x + {\tan ^{ - 1}}x\) and \(g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]\). Then

If \(f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)\) and \(g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)\), then the value of \(f(4)-g(4)\) is equal to ____________.

Let $$f(x) = \left| {\matrix{
{1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr
{{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr

} } \right|,\,x \in \left[ {{\pi \ov...

Let \(S\) be the set of all solutions of the equation \(\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]\). Then \(\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)\) is equal to ...

The negation of the expression \(q \vee \left( {( \sim \,q) \wedge p} \right)\) is equivalent to

Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\(\lambda x+y+z=1\)
\(x+\lambda y+z=1\)
\(x+y+\lambda z=1\)
is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)...

The value of \(\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}\) is :

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ____________.

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ___________.

In a binomial distribution \(B(n,p)\), the sum and the product of the mean and the variance are 5 and 6 respectively, then \(6(n+p-q)\) is equal to :

For a triangle \(ABC\), the value of \(\cos 2A + \cos 2B + \cos 2C\) is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

Let \(S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}\). Then \(n(S)\) is equal to

The sum of 10 terms of the series
\({1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,....\) is

Let \(a_{1}=8, a_{2}, a_{3}, \ldots, a_{n}\) be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is ___________.

Let \(R\) be a relation on \(\mathbb{R}\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number \(\}\). Then \(R\) is

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

The combined equation of the two lines \(ax+by+c=0\) and \(a'x+b'y+c'=0\) can be written as
\((ax+by+c)(a'x+b'y+c')=0\).
The equation of the angle bisectors of the lines represented by the equation \(2x^2+xy-3y^2=0\) is :

If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is \((\alpha,\beta)\), then the quadratic equation whose roots are \(\alpha+4\beta\) and \(4\alpha+\beta\), is :

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\) and \(\vec{u}\) be a vector such that \(|\vec{u}|=\alpha>0\). If the minimum value of the scalar triple product $$\left[ {\matrix{
{\overrightar...

\(A(2,6,2), B(-4,0, \lambda), C(2,3,-1)\) and \(D(4,5,0),|\lambda| \leq 5\) are the vertices of a quadrilateral \(A B C D\). If its area is 18 square units, then \(5-6 \lambda\) is equal to __________.