JEE Main 2023 (Online) 10th April Morning Shift

The shortest distance between the lines \({{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2}\) and \({{x - 4} \over 1} = {{y - 1} \over 2} = {{z + 3} \over 0}\) is :

Let two vertices of a triangle ABC be (2, 4, 6) and (0, \(-\)2, \(-\)5), and its centroid be (2, 1, \(-\)1). If the image of the third vertex in the plane \(x+2y+4z=11\) is \((\alpha,\beta,\gamma)\), then $$\alpha\beta+\beta\gamma+\gamma\al...

Let P be the point of intersection of the line \({{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2}\) and the plane \(x+y+z=2\). If the distance of the point P from the plane \(3x - 4y + 12z = 32\) is q, then q and 2q are the roots o...

The slope of tangent at any point (x, y) on a curve \(y=y(x)\) is \({{{x^2} + {y^2}} \over {2xy}},x > 0\). If \(y(2) = 0\), then a value of \(y(8)\) is :

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm\(^2\)) is equal to :

Let \(y = p(x)\) be the parabola passing through the points \(( - 1,0),(0,1)\) and \((1,0)\). If the area of the region \(\{ (x,y):{(x + 1)^2} + {(y - 1)^2} \le 1,y \le p(x)\}\) is A, then \(12(\pi - 4A)\) is equal to ___________.

If the coefficient of \({x^7}\) in \({\left( {ax - {1 \over {b{x^2}}}} \right)^{13}}\) and the coefficient of \({x^{ - 5}}\) in \({\left( {ax + {1 \over {b{x^2}}}} \right)^{13}}\) are equal, then \({a^4}{b^4}\) is equal to :

The coefficient of \(x^7\) in \({(1 - x + 2{x^3})^{10}}\) is ___________.

A line segment AB of length \(\lambda\) moves such that the points A and B remain on the periphery of a circle of radius \(\lambda\). Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

Let the complex number \(z = x + iy\) be such that \({{2z - 3i} \over {2z + i}}\) is purely imaginary. If \({x} + {y^2} = 0\), then \({y^4} + {y^2} - y\) is equal to :

Let \(f\) be a differentiable function such that \({x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}\), \(f(1) = {2 \over 3}\). Then \(18f(3)\) is equal to :

Let the ellipse \(E:{x^2} + 9{y^2} = 9\) intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. I...

If \(f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0\), then the least value of \(f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right)\) is :

If \(I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx}\) and \(I(0) = 1\), then \(I\left( {{\pi \over 3}} \right)\) is equal to :

Let \(f:( - 2,2) \to R\) be defined by \(f(x) = \left\{ {\matrix{ {x[x],} & { - 2 < x < 0} \cr {(x - 1)[x],} & {0 \le x \le 2} \cr } } \right.\) where \([x]\) denotes the greatest integer function. If m and n respectively are th...

Let a, b, c be three distinct positive real numbers such that \({(2a)^{{{\log }_e}a}} = {(bc)^{{{\log }_e}b}}\) and \({b^{{{\log }_e}2}} = {a^{{{\log }_e}c}}\).Then, 6a + 5bc is equal to ___________.

The negation of the statement \((p \vee q) \wedge (q \vee ( \sim r))\) is :

If A is a 3 \(\times\) 3 matrix and \(|A| = 2\), then \(|3\,adj\,(|3A|{A^2})|\) is equal to :

For the system of linear equations
\(2x - y + 3z = 5\)
\(3x + 2y - z = 7\)
\(4x + 5y + \alpha z = \beta\),
which of the following is NOT correct?

Let a common tangent to the curves \({y^2} = 4x\) and \({(x - 4)^2} + {y^2} = 16\) touch the curves at the points P and Q. Then \({(PQ)^2}\) is equal to __________.

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ___________.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons, who participated in the tournament, is ___________.

Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that \({2^N} < N!\) is \({m \over n}\), where m and n are coprime, then \(4m-3n\) is equal to :

Let the first term \(\alpha\) and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.

The number of elements in the set \(\{ n \in Z:|{n^2} - 10n + 19| < 6\}\) is _________.

If the mean of the frequency distribution

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\(96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}\) is equal to :

Let O be the origin and the position vector of the point P be \(- \widehat i - 2\widehat j + 3\widehat k\). If the position vectors of the points A, B and C are $$ - 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\wide...

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If \(\overrightarrow {OP} = \overrightarrow u ,\overrightarrow {OR} = \overrightarrow v\), and $$\overrightarrow {OQ} = \alpha \overrightarr...