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Let \( O \) be the origin and the position vector of the point \( P \) be \( -\hat{i}-2\hat{j}+3\hat{k} \). If the position vectors of the points \( A \), \( B \) and \( C \) are \( -2\hat{i}+\hat{j}-3\hat{k} \), \( 2\hat{i}+4\hat{j}-2\hat{k} \) and \( -4\hat{i}+2\hat{j}-\hat{k} \) respectively, then the projection of the vector \( \overrightarrow{OP} \) on a vector perpendicular to the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) is:

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

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

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

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

Let the ellipse \( E:x^{2}+9y^{2}=9 \) 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 \( P \). If the area of the triangle with vertices \( A \), \( P \) and the origin \( O \) is \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m-n \) is equal to:

Let P be the point of intersection of the line \( \dfrac{x+3}{3}=\dfrac{y+2}{1}=\dfrac{1-z}{2} \) 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 of the equation.

Let the first term \( a \) 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:

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

A square piece of tin of side \( 30 \, \text{cm} \) 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 \( \text{cm}^2 \)) is equal to:

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

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

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?

An arc \( PQ \) of a circle subtends a right angle at its centre \( O \). The midpoint of the arc \( PQ \) is \( R \). If \(\overrightarrow{ OP} =\vec{u} \), \(\overrightarrow{ OR} = \vec{v} \) and \(\overrightarrow{ OQ} = \ {\alpha \vec{u}} + \beta \vec{v} \), then \( \alpha, \ \beta^2 \) are the roots of equation.

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

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 \( \alpha\beta + \beta\gamma + \gamma\alpha \) is equal to:

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

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

Let \( f \) be a differentiable function such that \(\displaystyle x^{2}f(x)-x = 4\!\int_{0}^{x} t f(t)\,dt \), \( f(1)=\dfrac{2}{3} \). Then \( 18f(3) \) is equal to:

Let \( f:(-2,2)\rightarrow \mathbb{R} \) be defined by

where \([x]\) denotes the greatest integer function. If \(m\) and \(n\) respectively are the number of points in \((-2,2)\) at which \(y=| f(x)|\) is not continuous and not differentiable then \(m+n\) is equal to ____.

The number of permutations, of the digits \( 1,2,3,\ldots,7 \) without repetition, which neither contain the string \( 153 \) nor the string \( 2467 \), is ____________.

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

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 ________.

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 ________.

The sum of all those terms, of the arithmetic progression \( 3,\,8,\,13,\,\ldots,\,373 \), which are not divisible by \( 3 \), is equal to ________.

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 ________.

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 numbers of persons, who participated in the tournament, is ________.

If the mean of the frequency distribution

is \( 28 \), then its variance is __________.

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

JEE Main 2023 10th April Morning Shift

Physics Chemistry Mathematics

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