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When vector \( \vec{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} \) is subtracted from vector \( \vec{B} \), it gives a vector equal to \( 2\hat{j} \). Then the magnitude of vector \( \vec{B} \) will be :

The sum of the coefficients of three consecutive terms in the binomial expansion of \( (1 + x)^{n+2} \), which are in the ratio \( 1:3:5 \), is equal to:

Let the mean of \( 6 \) observations \( 1, 2, 4, 5, x \) and \( y \) be \( 5 \) and their variance be \( 10 \). Then their mean deviation about the mean is equal to:

For \( a \in \mathbb{C} \), let \( A=\{ z \in \mathbb{C} : \operatorname{Re}(a+\bar z)>\operatorname{Im}(\bar{a}+z) \} \) and \( B=\{ z \in \mathbb{C} : \operatorname{Re}(a+\bar z)<\operatorname{Im}(\bar{a}+z) \} \). Then among the two statements:
(\( \mathrm{S1} \)) : If \( \operatorname{Re}(a), \operatorname{Im}(a) > 0 \), then the set \( A \) contains all the real numbers
(\( \mathrm{S2} \)) : If \( \operatorname{Re}(a), \operatorname{Im}(a) < 0 \), then the set \( B \) contains all the real numbers

Let the function \( f:[0,2]\to \mathbb{R} \) be defined as where \( [t] \) denotes the greatest integer less then or equal to \( t \). Then the value of the integral \(\displaystyle \int_{0}^{2} x f(x)\,dx \) is:

If four distinct points with position vectors \( \vec{a}, \vec{b}, \vec{c} \) and \( \vec{d} \) are coplanar, then \( [\vec{a}\ \vec{b}\ \vec{c}] \) is equal to:

If the system of linear equations
\( 7x + 11y + \alpha z = 13 \)
\( 5x + 4y + 7z = \beta \)
\( 175x + 194y + 57z = 361 \)
has infinitely many solutions, then \( \alpha + \beta + 2 \) is equal to:

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is:

If \( f:\mathbb{R}\to\mathbb{R} \) be a continuous function satisfying \(\displaystyle \int_{0}^\dfrac{\pi}{2} f(\sin 2x)\sin x \, dx + \alpha \int_{0}^\dfrac{\pi}{4} f(\cos 2x)\cos x \, dx = 0 \), then the value of \( \alpha \) is:

The domain of the function \( f(x)=\dfrac{1}{\sqrt{[x]^2-3[x]-10}} \)is: (where \( [x] \) denotes the greatest integer less than or equal to \( x \))

The angle of elevation of the top \( P \) of a tower from the feet of one person standing due South of the tower is \( 45^\circ \) and from the feet of another person standing due west of the tower is \( 30^\circ \). If the height of the tower is \( 5 \, \text{meters} \), then the distance (in meters) between the two persons is equal to:

Let \( P \) be the plane passing through the points \( (5,3,0) \), \( (13,3,-2) \) and \( (1,6,2) \). For \( \alpha \in \mathbb{N} \); if the distances of the points \( A(3,4,\alpha) \) and \( B(2,\alpha,a) \) from the plane \( P \) are \( 2 \) and \( 3 \) respectively, then the positive value of \( a \) is:

Let \( A=\{1,3,4,6,9\} \) and \( B=\{2,4,5,8,10\} \). Let \( R \) be a relation defined on \( A\times B \) such that \( R=\{((a_1,b_1),(a_2,b_2)):\; a_1\le b_2 \;\&\; b_1\le a_2\} \). Then the number of elements in the set \( R \) is:

If the \( 1011^{\text{th}} \) term from the end in the binomial expansion of \( \left( \dfrac{4x}{5} - \dfrac{5}{2x} \right)^{2022} \) is \( 1024 \) times \( 1011^{\text{th}} \) term from the beginning, then \( |x| \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \(\displaystyle \frac{dy}{dx} + \frac{5}{x(x^5 + 1)}y = \frac{(x^5 + 1)^2}{x^7} , \; x>0 \). If \( y(1) = 2 \), then \( y(2) \) is equal to:

If the radius of the largest circle with centre \( (2,0) \) inscribed in the ellipse \( x^2 + 4y^2 = 36 \) is \( r \), then \( 12r^2 \) is equal to:

Let the line passing through the points \( P(2,-1,2) \) and \( Q(5,3,4) \) meet the plane \( x-y+z=4 \) at the point \( R \). Then the distance of the point \( R \) from the plane \( x+2y+3z+2=0 \) measured parallel to the line \(\displaystyle \frac{x-7}{2}=\frac{y+3}{2}=\frac{z-2}{1} \) is equal to:

Let \( a, b, c \) and \( d \) be positive real numbers such that \( a+b+c+d=11 \). If the maximum value of \( a^5 b^3 c^2 d \) is \( 3750\beta \), then the value of \( \beta \) is:

The converse of \( ((\sim p)\wedge q) \Rightarrow r \) is:

Let \( f \) and \( g \) be two functions defined by and . Then \( (g \circ f)(x) \) is:

, then \( \lambda, \dfrac{\lambda}{3} \) are the roots of the equation.

If the line \( \ell_{1}: 3y-2x=3 \) is the angular bisector of the lines \( \ell_{2}: x-y+1=0 \) and \( \ell_{3}: \alpha x+\beta y+17=0 \), then \( \alpha^{2}+\beta^{2}-\alpha-\beta \) is equal to ______.

Let \( \vec{a}=\hat{i}+2\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+\hat{j}-\hat{k} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c}=11 \), \( \vec{b}\cdot(\vec{a}\times\vec{c})=27 \) and \( \vec{b}\cdot\vec{c}=-\sqrt{3}\,|\vec{b}| \), then \( \left|\vec{a}\times\vec{c}\right|^{2} \) is equal to ______.

Let the tangent to the parabola \( y^{2}=12x \) at the point \( (3,\alpha) \) be perpendicular to the line \( 2x+2y=3 \). Then the square of distance of the point \( (6,-4) \) from the normal to the hyperbola \( \alpha^{2}x^{2}-9y^{2}=9\alpha^{2} \) at its point \( (\alpha-1,\alpha+2) \) is equal to ______.

Let \( S=\left\{ z\in\mathbb{C}-\{i,2i\}:\dfrac{z^{2}+8iz-15}{z^{2}-3iz-2}\in\mathbb{R}\right\}. \) If \( \alpha-\dfrac{13}{11}i\in S,\; \alpha\in\mathbb{R}-\{0\} \), then \( 242\alpha^{2} \) is equal to ______.

Let \( A = \{1,2,3,4,5\} \) and \( B = \{1,2,3,4,5,6\} \). Then the number of functions \( f : A \to B \) satisfying \( f(1) + f(2) = f(4) - 1 \) is equal to _______.

Let the probability of getting head for a biased coin be \( \dfrac{1}{4} \). It is tossed repeatedly until a head appears. Let \( N \) be the number of tosses required. If the probability that the equation \( 64x^2 + 5Nx + 1 = 0 \) has no real root is \( \dfrac{p}{q} \), where \( p \) and \( q \) are coprime, then \( q - p \) is equal to _______.

For \( k \in \mathbb{N} \), if the sum of the series \(\displaystyle 1 + \frac{4}{k} + \frac{8}{k^{2}} + \frac{13}{k^{3}} + \frac{19}{k^{4}} + \ldots \) is \( 10 \), then the value of \( k \) is _______.

If \( A \) is the area in the first quadrant enclosed by the curve \( C : 2x^{2} - y + 1 = 0 \), the tangent to \( C \) at the point \( (1,3) \) and the line \( x + y = 1 \), then the value of \( 60A \) is _______.

The number of points, where the curve \( f(x) = e^{8x} - 6^{6x} - 3e^{4x} - e^{2x} + 1, \ x \in \mathbb{R} \) cuts x-axis is equal to _______.

Let the line \( l : x = \dfrac{1-y}{-2} = \dfrac{z-3}{\lambda}, \ \lambda \in \mathbb{R} \) meet the plane \( P : x + 2y + 3z = 4 \) at the point \( (\alpha,\beta,\gamma) \). If the angle between the line \( l \) and the plane \( P \) is \( \cos^{-1}\!\left(\sqrt{\dfrac{5}{14}}\right) \), then \( \alpha + 2\beta + 6\gamma \) is equal to __________.

JEE Main 2023 11th April Evening Shift

Physics Chemistry Mathematics

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Let \( a, b, c \) and \( d \) be positive real numbers such that \( a+b+c+d=11 \). If the maximum value of \( a^5 b^3 c^2 d \) is \( 3750\beta \), then the value of \( \beta \) is: