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Let \( f(x)=2x+\tan^{-1}x \) and \( g(x)=\log_{e}\!\left(\sqrt{1+x^{2}}+x\right) \), \( x\in[0,3] \).
Then:

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:

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

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

If \( y=y(x) \) is the solution curve of the differential equation \( \dfrac{dy}{dx}+y\tan x = x\sec x \), \( 0 \le x \le \dfrac{\pi}{3} \), \( y(0)=1 \)
Then \( y\!\left(\dfrac{\pi}{6}\right) \) is equal to:

$\displaystyle \lim_{n\to\infty} \left[ \frac{1}{1+n} + \frac{1}{2+n} + \frac{1}{3+n} + \ldots + \frac{1}{2n} \right] \text{ is equal to :}$

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

The negation of the expression \( q \vee \big((\sim q) \wedge p\big) \) is equivalent to:

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

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

The area enclosed by the closed curve \( C \) given by the differential equation \( \dfrac{dy}{dx}+\dfrac{x+a}{y-a}=0,\ y(1)=0 \) is \( 4\pi \).
Let \( P \) and \( Q \) be the points of interaction of the curve \( C \) and the y-axis. If normal at \( P \) and \( Q \) on the curve \( C \) intersect x-axis at points \( R \) and \( S \) respectively, then the length of the line segment \( RS \) is :

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

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 \( \displaystyle \sum_{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right) \) is equal to:

For a triangle \( \triangle ABC \), 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?

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:

The sum to \( 10 \) terms of the series \( \dfrac{1}{1+1^{2}+1^{4}}+\dfrac{2}{1+2^{2}+2^{4}}+\dfrac{3}{1+3^{2}+3^{4}}+\ldots \) 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:

If \( \alpha \) and \( \beta \) respectively are the maximum and the minimum value of \( f \), then:

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

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

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 ________.

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

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 \( [\vec{u}\, \vec{v}\, \vec{w}] \) is \( -\alpha \sqrt{3401} \), and \( \left| \vec{u}\, \hat{i} \right| = \frac{m}{n} \) where \( m \) and \( n \) are coprime natural numbers, then \( m + n \) is equal to ________.

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

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

If \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x) + f(x) = \int_{0}^{2} f(t) \, dt \). If \( f(0) = e^{-2} \), then \( 2f(0) - f(2) \) is equal to __________.

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

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

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

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

JEE Main 2023 1st Feb Morning Shift

Physics Chemistry Mathematics

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