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Let the plane \( P \) pass through the intersection of the planes \( 2x+3y-z=2 \) and \( x+2y+3z=6 \), and be perpendicular to the plane \( 2x+y-z+1=0 \). If \( d \) is the distance of \( P \) from the point \( (-7,1,1) \), then \( d^{2} \) is equal to:

Which of the following statements is a tautology?

$\text{The value of the integral } \int_{-\pi/4}^{\pi/4} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} \, dx \text{ is:}$

Two dice are thrown independently. Let \( A \) be the event that the number appeared on the \( 1^{\text{st}} \) die is less than the number appeared on the \( 2^{\text{nd}} \) die, \( B \) be the event that the number appeared on the \( 1^{\text{st}} \) die is even and that on the second die is odd, and \( C \) be the event that the number appeared on the \( 1^{\text{st}} \) die is odd and that on the \( 2^{\text{nd}} \) is even. Then:

Let \( \vec a=2\hat i-7\hat j+5\hat k \), \( \vec b=\hat i+\hat k \) and \( \vec c=\hat i+2\hat j-3\hat k \) be three given vectors. If \( \vec r \) is a vector such that \( \vec r\times \vec a=\vec c\times \vec a \) and \( \vec r\cdot \vec b=0 \), then \( |\vec r| \) is equal to:

Let \( P(S) \) denote the power set of \( S=\{1,2,3,\ldots,10\} \). Define the relations \( R_{1} \) and \( R_{2} \) on \( P(S) \) as \( A R_{1} B \) if \( (A\cap B^{c})\cup(B\cap A^{c})=\varnothing \) and \( A R_{2} B \) if \( A\cup B^{c}=B\cup A^{c} \), \( \forall A,B\in P(S) \). Then:

Let \( \alpha x=\exp \left(x^{\beta} y^{\gamma}\right) \) be the solution of the differential equation \( 2x^{2}y\,dy-(1-xy^{2})\,dx=0, \ x>0, \ y(2)=\sqrt{\log_{e}2}. \) Then \( \alpha+\beta-\gamma \) equals:

Let \( P(x_{0},y_{0}) \) be the point on the hyperbola \( 3x^{2}-4y^{2}=36 \), which is nearest to the line \( 3x+2y=1 \). Then \( \sqrt{2}\,(y_{0}-x_{0}) \) is equal to:

Let $S = \left\{ x \in R : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \right\}$. If $n(S)$ denotes the number of elements in $S$ then:

For the system of linear equations \( \alpha x+y+z=1, \ x+\alpha y+z=1, \ x+y+\alpha z=\beta \), which one of the following statements is NOT correct?

The number of integral values of \( k \), for which one root of the equation \( 2x^{2}-8x+k=0 \) lies in the interval \( (1,2) \) and its other root lies in the interval \( (2,3) \), is:

If \( y(x) = x^x, \, x > 0 \), then \( y''(2) - 2y'(2) \) is equal to:

Let \( 9 = x_1 < x_2 < \ldots < x_7 \) be in an A.P. with common difference \( d \). If the standard deviation of \( x_1, x_2, \ldots, x_7 \) is \( 4 \) and the mean is \( \bar{x} \), then \( \bar{x} + x_6 \) is equal to:

Let \( f : ℝ-\{0,1\} \to \) ℝ be a function such that \( f(x) + f\!\left(\dfrac{1}{1-x}\right) = 1 + x \). Then \( f(2) \) is equal to:

The area of the region given by \( \{(x, y) : xy \le 8, \, 1 \le y \le x^2\} \) is:

Let \( \vec{a} = 5\hat{i} - \hat{j} - 3\hat{k} \) and \( \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k} \) be two vectors. Then which one of the following statements is TRUE?

The sum \( \displaystyle \sum_{n=1}^{\infty} \frac{2n^2 + 3n + 4}{(2n)!} \) is equal to:

The sum of the absolute maximum and minimum values of the function \( f(x) = \left| x^2 - 5x + 6 \right| - 3x + 2 \) in the interval \( [-1, 3] \) is equal to:

Let \( a, b \) be two real numbers such that \( ab < 0 \). If the complex number \( \frac{1 + ai}{\, b + i \,} \) is of unit modulus and \( a + ib \) lies on the circle \( |z - 1| = |2z| \), then a possible value of \( \frac{1 + [a]}{4b} \), where \( [t] \) is greatest integer function, is:

Number of integral solutions to the equation \( x + y + z = 21 \), where \( x \ge 1 \), \( y \ge 3 \), \( z \ge 4 \), is equal to \( \_\_\_\_ \).

If the \( x \)-intercept of a focal chord of the parabola \( y^2 = 8x + 4y + 4 \) is \( 3 \), then the length of this chord is equal to \( \_\_\_\_ \).

The line \( x = 8 \) is the directrix of the ellipse \( E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with the corresponding focus \( (2, 0) \). If the tangent to \( E \) at the point \( P \) in the first quadrant passes through the point \( (0, 4\sqrt{3}) \) and intersects the \( x \)-axis at \( Q \), then \( (3PQ)^{2} \) is equal to ______.

Let the sixth term in the binomial expansion of \( \left( \sqrt{2^{\log_{2}(10-3^{x})}} + \sqrt[5]{2^{(x-2)\log_{2} 3}} \right)^{m} \), in the increasing powers of \( 2^{(x-2)\log_{2} 3} \), be \( 21 \). If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of \( x \) is _______.

Let \( \alpha x + \beta y + yz = 1 \) be the equation of the plane passing through the point \( (3, -2, 5) \) and perpendicular to the line joining the points \( (1, 2, 3) \) and \( (-2, 3, 5) \). Then the value of \( \alpha \beta \gamma \) is equal to ______.

If the term without \( x \) in the expansion of \( \left(x^{2/3} + \frac{\alpha}{x^{3}}\right)^{22} \) is \( 7315 \), then \( |\alpha| \) is equal to ______.

The point of intersection \( C \) of the plane \( 8x + y + 2z = 0 \) and the line joining the points \( A(-3, -6, 1) \) and \( B(2, 4, -3) \) divides the line segment \( AB \) internally in the ratio \( k : 1 \). If \( a, b, c \) ( \( |a|, |b|, |c| \) are coprime) are the direction ratios of the perpendicular from the point \( C \) on the line \( \frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3} \), then \( |a + b + c| \) is equal to ______.

The total number of six digit numbers, formed using the digits \( 4, 5, 9 \) only and divisible by \( 6 \), is ______.

$\text{If } \int_{0}^{\pi} \frac{5^{\cos x} (1 + \cos x \cos 3x + \cos^2 x + \cos^3 x \cos 3x) dx}{1 + 5^{\cos x}} = \frac{k\pi}{16}, \text{ then } k \text{ is equal to } \_\_\_\_\_\_\_\_.$

The sum of the common terms of the following three arithmetic progressions.
\( 3, 7, 11, 15, \ldots, 399, \)
\( 2, 5, 8, 11, \ldots, 359 \) and
\( 2, 7, 12, 17, \ldots, 197, \)
is equal to ______.

JEE Main 2023 1st Feb Evening Shift

Physics Chemistry Mathematics

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Let \( \alpha x=\exp \left(x^{\beta} y^{\gamma}\right) \) be the solution of the differential equation \( 2x^{2}y\,dy-(1-xy^{2})\,dx=0, \ x>0, \ y(2)=\sqrt{\log_{e}2}. \) Then \( \alpha+\beta-\gamma \) equals: