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Let \( \alpha \) be a root of the equation \( (a-c)x^2 + (b-a)x + (c-b) = 0 \) where \( a,b,c \) are distinct real numbers such that the matrix is singular. Then, the value of \( \dfrac{(a-c)^2}{(b-a)(c-b)} + \dfrac{(b-a)^2}{(a-c)(c-b)} + \dfrac{(c-b)^2}{(a-c)(b-a)} \) is:

Let \( \vec{u} = \hat{i} - \hat{j} - 2\hat{k} \), \( \vec{v} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{v} \cdot \vec{w} = 2 \) and \( \vec{v} \times \vec{w} = \vec{u} + \lambda \vec{v} \). Then \( \vec{u} \cdot \vec{w} \) is equal to:

$\lim_{t \to 0} \left( 1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + \dots + n^{\frac{1}{\sin^2 t}} \right)^{\sin^2 t} \text{ is equal to:}$

The relation \( R=\{(a,b):\gcd(a,b)=1,\, 2a\neq b,\, a,b\in \mathbb{Z}\} \) is:

Let \( y=y(x) \) be the solution of the differential equation \( x^{3}dy+(xy-1)dx=0,\, x>0,\, y\!\left(\dfrac{1}{2}\right)=3-e \). Then \( y(1) \) is equal to:

The equation \( x^{2}-4x+[x]+3=x[x] \), where \( [x] \) denotes the greatest integer function, has:

Let \( p, q \in \) ℝ and \( (1-\sqrt{3}i)^{200} = 2^{199}(p+iq), \; i=\sqrt{-1} \). Then \( p+q+q^{2} \) and \( p-q+q^{2} \) are roots of the equation.

Let \( PQR \) be a triangle. The points \( A, B \) and \( C \) are on the sides \( QR, RP \) and \( PQ \) respectively such that \( \frac{QA}{AR}=\frac{RB}{BP}=\frac{PC}{CQ}=\frac{1}{2} \). Then \( \frac{\text{Area}(\triangle PQR)}{\text{Area}(\triangle ABC)} \) is equal to:

Then at \( x=0 \)

The distance of the point \( (-1,9,-16) \) from the plane \( 2x+3y-z=5 \) measured parallel to the line \( \frac{x+4}{3}=\frac{2-y}{4}=\frac{z-3}{12} \) is:

The value of $\displaystyle \sum_{r=0}^{22} {}^{22}C_r \ {}^{23}C_r$ is:

If \( A \) and \( B \) are two non-zero \( n\times n \) matrices such that \( A^{2}+B=A^{2}B \), then:

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event.
Given below are two statements:
Statement I :If $\text{P}(A) = 0$, then $A = \phi$
Statement II : If $\text{P}(A) = 1$, then $A = \Omega$
Then

Let \( N \) denote the number that turns up when a fair die is rolled. If the probability that the system of equations
\( x+y+z=1 \)
\( 2x+Ny+2z=2 \)
\( 3x+3y+Nz=3 \)
has unique solution is \( \dfrac{k}{6} \), then the sum of value of \( k \) and all possible values of \( N \) is:

For three positive integers \( p,q,r \), \( x^{pq^{2}} = y^{qr} = z^{p^{2}r} \) and \( r=pq+1 \) such that \( 3, \; 3\log_{y}x, \; 3\log_{z}y, \; 7\log_{x}z \) are in A.P. with common difference \( \dfrac{1}{2} \). Then \( r-p-q \) is equal to:

The distance of the point \( (7,-3,-4) \) from the plane passing through the points \( (2,-3,1) \), \( (-1,1,-2) \) and \( (3,-4,2) \) is:

Let a tangent to the curve \( y^{2}=24x \) meet the curve \( xy=2 \) at the points \( A \) and \( B \). Then the mid-points of such line segment \( AB \) lie on a parabola with the:

The area enclosed by the curves \( y^{2}+4x=4 \) and \( y-2x=2 \) is:

The compound statement \( (\sim(P \land Q)) \lor ((\sim P) \land Q) \Rightarrow ((\sim P) \land (\sim Q)) \) is equivalent to:

\( \tan^{-1}\!\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\!\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \) is equal to:

Suppose \( \displaystyle \sum_{r=0}^{2023} r^{2} \, {^{2023}\!C_{r}} = 2023 \times \alpha \times 2^{2022} \). Then the value of \( \alpha \) is _______.

A boy needs to select five courses from \( 12 \) available courses, out of which \( 5 \) courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is _______.

The value of \( 12\!\int_{0}^{3} \left| x^{2}-3x+2 \right| dx \) is _______.

Let a tangent to the curve \( 9x^{2} + 16y^{2} = 144 \) intersect the coordinate axes at the points \( A \) and \( B \). Then, the minimum length of the line segment \( AB \) is _______.

The \( 4^{\text{th}} \) term of GP is \( 500 \) and its common ratio is \( \frac{1}{m} \), \( m \in \mathbb{N} \). Let \( S_n \) denote the sum of the first \( n \) terms of this GP. If \( S_6 > S_5 + 1 \) and \( S_7 < S_6 + \frac{1}{2} \), then the number of possible values of \( m \) is ______.

The number of \( 9 \)-digit numbers, that can be formed using all the digits of the number \( 123412341 \) so that the even digits occupy only even places, is ______.

Let \( \lambda \in \mathbb{R} \) and let the equation \( E \) be \( |x|^{2} - 2|x| + |\lambda - 3| = 0 \). Then the largest element in the set \( S = \{x + \lambda : x \text{ is an integer solution of } E\} \) is ______.

The value of \( 8\int_{0}^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} \, dx \) is ______.

The shortest distance between the lines \( \frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2} \) and \( \frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0} \) is equal to ________.

Let \( C \) be the largest circle centred at \( (2, 0) \) and inscribed in the ellipse \( \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 \). If \( (1, \alpha) \) lies on \( C \), then \( 10\alpha^{2} \) is equal to ______.

JEE Main 2023 24th Jan Morning Shift

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Let a tangent to the curve \( 9x^{2} + 16y^{2} = 144 \) intersect the coordinate axes at the points \( A \) and \( B \). Then, the minimum length of the line segment \( AB \) is _______.