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If \(\tan 15^\circ + \dfrac{1}{\tan 75^\circ} + \dfrac{1}{\tan 105^\circ} + \tan 195^\circ = 2a\), Then the value of \(\left(a + \dfrac{1}{a}\right)\) is:

If an unbiased die, marked with \(-2, -1, 0, 1, 2, 3\) on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

If the solution of the equation \(\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1,\; x \in \left(0,\dfrac{\pi}{2}\right),\) is \(\sin^{-1}\!\left(\dfrac{\alpha + \sqrt{\beta}}{2}\right),\) where \(\alpha, \beta\) are integers, then \(\alpha + \beta\) is equal to:

Suppose \(f : \mathbb{R} \rightarrow (0, \infty)\) be a differentiable function such that \(5 f(x+y) = f(x)\cdot f(y), \; \forall \, x, y \in \mathbb{R}\). If \(f(3) = 320\), then \(\displaystyle \sum_{n=0}^{5} f(n)\) is equal to:

If \([t]\) denotes the greatest integer \(\le t\), then the value of \(\dfrac{3(e-1)}{e}\displaystyle \int_{1}^{2} x^{2} e^{[x]+[x^{3}]} dx\) is:

The number of points on the curve \( y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x \) at which the normal lines are parallel to \( x + 90y + 2 = 0 \) is:

, \( d = |A| \neq 0 \) and \( |A - d(\operatorname{Adj} A)| = 0 \). Then:

Let the system of linear equations \( x+y+kz=2 \), \( 2x+3y-z=1 \), \( 3x+4y+2z=k \) have infinitely many solutions. Then the system
\( (k+1)x+(2k-1)y=7 \)
\( (2k+1)x+(k+5)y=10 \)
has:

Among the statements:
\( (S1) \; ((p \lor q) \Rightarrow r) \Leftrightarrow (p \Rightarrow r) \)
\( (S2) \; ((p \lor q) \Rightarrow r) \Leftrightarrow ((p \Rightarrow r) \lor (q \Rightarrow r)) \)

Let \( y=x+2 \), \( 4y=3x+6 \) and\( 3y=4x+1 \) be three tangent lines to the circle \( (x-h)^2+(y-k)^2=r^2 \). Then \( h+k \) is equal to:

Let the solution curve \( y=y(x) \) of the differential equation\( \dfrac{dy}{dx}+\left(\dfrac{-3x^5\tan^{-1}(x^3)}{(1+x^6)^{3/2}}\right)y=2x\,exp{\left\{\dfrac{x^3-\tan^{-1}(x^3)}{\sqrt{1+x^6}}\right\}}\) pass through origin. Then \( y(1) \) is equal to:

Let a unit vector $\widehat{OP}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $OX, OY, OZ$ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. If $\widehat{OP}$ is perpendicular to the plane through points $(1, 2, 3)$, $(2, 3, 4)$ and $(1, 5, 7)$, then which one of the following is true?

If \( a_n = \frac{-2}{4n^2 - 16n + 15} \), then \( a_1 + a_2 + \ldots + a_{25} \) is equal to:

If \( P(h, k) \) be a point on the parabola \( x = 4y^{2} \), which is nearest to the point \( Q(0, 33) \), then the distance of \( P \) from the directrix of the parabola \( y^{2} = 4(x + y) \) is equal to:

The line \( l_{1} \) passes through the point \( (2, 6, 2) \) and is perpendicular to the plane \( 2x + y - 2z = 10 \). Then the shortest distance between the line \( l_{1} \) and the line \( \frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2} \) is:

A straight line cuts off the intercepts \( OA = a \) and \( OB = b \) on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin \( O \) to this makes an angle of \( \frac{\pi}{6} \) with positive direction of y-axis and the area of \( \triangle OAB \) is \( \frac{98}{3}\sqrt{3} \), then \( a^{2}-b^{2} \) is equal to:

If the coefficient of \( x^{15} \) in the expansion of \( \left(ax^{3}+\frac{1}{bx^{1/3}}\right)^{15} \) is equal to the coefficient of \( x^{-15} \) in the expansion of \( \left(ax^{1/3}-\frac{1}{bx^{3}}\right)^{15} \), where \( a \) and \( b \) are positive real numbers, then for each such ordered pair \( (a,b) \),

The coefficient of \( x^{301} \) in \( (1+x)^{500} + x(1+x)^{499} + x^{2}(1+x)^{498} + \ldots + x^{500} \) is:

The minimum number of elements that must be added to the relation \( R=\{(a,b),(b,c)\} \) on the set \( \{a,b,c\} \) so that it becomes symmetric and transitive is:

If \( \vec{a}, \vec{b}, \vec{c} \) are three non-zero vectors and \( \hat{n} \) is a unit vector perpendicular to \( \vec{c} \) such that \( \vec{a}=\alpha \vec{b}-\hat{n} \), \( (\alpha \neq 0) \) and \( \vec{b}\cdot \vec{c}=12 \), then \( \left\lvert \vec{c}\times(\vec{a}\times\vec{b}) \right\rvert \) is equal to:

The mean and variance of \( 7 \) observations are \( 8 \) and \( 16 \) respectively. If one observation \( 14 \) is omitted and \( a \) and \( b \) are respectively mean and variance of remaining \( 6 \) observation, then \( a+3b-5 \) is equal to \( \_\_\_\_\_ \).

If \( \lambda_{1}<\lambda_{2} \) are two values of \( \lambda \) such that the angle between the planes \( P_{1}:\ \vec{r}\cdot(3\hat{i}-5\hat{j}+\hat{k})=7 \) and \( P_{2}:\ \vec{r}\cdot(\lambda \hat{i}+\hat{j}-3\hat{k})=9 \) is \( \sin^{-1}\!\left(\dfrac{2\sqrt{6}}{5}\right) \), then the square of the length of perpendicular from the point \( (38\lambda_{1},\,10\lambda_{2},\,2) \) to the plane \( P_{1} \) is \( \_\_\_\_\_ \).

\( \displaystyle \lim_{x\to 0} \dfrac{48}{x^{4}}\int_{0}^{x}\dfrac{t^{3}}{t^{6}+1}\,dt \) is equal to \( \_\_\_\_\_ \).

Let \( z = 1 + i \) and \( z_{1} = \dfrac{1 + i\overline{z}}{\overline{z}(1 - z) + \dfrac{1}{\overline{z}}} \). Then \( \dfrac{12}{\pi} \arg(z_{1}) \) is equal to _______.

Let \( S = \{1, 2, 3, 4, 5, 6\} \). Then the number of one-one functions \( f : S \to P(S) \), where \( P(S) \) denotes the power set of \( S \), such that \( f(n) \subset f(m) \) where \( n < m \) is ________.

Number of \( 4 \)-digit numbers (the repeation of digits is allowed) which are made using the digits \( 1,2,3 \) and \( 5 \), and are divisible by \( 15 \), is equal to ________.

Let \( \displaystyle \sum_{n=0}^{\infty} \dfrac{n^{3}\,(2n)! + (2n-1)(n!)}{(n!)((2n)!)} = a e + \dfrac{b}{e} + c \), where \( a, b, c \in \mathbb{Z} \) and \( e = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{n!} \). Then \( a^{2} - b + c \) is equal to ________.

Let \( \alpha \) be the area of the larger region bounded by the curve \( y^{2} = 8x \) and the \( y = x \) and \( x = 2 \), which lies in the first quadrant. Then the value of \( 3\alpha \) is equal to ________.

If the equation of the plane passing through the point \( (1, 1, 2) \) and perpendicular to the line \( x - 3y + 2z - 1 = 0 = 4x - y + z \) is \( Ax + By + Cz = 1 \), then \( 140(C - B + A) \) is equal to _______.

Let \( f^1(x) = \frac{3x + 2}{2x + 3}, \; x \in \mathbb{R} - \left\{ \frac{-3}{2} \right\} \). For \( n \ge 2 \), define \( f^n(x) = f^1 \circ f^{n-1}(x) \). If \( f^5(x) = \frac{ax + b}{bx + a} \), \( \gcd(a,b) = 1 \), then \( a + b \) is equal to _______.

JEE Main 2023 30th Jan Morning Shift

Physics Chemistry Mathematics

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If the equation of the plane passing through the point \( (1, 1, 2) \) and perpendicular to the line \( x - 3y + 2z - 1 = 0 = 4x - y + z \) is \( Ax + By + Cz = 1 \), then \( 140(C - B + A) \) is equal to _______.