JEE Main 2023 (Online) 29th January Morning Shift

Let the equation of the plane P containing the line \(x+10=\frac{8-y}{2}=z\) be \(ax+by+3z=2(a+b)\) and the distance of the plane \(P\) from the point (1, 27, 7) be \(c\). Then \(a^2+b^2+c^2\) is equal to __________.

Let the co-ordinates of one vertex of \(\Delta ABC\) be \(A(0,2,\alpha)\) and the other two vertices lie on the line \({{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}\). For \(\alpha \in \mathbb{Z}\), if the area of $$\Delta...

Let \(\Delta\) be the area of the region \(\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}\). Then \({1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)\) is equal to

Let \([x]\) denote the greatest integer \(\le x\). Consider the function \(f(x) = \max \left\{ {{x^2},1 + [x]} \right\}\). Then the value of the integral \(\int\limits_0^2 {f(x)dx}\) is

Let \(A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}\) and \(B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^{2}}\right\}\right\} \text {. }\).
Then t...

Let the coefficients of three consecutive terms in the binomial expansion of \((1+2x)^n\) be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is __________.

If the co-efficient of \(x^9\) in \({\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}\) and the co-efficient of \(x^{-9}\) in \({\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}}\) are equal, then \((\alpha\beta)^2\) is equ...

Let the tangents at the points \(A(4,-11)\) and \(B(8,-5)\) on the circle \(x^{2}+y^{2}-3 x+10 y-15=0\), intersect at the point \(C\). Then the radius of the circle, whose centre is \(C\) and the line joining \(A\) and \(B\) is its tangent,...

For two non-zero complex numbers \(z_{1}\) and \(z_{2}\), if \(\operatorname{Re}\left(z_{1} z_{2}\right)=0\) and \(\operatorname{Re}\left(z_{1}+z_{2}\right)=0\), then which of the following are possible?
A. $$\operatorname{Im}\left(z_{1}\ri...

Let \(f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R\) be a function which satisfies \(f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy}\). then \((a+b)\) is equal to

Let \(y=f(x)\) be the solution of the differential equation \(y(x+1)dx-x^2dy=0,y(1)=e\). Then \(\mathop {\lim }\limits_{x \to {0^ + }} f(x)\) is equal to

Let \(f:\mathbb{R}\to\mathbb{R}\) be a differentiable function that satisfies the relation \(f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}\). If \(f'(0)=2\), then \(|f(-2)|\) is equal to ___________.

The domain of \(f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}\) is

Let \(f:R \to R\) be a function such that \(f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}\). Then

Suppose \(f\) is a function satisfying \(f(x + y) = f(x) + f(y)\) for all \(x,y \in N\) and \(f(1) = {1 \over 5}\). If \(\sum\limits_{n = 1}^m {{{f(n)} \over {n(n + 1)(n + 2)}} = {1 \over {12}}}\), then \(m\) is equal to __________.

Let \(x=2\) be a root of the equation \(x^2+px+q=0\) and \(f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.\)
Then $$\mathop {\lim }...

If \(p,q\) and \(r\) are three propositions, then which of the following combination of truth values of \(p,q\) and \(r\) makes the logical expression $$\left\{ {(p \vee q) \wedge \left( {( \sim p) \vee r} \right)} \right\} \to \left( {( \s...

Let \(\alpha\) and \(\beta\) be real numbers. Consider a 3 \(\times\) 3 matrix A such that \(A^2=3A+\alpha I\). If \(A^4=21A+\beta I\), then

Consider the following system of equations
\(\alpha x+2y+z=1\)
\(2\alpha x+3y+z=1\)
\(3x+\alpha y+2z=\beta\)
for some \(\alpha,\beta\in \mathbb{R}\). Then which of the following is NOT correct.

If all the six digit numbers \(x_1\,x_2\,x_3\,x_4\,x_5\,x_6\) with \(0< x_1 < x_2 < x_3 < x_4 < x_5 < x_6\) are arranged in the increasing order, then the sum of the digits in the \(\mathrm{72^{th}}\) number is _____________.

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____________.

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

Let \(\lambda \ne 0\) be a real number. Let \(\alpha,\beta\) be the roots of the equation \(14{x^2} - 31x + 3\lambda = 0\) and \(\alpha,\gamma\) be the roots of the equation \(35{x^2} - 53x + 4\lambda = 0\). Then $${{3\alpha } \over \bet...

Let \(a_1,a_2,a_3,...\) be a \(GP\) of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then \(a_1a_9+a_2a_4a_9+a_5+a_7\) is equal to __________.

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If \(\mu\) and \(\sigma^2\) represent mean and variance ...

Let \(B\) and \(C\) be the two points on the line \(y+x=0\) such that \(B\) and \(C\) are symmetric with respect to the origin. Suppose \(A\) is a point on \(y-2 x=2\) such that \(\triangle A B C\) is an equilateral triangle. Then, the area...

A light ray emits from the origin making an angle 30\(^\circ\) with the positive \(x\)-axis. After getting reflected by the line \(x+y=1\), if this ray intersects \(x\)-axis at Q, then the abscissa of Q is :

Let \(f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )\) and $$S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}...

If the vectors \(\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k\), \(\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k\) and \(\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k\) are coplanar ...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three non-zero non-coplanar vectors. Let the position vectors of four points \(A,B,C\) and \(D\) be $$\overrightarrow a - \overrightarrow b + \overrightarrow...