JEE Main 2024 (Online) 9th April Morning Shift

Let the line \(\mathrm{L}\) intersect the lines \(x-2=-y=z-1,2(x+1)=2(y-1)=z+1\) and be parallel to the line \(\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}\). Then which of the following points lies on \(\mathrm{L}\) ?

The shortest distance between the lines \(\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}\) and \(\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}\) is:

Let the set of all positive values of \(\lambda\), for which the point of local minimum of the function \((1+x(\lambda^2-x^2))\) satisfies \(\frac{x^2+x+2}{x^2+5 x+6}<0\), be \((\alpha, \beta)\). Then \(\alpha^2+\beta^2\) is equal to ______...

The parabola \(y^2=4 x\) divides the area of the circle \(x^2+y^2=5\) in two parts. The area of the smaller part is equal to :

The coefficient of \(x^{70}\) in \(x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}\) is \({ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}\). Then a possible value of \(\mathrm{p}+\mathrm{q}\) is :

The remainder when \(428^{2024}\) is divided by 21 is __________.

Let a circle passing through \((2,0)\) have its centre at the point \((\mathrm{h}, \mathrm{k})\). Let \((x_{\mathrm{c}}, y_{\mathrm{c}})\) be the point of intersection of the lines \(3 x+5 y=1\) and \((2+\mathrm{c}) x+5 \mathrm{c}^2 y=1\). ...

Let the centre of a circle, passing through the points \((0,0),(1,0)\) and touching the circle \(x^2+y^2=9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k), 4\left(h^2+k^2\right)\) is equal to _______...

The sum of the square of the modulus of the elements in the set \(\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}\) is __________.

Let \(\lim _\limits{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.\) $$\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2...

The solution of the differential equation \((x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0\), is :

The solution curve, of the differential equation \(2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}\), passing through the point \((0,1)\) is a conic, whose vertex lies on the line :

Let \(f(x)=a x^3+b x^2+c x+41\) be such that \(f(1)=40, f^{\prime}(1)=2\) and \(f^{\prime \prime}(1)=4\). Then \(a^2+b^2+c^2\) is equal to:

Let \(f(x)=x^2+9, g(x)=\frac{x}{x-9}\) and \(\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)\). If \(\mathrm{e}\) and \(l\) denote the eccentricity and the length of the latus rectum of the ellipse $$\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mat...

If the domain of the function \(f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)\) is \(\mathbf{R}-(\alpha, \beta)\), then \(12 \alpha \beta\) is equal to :

If a function \(f\) satisfies \(f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})\) for all \(\mathrm{m}, \mathrm{n} \in \mathbf{N}\) and \(f(1)=1\), then the largest natural number \(\lambda\) such that $$\sum_\limits{\mathrm{k}=1}^{2...

Let \(\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C\), where \(C\) is the constant of integration. Then \(\alpha+\frac{\gamma}{\beta}\) is equal to :

Let \(f:(0, \pi) \rightarrow \mathbf{R}\) be a function given by $$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0< x<\frac{\pi}{2} \\ \mathrm{a}-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{\mathrm{...

Let \(\lambda, \mu \in \mathbf{R}\). If the system of equations
$$\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}$$
has infinitely many solutions, then \(\mu+2 \lambda\) is equal to :

Let \(A\) be a non-singular matrix of order 3. If \(\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}\) and $$\operatorname{det}(3\operatorname{adj}(2 \mathrm{A}))=2^{\mathrm{m}} ...

Let \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked \(1,2,3,4\). If the probability that \(a x^2+b x+c=0\) has all real roots is $$\frac{m}{n...

Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}(\alpha^6+\beta^6)\), is:

If the sum of the series \(\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}\) is equal to 5, then \(50 \mathrm{~d}\) is equal to :

Let \(A=\{2,3,6,7\}\) and \(B=\{4,5,6,8\}\). Let \(R\) be a relation defined on \(A \times B\) by \((a_1, b_1) R(a_2, b_2)\) if and only if \(a_1+a_2=b_1+b_2\). Then the number of elements in \(R\) is __________.

The frequency distribution of the age of students in a class of 40 students is given below.

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A variable line \(\mathrm{L}\) passes through the point \((3,5)\) and intersects the positive coordinate axes at the points \(\mathrm{A}\) and \(\mathrm{B}\). The minimum area of the triangle \(\mathrm{OAB}\), where \(\mathrm{O}\) is the or...

A ray of light coming from the point \(\mathrm{P}(1,2)\) gets reflected from the point \(\mathrm{Q}\) on the \(x\)-axis and then passes through the point \(R(4,3)\). If the point \(S(h, k)\) is such that \(P Q R S\) is a parallelogram, then...

Let \(|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]\). Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :

Let three vectors ,\(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}\) form a triangle such that $$\vec{c}=...

Let \(\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}\) and \(\overrightarrow{O C}=3 \vec{b}\), where \(O\) is the origin. If the area of the parallelogram with adjacent sides \(\overrightarrow{O A}\) and $$\overrig...