If the system of equations $x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0$ $x+(\cos\alpha)y+(\sin\alpha)z=0$ $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha\in\left(0,\frac{\pi}{2}\right)$ is equal to:

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\overline{z})^2 + |z| = 0,\; z \in \mathbb{C}$. Then $4(\alpha^2 + \beta^2)$ is equal to:

Let $f:\mathbb{R}\to\mathbb{R}$ be a function given by $ f(x)= \begin{cases} \dfrac{1-\cos 2x}{x^2}, & x<0,\\[6pt] \alpha, & x=0,\\[6pt] \dfrac{\beta\sqrt{\,1-\cos x\,}}{x}, & x>0, \end{cases} $ where $\alpha,\beta\in\mathbb{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to:

Consider a hyperbola $\text{H}$ having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola $\text{H}$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $\text{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of $\text{H}$ is:

If the coefficients of $x^4$, $x^5$, and $x^6$ in the expansion of $(1+x)^n$ are in arithmetic progression, then the maximum value of $n$ is:

The value of $\dfrac{1\times2^2+2\times3^2+\ldots+100\times(101)^2}{1\times3+2\times4+3\times5+\ldots+100\times101}$ is:

If the function $ f(x)= \begin{cases} \dfrac{7^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos^{2}x}}, & x\neq0,\\[6pt] a\log_{e}2\log_{e}3, & x=0 \end{cases} $ is continuous at $x=0$, then the value of $a^{2}$ is equal to:

Let $P$ be the point of intersection of the lines $\dfrac{x-2}{1}=\dfrac{y-4}{5}=\dfrac{z-2}{1}$ and $\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}$. Then, the shortest distance of $P$ from the line $4x=2y=z$ is:

Let $f(x)=3\sqrt{x-2}+\sqrt{4-x}$ be a real-valued function. If $\alpha$ and $\beta$ are respectively the minimum and maximum values of $f$, then $\alpha^2+2\beta^2$ is equal to:

Let $f(x)=\int_{0}^{x}\left(t+\sin(1-e^{t})\right)dt,\;x\in\mathbb{R}$. Then, $\displaystyle\lim_{x\to0}\dfrac{f(x)}{x^{3}}$ is equal to:

The area (in sq. units) of the region described by $\{(x,y):y^{2}\le2x,\;y\ge4x-1\}$ is:

Given that the inverse trigonometric functions assume principal values only. Let $x,y\in[-1,1]$ such that $\cos^{-1}x-\sin^{-1}y=\alpha$, with $-\dfrac{\pi}{2}\le\alpha\le\pi$. Then, the minimum value of $x^{2}+y^{2}+2xy\sin\alpha$ is:

Let $y=y(x)$ be the solution of the differential equation $(x^{2}+4)^{2}\,dy+(2x^{3}y+8xy-2)\,dx=0.$ If $y(0)=0$, then $y(2)$ is equal to:

The mean of the following probability distribution of a random variable $X$ is $\dfrac{46}{9}$.

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersect the circle $C$ at the points $P$ and $Q$. Let $MN$ be a chord of $C$ of length $2$ units and slope $-1$. Then, the distance (in units) between the chord $PQ$ and the chord $MN$ is:

Let $PQ$ be a chord of the parabola $y^{2}=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following points lies on the line passing through the points $P$ and $Q$?

The area (in sq. units) of the region $S = \{\, z \in \mathbb{C} : |z - 1| \le 2,\ (z + \bar{z}) + i(z - \bar{z}) \le 2,\ \operatorname{Im}(z) \ge 0 \,\}$ is :

If $\displaystyle \int_{-1}^{1}\frac{\cos(\alpha x)}{1+3x^{2}},dx=\frac{2}{\pi}$, then a value of $\alpha$ is:

If $a,b,c$ are in AP and $a+1,; b,; c+3$ are in GP. Given $a>10$ and the arithmetic mean of $a,b,c$ is $8$, then the cube of the geometric mean of $a,b,c$ is:

Let a relation $R$ on $\mathbb N\times\mathbb N$ be defined by $(x_1,y_1),R,(x_2,y_2)$ iff $x_1\le x_2$ or $y_1\le y_2$. Consider: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive. Which of the following is true?

Let $\vec a=\hat i+\hat j+\hat k,;\vec b=2\hat i+4\hat j-5\hat k$ and $\vec c=x\hat i+2\hat j+3\hat k,;x\in\mathbb R$. If $\vec d$ is the unit vector in the direction of $(\vec b+\vec c)$ such that $\vec a\cdot\vec d=1$, then $(\vec a\times\vec b)\cdot\vec c$ equals:

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix

Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of $ABCD$ lie on the sides of $PQRS$. Let $a$ and $b$ be the sides of $PQRS$ when its area is maximum. Then $(a+b)^2$ is equal to:

Let $A={1,3,7,9,11}$ and $B={2,4,5,7,8,10,12}$. The total number of one–one maps $f:A\to B$ such that $f(1)+f(3)=14$ is:

Two straight lines through the origin $O$ intersect the line $3x+4y=12$ at points $P$ and $Q$ such that $\triangle OPQ$ is isosceles and $\angle POQ=90^\circ$. If $I=OP^2+PQ^2+QO^2$, then the greatest integer $\le I$ is:

If the line $\dfrac{2-x}{3}=\dfrac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\dfrac{x+3}{3\mu}=\dfrac{1-2y}{6}=\dfrac{5-z}{7}$, then $4\lambda+9\mu$ equals:

Consider the following two statements: Statement I: For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\dfrac{z_1}{|z_1|}+\dfrac{z_2}{|z_2|}\right|\le 2(|z_1|+|z_2|)$. Statement II: If $x,y,z$ are three distinct complex numbers and $a,b,c$ are positive real numbers such that $\dfrac{a}{|,y-z,|}=\dfrac{b}{|,z-x,|}=\dfrac{c}{|,x-y,|}$, then $\dfrac{a^{2}}{,y-z,}+\dfrac{b^{2}}{,z-x,}+\dfrac{c^{2}}{,x-y,}=1$. Between the above two statements:

Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then $\left|,A^{T}A,( \operatorname{adj}(2A))^{-1},(\operatorname{adj}(4B)),(\operatorname{adj}(AB))^{-1},A A^{T}\right|$ is equal to:

Let a circle $C$ of radius $1$ and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $C$ from the point $(5,5)$ is:

Let $f(x)=x^{5}+2x^{3}+3x+1,; x\in\mathbb{R}$, and let $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{R}$. Then $\dfrac{g'(7)}{g'(7)}$ is equal to:

The coefficients $a,b,c$ in the quadratic $ax^{2}+bx+c=0$ are chosen from the set ${1,2,3,4,5,6,7,8}$. The probability that the equation has repeated roots is:

If the system $11x+y+\lambda z=-5,\quad 2x+3y+5z=3,\quad 8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^{4}-\mu$ equals:

The integral $\displaystyle \int_{0}^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x},dx$ is equal to:

If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}}\,dx = a + b\sqrt{2} + c\sqrt{3}$, where $a, b, c$ are rational numbers, then $2a + 3b - 4c$ is equal to:

If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\cdots+\dfrac{1}{99\cdot 100}=n$, then the point $(m,n)$ lies on the line:

If $S = \{ z \in \mathbb{C} : |z - i| = |z + i| = |z - 1| \}$, then $n(S)$ is:

Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2}$ from the point $(7,8,9)$. Then $d^{2}+6$ is equal to:

Let $S=\{1,2,3,\ldots,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A,B): A\cap B\ne \phi;\ A,B\in M\}$ is:

If the function $f(x)=\dfrac{\sin 3x+\alpha\sin x-\beta\cos 3x}{x^{3}},; x\in\mathbb{R},$ is continuous at $x=0$, then $f(0)$ is equal to:

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is $d$, then $d^2$ is equal to:

Let the line $2x+3y-k=0,\ k>0$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. If the circle having $AB$ as a diameter is $x^{2}+y^{2}-3x-2y=0$ and the length of the latus rectum of the ellipse $x^{2}+9y^{2}=k^{2}$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $2m+n$ is equal to:

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and $I_1=\displaystyle\int_a^b x\sin(4x-x^2)\,dx,\ \ I_2=\displaystyle\int_a^b \sin(4x-x^2)\,dx,$ then $36\,\dfrac{I_1}{I_2}$ is equal to:

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y},dy$ is:

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\dfrac{dx}{dt}+ax=0$ and $\dfrac{dy}{dt}+by=0$ respectively, $a,b\in\mathbb{R}$. Given that $x(0)=2$, $y(0)=1$ and $3y(1)=2x(1)$, the value of $t$ for which $x(t)=y(t)$ is:

Suppose $\theta\in[0,\tfrac{\pi}{4}]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:

The distance of the point $(7,-2,11)$ from the line $\dfrac{x-6}{1}=\dfrac{y-4}{0}=\dfrac{z-8}{3}$ along the line $\dfrac{x-5}{2}=\dfrac{y-1}{-3}=\dfrac{z-5}{6}$ is:

For $f(x)=\sin x+3x-\dfrac{2}{\pi}(x^{2}+x)$, where $x\in\left[0,\tfrac{\pi}{2}\right]$, consider: (I) $f$ is increasing in $\left(0,\tfrac{\pi}{2}\right)$. (II) $f'$ is decreasing in $\left(0,\tfrac{\pi}{2}\right)$.

The length of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose midpoint is $\left(1,\dfrac{9}{5}\right)$, is equal to:

Suppose $\theta\in\left[0,\tfrac{\pi}{4}\right]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:

Consider the function $ f(x)= \begin{cases} \dfrac{a\,(7x-12-x^{2})}{\,b\,\lfloor x^{2}-7x+12\rfloor\,}, & x<3,\\[6pt] \dfrac{\sin(x-3)}{2^{\,x-1}}, & x>3,\\[6pt] b, & x=3, \end{cases} $ where $\lfloor x\rfloor$ denotes the greatest integer $\le x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is:

If $y=y(x)$ solves the differential equation $\dfrac{dy}{dx}+2y=\sin(2x)$ with $y(0)=\dfrac{3}{4}$, then $y!\left(\dfrac{\pi}{8}\right)$ is:

If $a=\displaystyle\lim_{x\to 0}\dfrac{\sqrt{\,1+\sqrt{\,1+x^{2}\,}\,}-\sqrt{2}}{x^{2}}$ and $b=\displaystyle\lim_{x\to 0}\dfrac{\sin^{2}x}{\sqrt{2}-\sqrt{\,1+\cos x\,}}$, then the value of $ab^{3}$ is:

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle with centre $(-1,0)$ and radius $2$. If the line of the common chord of $C_1$ and $C_2$ meets the $y$-axis at the point $P$, then the square of the distance of $P$ from the centre of $C_1$ is:

Consider the matrix $f(x)=\begin{bmatrix} \cos x & -\sin x & 0\\ \sin x & \cos x & 0\\ 0 & 0 & 1 \end{bmatrix}$. Given below are two statements: Statement I : $f(-x)$ is the inverse of the matrix $f(x)$. Statement II : $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer:

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, & x \geq 0 \\ x+1, & x \leq 0 .\end{cases}$

Then the function $f(g(x))$ is

Four distinct points $(2k,3k)$, $(1,0)$, $(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to:

Let $ABCD$ and $AEFG$ be squares of side $4$ and $2$ units, respectively. The point $E$ is on the line segment $AB$ and the point $F$ is on the diagonal $AC$. Then the radius $r$ of the circle passing through the point $F$ and touching the line segments $BC$ and $CD$ satisfies:

If the shortest distance between the lines $\dfrac{x-4}{2}=\dfrac{y+1}{3}=\dfrac{z-\lambda}{2}$ and $\dfrac{x-2}{1}=\dfrac{y+1}{4}=\dfrac{z-2}{-3}$ is $\dfrac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is:

Let $(\alpha,\beta,\gamma)$ be the image of the point $(8,5,7)$ in the line $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5}$. Then $\alpha+\beta+\gamma$ is:

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of the angle between the lines $L_1$ and $L_2$ is:

If the constant term in the expansion of $\left(\dfrac{\sqrt{3}}{x}+\dfrac{2x}{\sqrt{5}}\right)^{12}$, $x\ne 0$, is $\alpha\times 2^{8}\times\sqrt{3}$, then $25\alpha$ is:

${}^{n-1}C_r \;=\; (k^2-8)\, {}^{n}C_{r+1}$ if and only if:

Let $\vec a=2\hat i+5\hat j-\hat k$, $\vec b=2\hat i-2\hat j+2\hat k$ and $\vec c$ be three vectors such that $(\vec c+\hat i)\times(\vec a+\vec b+\hat i)=\vec a\times(\vec c+\hat i)$. If $\vec a\cdot\vec c=-29$, then $\vec c\cdot(-2\hat i+\hat j+\hat k)$ is equal to:

The number of common terms in the progressions $4,\,9,\,14,\,19,\ldots,$ up to $25^{\text{th}}$ term and $3,\,6,\,9,\,12,\ldots,$ up to $37^{\text{th}}$ term is:

Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is $10$. If the locus of $P$ is $ax+by=15$, then $5a+2b$ is:

Let $a_1,a_2,\ldots,a_{10}$ be $10$ observations such that $\displaystyle \sum_{k=1}^{10} a_k = 50$ and $\displaystyle \sum_{i
1 5
2 $\sqrt{115}$
3 10
4 $\sqrt{5}$
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

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Solution