If the angle of intersection at a point where two circles with radii $5\text{ cm}$ and $12\text{ cm}$ intersect is $90^\circ$, then the length (in cm) of their common chord is:

The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}(1+x+x^{2})^{9}$ is:

The curve amongst the family of curves represented by the differential equation $(x^2 - y^2)dx + 2xy\,dy = 0$ which passes through $(1, 1)$ is :

If $m$ is the minimum value of $k$ for which the function $f(x)=x\sqrt{kx-x^{2}}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m$, then the ordered pair $(m,M)$ is equal to:

Let $z = \left(\dfrac{\sqrt{3}}{2} + \dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2} - \dfrac{i}{2}\right)^5.$ If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$, then :

Let $A, B$ and $C$ be sets such that $\varnothing \ne A\cap B \subseteq C$. Which of the following statements is not true?

The positive value of $\lambda$ for which the coefficient of $x^2$ in the expression $x^2 \left( \sqrt{x} + \dfrac{\lambda}{x^2} \right)^{10}$ is $720$, is –

If the area (in sq. units) bounded by the parabola $y^{2}=4\lambda x$ and the line $y=\lambda x,\ \lambda>0$, is $\dfrac{1}{9}$, then $\lambda$ is equal to:

The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x + 2 = \lambda$ has the least value, is –

The general solution of the differential equation (y² – x³)dx – xydy = 0 (x $ \ne $ 0) is : (where c is a constant of integration)

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2 + 1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum value of $\dfrac{\det(A)}{b}$ is –

Let $a \in \left(0, \dfrac{\pi}{2}\right)$ be fixed. If $\displaystyle \int \dfrac{\tan x + \tan a}{\tan x - \tan a} , dx = A(x)\cos 2a + B(x)\sin 2a + C,$ where $C$ is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively:

If $\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$ and $\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$ be two given vectors $\vec{a}$ and $\vec{b}$ which are non-collinear, then the value of $\lambda$ for which vectors $\vec{\alpha}$ and $\vec{\beta}$ are collinear, is –

The term independent of $x$ in the expansion of $\left(\dfrac{1}{60} - \dfrac{x^{8}}{81}\right)\left(2x^{2} - \dfrac{3}{x^{2}}\right)^{6}$ is equal to:

Let $S = \{(x, y) \in \mathbb{R}^2 : \dfrac{y^2}{1 + r} - \dfrac{x^2}{1 - r} = 1 ; \, r \neq \pm 1 \}$. Then $S$ represents :

If $a_{1}, a_{2}, a_{3}, \dots$ are in A.P. such that $a_{1} + a_{7} + a_{16} = 40$, then the sum of the first 15 terms of this A.P. is:

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x] + [\sin x] + 4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is :

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^{2} + 2\beta x + \gamma = 0$ and $x^{2} + x - 1 = 0$ have a common root, then $\alpha(\beta + \gamma)$ is equal to:

Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocenter is at the origin, then its third vertex lies in which quadrant:

A straight line $L$ at a distance of $4$ units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^\circ$ with the line $x + y = 0$. Then an equation of the line $L$ is:

Let $a_1,a_2,\dots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\dots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\mathbb{N}$, for which $ \begin{vmatrix} \log_e(a_1^{\,r}a_2^{\,k}) & \log_e(a_2^{\,r}a_3^{\,k}) & \log_e(a_3^{\,r}a_4^{\,k})\\ \log_e(a_4^{\,r}a_5^{\,k}) & \log_e(a_5^{\,r}a_6^{\,k}) & \log_e(a_6^{\,r}a_7^{\,k})\\ \log_e(a_7^{\,r}a_8^{\,k}) & \log_e(a_8^{\,r}a_9^{\,k}) & \log_e(a_9^{\,r}a_{10}^{\,k}) \end{vmatrix} =0. $ Then the number of elements in $S$, is –

Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x $ \in $ R. If f(x) attains maximum value at $\alpha $ and g(x) attains minimum value at $\beta $, then $\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)} \over {{x^2} - 6x + 8}}$ is equal to :

If $\displaystyle \sum_{r=0}^{25} \left\{ {^{50}C_{r}} \cdot {^{\,50-r}C_{\,25-r}} \right\} = K \binom{50}{25}$, then $K$ is equal to :

Let z $ \in $ C with Im(z) = 10 and it satisfies ${{2z - n} \over {2z + n}}$ = 2i - 1 for some natural number n. Then :

The value of $\cos \dfrac{\pi}{22}\cdot \cos \dfrac{\pi}{23}\cdot \ldots \cdot \cos \dfrac{\pi}{210}\cdot \sin \dfrac{\pi}{210}$ is –

A value of $\theta \in \left( {0,{\pi \over 3}} \right)$, for which
$\left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr } } \right| = 0$, is :

If mean and standard deviation of 5 observations $x_1,x_2,x_3,x_4,x_5$ are $10$ and $3$ respectively, then the variance of 6 observations $x_1,x_2,\ldots,x_5$ and $-50$ is equal to :

$\lim_{x\to 0}\dfrac{x+2\sin x}{\sqrt{x^{2}+2\sin x+1}-\sqrt{\sin^{2}x-x+1}}$ is:

The value of $\cot\!\left(\displaystyle\sum_{n=1}^{19}\cot^{-1}\!\left(1+\sum_{p=1}^{n}2p\right)\right)$ is :

The derivative of ${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$, with respect to ${x \over 2}$ , where $\left( {x \in \left( {0,{\pi \over 2}} \right)} \right)$ is :

Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$, then one of its vertices is :

A value of $\alpha$ such that $\displaystyle \int_{\alpha}^{\alpha+1} \dfrac{dx}{(x+\alpha)(x+\alpha+1)} = \log_e\left(\dfrac{9}{8}\right)$ is:

A helicopter is flying along the curve given by $y - x^{3/2} = 7,\ (x \ge 0)$. A soldier positioned at the point $\left(\dfrac{1}{2},\,7\right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is –

An ellipse, with foci at $(0, 2)$ and $(0, -2)$ and minor axis of length $4$, passes through which of the following points?

If $\displaystyle \int x^{5}\,e^{-4x^{3}}\,dx=\dfrac{1}{48}\,e^{-4x^{3}}\,f(x)+C$, where $C$ is a constant of integration, then $f(x)$ is equal to –

A group of students comprises of $5$ boys and $n$ girls. If the number of ways in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team is $1750$, then $n$ is equal to:

If the area of an equilateral triangle inscribed in the circle $x^{2}+y^{2}+10x+12y+c=0$ is $27\sqrt{3}$ sq units, then $c$ is equal to :

A person throws two fair dice. He wins Rs. $15$ for throwing a doublet (same numbers on the two dice), wins Rs. $12$ when the throw results in the sum of $9$, and loses Rs. $6$ for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is:

The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt{2}\,y+4\sqrt{2}=0$ is –

A circle touching the $x$-axis at $(3,0)$ and making an intercept of length $8$ on the $y$-axis passes through the point:

Let $\mathbb{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:\mathbb{N}\to\mathbb{N}$ such that $$ f(n)= \begin{cases} \dfrac{n+1}{2}, & \text{if $n$ is odd},\\[4pt] \dfrac{n}{2}, & \text{if $n$ is even}, \end{cases} \qquad g(n)=n-(-1)^n. $$ Then $f\circ g$ is –

The outcome of each of 30 items was observed; 10 items gave an outcome $\dfrac{1}{2}-d$ each, 10 items gave outcome $\dfrac{1}{2}$ each and the remaining 10 items gave outcome $\dfrac{1}{2}+d$ each. If the variance of this outcome data is $\dfrac{4}{3}$ then $|d|$ equals :

If $x\log_e(\log_e x)-x^2+y^2=4\ (y>0)$, then $\left.\dfrac{dy}{dx}\right|_{x=e}$ is equal to :

The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is :

Let $A=\begin{pmatrix} 0 & 2q & r\\ p & q & -r\\ p & -q & r \end{pmatrix}$. If $AA^{T}=I_{3}$, then $|p|$ is :

Let $[x]$ denote the greatest integer less than or equal to $x$. Then $\displaystyle \lim_{x\to 0}\frac{\tan(\pi\sin^{2}x)+\left(|x|-\sin(x[x])\right)^{2}}{x^{2}}$ :

If $y(x)$ is the solution of the differential equation $\dfrac{dy}{dx}+\left(\dfrac{2x+1}{x}\right)y=e^{-2x},\ x>0,$ where $y(1)=\dfrac{1}{2}e^{-2}$, then

A square is inscribed in the circle $x^{2}+y^{2}-6x+8y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is :

If one real root of the quadratic equation $81x^{2}+kx+256=0$ is cube of the other root, then a value of $k$ is :

The straight line $x+2y=1$ meets the coordinate axes at $A$ and $B$. A circle is drawn through $A$, $B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is :

The maximum value of the function $f(x)=3x^{3}-18x^{2}+27x-40$ on the set $S=\{x\in\mathbb{R}: x^{2}+30\le 11x\}$ is :

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\dfrac{x^{3}}{3}+\dfrac{3}{x}\right)^{8}$ equals $5670$ is :

Let $\vec{a}=\hat{i}+2\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}+\lambda\hat{j}+4\hat{k}$ and $\vec{c}=2\hat{i}+4\hat{j}+(\lambda^{2}-1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\vec{a}\times\vec{c}$ is :

If $\displaystyle \int \frac{\sqrt{\,1-x^{2}\,}}{x^{4}}\,dx = A(x)\left(\sqrt{\,1-x^{2}\,}\right)^{m} + C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant of integration, then $(A(x))^{m}$ equals :

Two integers are selected at random from the set $\{1,2,\ldots,11\}$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

Let $f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$ and

$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$

Then, in the interval (–2, 2), g is :

Let $\left(-2-\dfrac{1}{3}i\right)^{3}=e^{\frac{x+iy}{2\pi i}}\ (i=\sqrt{-1})$, where $x$ and $y$ are real numbers, then $\,y-x\,$ equals :

Let $f_k(x)=\dfrac{1}{k}\left(\sin^{k}x+\cos^{k}x\right)$ for $k=1,2,3,\ldots$ Then for all $x\in\mathbb{R}$, the value of $f_4(x)-f_6(x)$ is equal to

Let $a_{1},a_{2},\ldots,a_{10}$ be a G.P. If $\dfrac{a_{3}}{a_{1}}=25$, then $\dfrac{a_{9}}{a_{5}}$ equals

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\dfrac{x}{1+x^{2}},\ x\in\mathbb{R}$. Then the range of $f$ is :

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\dfrac{x^m y^n}{(1+x^{2m})(1+y^{2n})}$ is :

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :

Let $S=\{1,2,\ldots,20\}$. A subset $B$ of $S$ is said to be “nice”, if the sum of the elements of $B$ is $203$. Then the probability that a randomly chosen subset of $S$ is “nice” is :

Let the length of the latus rectum of an ellipse with its major axis along the $x$-axis and centre at the origin be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

Let $f(x)=\dfrac{x}{\sqrt{a^{2}+x^{2}}}-\dfrac{d-x}{\sqrt{b^{2}+(d-x)^{2}}},\ x\in\mathbb{R}$, where $a,b,d$ are non-zero real constants. Then:

The integral $\displaystyle \int_{\pi/6}^{\pi/4}\frac{dx}{\sin 2x\,(\tan^{5}x+\cot^{5}x)}$ equals :

Let a function $f:(0,\infty)\to(0,\infty)$ be defined by $f(x)=\left|1-\dfrac{1}{x}\right|$. Then $f$ is :

If $\displaystyle \int \frac{x+1}{\sqrt{2x-1}}\,dx = f(x)\,\sqrt{2x-1}+C$, where $C$ is a constant of integration, then $f(x)$ is equal to :

The area (in sq. units) in the first quadrant bounded by the parabola $y=x^{2}+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :

$\displaystyle \lim_{x\to 0}\frac{x\cot(4x)}{\sin^{2}x\;\cot^{2}(2x)}$ is equal to :

If \[ \begin{vmatrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b\\ 2c & 2c & c-a-b \end{vmatrix} =(a+b+c)\,(x+a+b+c)^{2},\ x\ne 0, \] then $x$ is equal to :

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2}\sin\theta-x(\sin\theta\cos\theta+1)+\cos\theta=0$ $(0<\theta<45^\circ)$, and $\alpha<\beta$. Then $\displaystyle\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$ is equal to :

Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to :

If $19^{\text{th}}$ term of a non-zero A.P. is zero, then its $(49^{\text{th}}\ \text{term}) : (29^{\text{th}}\ \text{term})$ is :

Let $K$ be the set of all real values of $x$ where the function $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable. Then the set $K$ is equal to:

The number of functions $f$ from $\{1,2,3,\ldots,20\}$ onto $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is:

he solution of the differential equation, $\dfrac{dy}{dx}=(x-y)^{2}$, when $y(1)=1$, is

If in a parallelogram $ABDC$, the coordinates of $A, B$ and $C$ are respectively $(1,2)$, $(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is :

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^{2}+4(x-a^{2})=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is $250$ sq. units, then a value of $a$ is :

A circle cuts a chord of length $4a$ on the $x$-axis and passes through a point on the $y$-axis, distant $2b$ from the origin. Then the locus of the centre of this circle, is :

All $x$ satisfying the inequality $(\cot^{-1}x)^2 - 7(\cot^{-1}x) + 10 > 0$ lie in the interval:

Let $\sqrt{3}\,\hat{i}+\hat{j}$, $\ \hat{i}+\sqrt{3}\,\hat{j}$ and $\ \beta\,\hat{i}+(1-\beta)\,\hat{j}$ respectively be the position vectors of the points $A$, $B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\dfrac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is:

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:

A ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the binomial expansion of $\left(2^{1/3}+\dfrac{1}{2\cdot 3^{1/3}}\right)^{10}$ is:

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$. Then $\displaystyle \int_{0}^{a} f(x)\,g(x)\,dx$ is equal to :

If $\displaystyle \frac{z-\alpha}{z+\alpha}\ (\alpha\in\mathbb{R})$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is:

Let $S=\{1,2,3,\ldots,100\}$. The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is:

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw is:

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola $y=12-x^{2}$, such that the rectangle lies inside the parabola, is:

If the vertices of a hyperbola are at $(-2,0)$ and $(2,0)$ and one of its foci is at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^{2}=4x$, and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex, such that the area of $\triangle PXQ$ is maximum. Then this maximum area (in sq. units) is:

Consider three boxes, each containing $10$ balls labelled $1,2,\ldots,10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{\text{th}}$ box ($i=1,2,3$). Then, the number of ways in which the balls can be chosen such that $n_1
1 164
2 240
3 82
4 120
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

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Solution