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 :

1

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…

Topic: JEE Main 2019 (11 January Evening Shift)

2

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

Topic: JEE Main 2019 (11 January Evening Shift)

3

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…

Topic: JEE Main 2019 (11 January Evening Shift)

4

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

Topic: JEE Main 2019 (11 January Evening Shift)

5

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

Topic: JEE Main 2019 (11 January Evening Shift)

6

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

Topic: JEE Main 2019 (11 January Evening Shift)

7

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-ze…

Topic: JEE Main 2019 (11 January Evening Shift)

8

Let the length of the latus rectum of an ellipse with its major axis along the $x$-axis and centre at the origin be $8$…

Topic: JEE Main 2019 (11 January Evening Shift)

9

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 t…

Topic: JEE Main 2019 (11 January Evening Shift)

10

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

Topic: JEE Main 2019 (11 January Evening Shift)

1

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

Topic: JEE Main 2019 (11 January Evening Shift)

2

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

Topic: JEE Main 2019 (11 January Evening Shift)

3

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 differentiabl…

Topic: JEE Main 2019 (11 January Evening Shift)

4

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 $3$…

Topic: JEE Main 2019 (11 January Evening Shift)

5

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

Topic: JEE Main 2019 (11 January Evening Shift)

6

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

Topic: JEE Main 2019 (11 January Evening Shift)

7

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 v…

Topic: JEE Main 2019 (11 January Evening Shift)

8

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 …

Topic: JEE Main 2019 (11 January Evening Shift)

9

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

Topic: JEE Main 2019 (11 January Evening Shift)

10

Let $\sqrt{3}\,\hat{i}+\hat{j}$, $\ \hat{i}+\sqrt{3}\,\hat{j}$ and $\ \beta\,\hat{i}+(1-\beta)\,\hat{j}$ respectively b…

Topic: JEE Main 2019 (11 January Evening Shift)