Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices on the coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is:

1

Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocentre of this …

Topic: JEE Main 2019 (9 January Evening Shift)

2

A data consists of $n$ observations: $x_1,x_2,\ldots,x_n$. If $\displaystyle \sum_{i=1}^{n}(x_i+1)^2=9n$ and $\displays…

Topic: JEE Main 2019 (9 January Evening Shift)

3

If $A=\begin{bmatrix} e^{t} & e^{-t}\cos t & e^{-t}\sin t\\[4pt] e^{t} & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t …

Topic: JEE Main 2019 (9 January Evening Shift)

4

If $f(x)=\displaystyle\int \frac{5x^{8}+7x^{6}}{(x^{2}+1+2x^{7})^{2}}\,dx,\ (x\ge 0)$ and $f(0)=0$, then the value of $…

Topic: JEE Main 2019 (9 January Evening Shift)

5

Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$ such that $|f(x)-f(y)|\le 2|x-y|^{3/2}$ for all …

Topic: JEE Main 2019 (9 January Evening Shift)

6

If $x=3\tan t$ and $y=3\sec t$, then the value of $\dfrac{d^{2}y}{dx^{2}}$ at $t=\dfrac{\pi}{4}$ is:

Topic: JEE Main 2019 (9 January Evening Shift)

7

The number of natural numbers less than 7000 which can be formed by using the digits 0,1,3,7,9 (repetition of digits al…

Topic: JEE Main 2019 (9 January Evening Shift)

8

A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length $4$ along …

Topic: JEE Main 2019 (9 January Evening Shift)

9

If the lines $x=ay+b,\ z=cy+d$ and $x=a'z+b',\ y=c'z+d'$ are perpendicular, then:

Topic: JEE Main 2019 (9 January Evening Shift)

10

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola $y^{2}=4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, wh…

Topic: JEE Main 2019 (9 January Evening Shift)

1

If both the roots of the quadratic equation $x^{2}-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, …

Topic: JEE Main 2019 (9 January Evening Shift)

2

Let $z_{0}$ be a root of the quadratic equation $x^{2}+x+1=0$. If $z=3+6i\,z_{0}^{81}-3i\,z_{0}^{93}$, then $\arg z$ is…

Topic: JEE Main 2019 (9 January Evening Shift)

3

If the system of linear equations$x-4y+7z=g$,$3y-5z=h$,$-2x+5y-9z=k$is consistent, then:

Topic: JEE Main 2019 (9 January Evening Shift)

4

If $x=\sin^{-1}(\sin 10)$ and $y=\cos^{-1}(\cos 10)$, then $y-x$ is equal to:

Topic: JEE Main 2019 (9 January Evening Shift)

5

For each $x\in\mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\displaystyle \lim_{x\t…

Topic: JEE Main 2019 (9 January Evening Shift)

6

Let $\vec a=\hat i+\hat j+\sqrt{2}\,\hat k$, $\vec b=b_1\hat i+b_2\hat j+\sqrt{2}\,\hat k$, $\vec c=5\hat i+\hat j+\sqr…

Topic: JEE Main 2019 (9 January Evening Shift)

7

An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a…

Topic: JEE Main 2019 (9 January Evening Shift)

8

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation $6x^{2}-1…

Topic: JEE Main 2019 (9 January Evening Shift)

9

Let $A=\{x\in\mathbb{R}:\ x\ \text{is not a positive integer}\}$. Define a function $f:A\to\mathbb{R}$ as $f(x)=\dfrac{…

Topic: JEE Main 2019 (9 January Evening Shift)

10

The area of the region $A=\{(x,y): 0\le y\le x|x|+1 \text{ and } -1\le x\le 1\}$ (in sq. units) is:

Topic: JEE Main 2019 (9 January Evening Shift)