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 :

1

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\ma…

Topic: JEE Main 2019 (10 January Evening Shift)

2

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

Topic: JEE Main 2019 (10 January Evening Shift)

3

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x] + [\sin x] + 4}$, where $[t]$ denotes the greatest i…

Topic: JEE Main 2019 (10 January Evening Shift)

4

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

Topic: JEE Main 2019 (10 January Evening Shift)

5

If $\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$ and $\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$ be two given ve…

Topic: JEE Main 2019 (10 January Evening Shift)

6

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2 + 1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum va…

Topic: JEE Main 2019 (10 January Evening Shift)

7

The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x …

Topic: JEE Main 2019 (10 January Evening Shift)

8

The positive value of $\lambda$ for which the coefficient of $x^2$ in the expression $x^2 \left( \sqrt{x} + \dfrac{\l…

Topic: JEE Main 2019 (10 January Evening Shift)

9

Let $z = \left(\dfrac{\sqrt{3}}{2} + \dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2} - \dfrac{i}{2}\right)^5.$ If $…

Topic: JEE Main 2019 (10 January Evening Shift)

10

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

Topic: JEE Main 2019 (10 January Evening Shift)

1

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

Topic: JEE Main 2019 (10 January Evening Shift)

2

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…

Topic: JEE Main 2019 (10 January Evening Shift)

3

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

Topic: JEE Main 2019 (10 January Evening Shift)

4

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 on…

Topic: JEE Main 2019 (10 January Evening Shift)

5

A helicopter is flying along the curve given by $y - x^{3/2} = 7,\ (x \ge 0)$. A soldier positioned at the point $\left…

Topic: JEE Main 2019 (10 January Evening Shift)

6

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

Topic: JEE Main 2019 (10 January Evening Shift)

7

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…

Topic: JEE Main 2019 (10 January Evening Shift)

8

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

Topic: JEE Main 2019 (10 January Evening Shift)

9

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}…

Topic: JEE Main 2019 (10 January Evening Shift)