Evaluate the sum: $\displaystyle \sum_{k=0}^{6} \binom{51-k}{3}$

1

Let $A=\begin{bmatrix}\dfrac{1}{\sqrt{10}} & \dfrac{3}{\sqrt{10}}\\[4pt]-\dfrac{3}{\sqrt{10}} & \dfrac{1}{\sqrt{10}}\e…

Topic: JEE Main 2023 (25 January Evening Shift)

2

Let $y=y(t)$ be a solution of the differential equation $\dfrac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where $\alpha>0$,…

Topic: JEE Main 2023 (25 January Evening Shift)

3

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition,…

Topic: JEE Main 2023 (25 January Evening Shift)

4

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\log_{\sqrt{m}}\!\left(\sqrt{2}(\sin x-\cos x)+m-2\righ…

Topic: JEE Main 2023 (25 January Evening Shift)

5

$ \text{The integral } 16 \int_{1}^{2} \frac{dx}{x^{3}(x^{2}+2)^{2}} \text{ is equal to:}$

Topic: JEE Main 2023 (25 January Evening Shift)

6

Let $z$ be a complex number such that $\left|\dfrac{z-2i}{z+i}\right|=2,\ z\ne -i$. Then $z$ lies on the circle of ra…

Topic: JEE Main 2023 (25 January Evening Shift)

7

Let the function $f(x)=2x^{3}+(2p-7)x^{2}+3(2p-9)x-6$ have a maxima for some value of $x0$. Then, the set of all valu…

Topic: JEE Main 2023 (25 January Evening Shift)

8

Let $T$ and $C$ respectively be the transverse and conjugate axes of the hyperbola $16x^{2}-y^{2}+64x+4y+44=0$. The…

Topic: JEE Main 2023 (25 January Evening Shift)

9

The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is:

Topic: JEE Main 2023 (25 January Evening Shift)

10

If the function $f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}}…

Topic: JEE Main 2023 (25 January Evening Shift)

1

Let $f(x)=2x^{n}+\lambda$, $\lambda\in \mathbb{R}$, $n\in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of al…

Topic: JEE Main 2023 (25 January Evening Shift)