If the system of linear equations $7x + 11y + \alpha z = 13$ $5x + 4y + 7z = \beta$ $175x + 194y + 57z = 361$ has infinitely many solutions, then $\alpha + \beta + 2$ is equal to:

1

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{\,n+2}$, which are in the r…

Topic: JEE Main 2023 (11 April Evening Shift)

1

For $a \in \mathbb{C}$, let $A = \{\, z \in \mathbb{C} : \Re(a + \bar z) > \Im(\bar a + z) \,\}$ and $B = \{\, z \in…

Topic: JEE Main 2023 (11 April Evening Shift)

2

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750\beta$,…

Topic: JEE Main 2023 (11 April Evening Shift)

3

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with ser…

Topic: JEE Main 2023 (11 April Evening Shift)

4

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A\times B$ such that $R=\{\,((a_1,b_1…

Topic: JEE Main 2023 (11 April Evening Shift)

5

If the $1011^{\text{th}}$ term from the end in the binomial expansion of \(\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2…

Topic: JEE Main 2023 (11 April Evening Shift)

6

If \[ \begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \dfrac{9}{8}\,(103x+81),…

Topic: JEE Main 2023 (11 April Evening Shift)

7

Let the mean of 6 observations $1, 2, 4, 5, x, y$ be $5$ and their variance be $10$. Then their mean deviation about t…

Topic: JEE Main 2023 (11 April Evening Shift)

8

If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying \[ \int_{0}^{\pi/2} f(\sin 2x)\,\sin x\,dx \;+\; \al…

Topic: JEE Main 2023 (11 April Evening Shift)

9

Let $f$ and $g$ be two functions defined by \[ f(x)= \begin{cases} x+1, & x

Topic: JEE Main 2023 (11 April Evening Shift)

10

Let $y=y(x)$ be the solution of the differential equation \[ \frac{dy}{dx}+\frac{5}{x(x^5+1)}\,y=\frac{(x^5+1)^2}{x^7},…

Topic: JEE Main 2023 (11 April Evening Shift)