If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :

1

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y…

Topic: JEE Main 2025 (28 January Evening Shift)

2

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, o…

Topic: JEE Main 2025 (28 January Evening Shift)

3

Let [x] denote the greatest integer less than or equal to x. Then the domain of $ f(x) = \sec^{-1}(2[x] + 1) $ is:

Topic: JEE Main 2025 (28 January Evening Shift)

4

Let $f:\mathbb{R}-{0}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\l…

Topic: JEE Main 2025 (28 January Evening Shift)

5

If the components of $\vec a=\alpha,\hat i+\beta,\hat j+\gamma,\hat k$ along and perpendicular to $\vec b=3\hat i+\hat …

Topic: JEE Main 2025 (28 January Evening Shift)

6

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ …

Topic: JEE Main 2025 (28 January Evening Shift)

7

Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then t…

Topic: JEE Main 2025 (28 January Evening Shift)

8

The square of the distance of the point $\left(\dfrac{15}{7},,\dfrac{32}{7},,7\right)$ from the line $\dfrac{x+1}{3}=\d…

Topic: JEE Main 2025 (28 January Evening Shift)

9

If the midpoint of a chord of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is $\left(\sqrt{2},,\dfrac{4}{3}\right)…

Topic: JEE Main 2025 (28 January Evening Shift)

10

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x\in\m…

Topic: JEE Main 2025 (28 January Evening Shift)

1

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\…

Topic: JEE Main 2025 (28 January Evening Shift)

2

>Let $A, B, C$ be three points in xy-plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt…

Topic: JEE Main 2025 (28 January Evening Shift)

3

If $f(x)=\displaystyle\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)},dx,; f(0)=-6$, then $f(1)$ is equal to:

Topic: JEE Main 2025 (28 January Evening Shift)

4

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin…

Topic: JEE Main 2025 (28 January Evening Shift)

5

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d…

Topic: JEE Main 2025 (28 January Evening Shift)

6

et $f:[0,3]\to A$ be defined by $,f(x)=2x^3-15x^2+36x+7,$ and $g:[0,\infty)\to B$ be defined by $,g(x)=\dfrac{x^{2025}}…

Topic: JEE Main 2025 (28 January Evening Shift)

7

For positive integers $n$, if $4a_n=(n^2+5n+6)$ and $S_n=\displaystyle\sum_{k=1}^{n}\frac{1}{a_k},$ then the value of $…

Topic: JEE Main 2025 (28 January Evening Shift)