Let $\cos(\alpha+\beta)=-\frac{1}{10}$ and $\sin(\alpha-\beta)=\frac{3}{8}$, where $0<\alpha<\frac{\pi}{3}$ and $0<\beta<\frac{\pi}{4}$. If $\tan2\alpha=\dfrac{3(1-r\sqrt5)}{\sqrt{11(s+\sqrt5)}}$, $r,s\in\mathbb{N}$, then $r+s$ is equal to ____ (Integer Type)

1

Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_0^{64}(x^{1/3}-[x^{1/3}])dx$, then $\dfrac{1}{\pi}\int_…

Topic: JEE Main 2026 (22 January Evening Shift)

2

Let a vector $\vec{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda\hat{k}$, $\lambda>0$, make an obtuse angle with the vector $\vec{…

Topic: JEE Main 2026 (22 January Evening Shift)

3

Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1+x)^n$, $n\in \mathbb N$, $0\le r\le n$. If $…

Topic: JEE Main 2026 (22 January Evening Shift)

4

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x)f(y)$, $f(1)=7$ and $g(x+y)=g(xy)$, $g(1)=1$, for all $x,y\in \math…

Topic: JEE Main 2026 (22 January Evening Shift)

5

Let $f(x)=[x]^2-[x+3]-3$, $x\in\mathbb R$, where $[\,\cdot\,]$ is the greatest integer function. Then

Topic: JEE Main 2026 (22 January Evening Shift)

6

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4x$ be the curve $S$. Let $P$ b…

Topic: JEE Main 2026 (22 January Evening Shift)

7

Let $S$ and $S'$ be the foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$ and $P(\alpha,\beta)$ be a point on the …

Topic: JEE Main 2026 (22 January Evening Shift)

8

Let $[\,\cdot\,]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt{2}\,x,x^2\}$. Let $S=\{x\in(-2,2):\te…

Topic: JEE Main 2026 (22 January Evening Shift)

9

Let the domain of the function $f(x)=\log_3\log_5\left(7-\log_2(x^2-10x+85)\right)+\sin^{-1}\left(\left|\dfrac{3x-7}{17…

Topic: JEE Main 2026 (22 January Evening Shift)

10

Let $\alpha,\beta$ be the roots of the quadratic equation $12x^2-20x+3\lambda=0$, $\lambda\in \mathbb{Z}$. If $\dfrac12…

Topic: JEE Main 2026 (22 January Evening Shift)

1

Suppose $a,b,c$ are in A.P. and $a^2,2b^2,c^2$ are in G.P. If $a

Topic: JEE Main 2026 (22 January Evening Shift)

2

Let $S$ be the set of the first $11$ natural numbers. Then the number of elements in $A=\{B\subseteq S:n(B)\ge 2 \text{…

Topic: JEE Main 2026 (22 January Evening Shift)