Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda\hat{j}+2\hat{k}$, $\lambda\in Z$. Let $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}$ be a vector of magnitude $2$ in $yz$-plane. If $|\vec{c}|=\sqrt{53}$, then the maximum possible value of $(\vec{c}\cdot\vec{d})^2$ is:

1

Among the statements: (S1): If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle whose orthocentre is $(0,0)$, the…

Topic: JEE Main 2026 (22 January Evening Shift)

2

If y = y(x) satisfies the differential equation 16(√x + 9√x)(4 + √9 + √x) cos y dy = (1 + 2 sin y) dx, x>0 and y(256)=π…

Topic: JEE Main 2026 (22 January Evening Shift)

3

If lim(x→0) [e^{(a-1)x} + 2cos(bx) + (c-2)e^{-x}] / [x cos x - log_e(1+x)] = 2, then a^2 + b^2 + c^2 is equal to:

Topic: JEE Main 2026 (22 January Evening Shift)

4

Let S = {z ∈ C : 4z^2 + z = 0}. Then Σ|z|^2 (z ∈ S) is equal to:

Topic: JEE Main 2026 (22 January Evening Shift)

5

The number of elements in the relation R = {(x,y): 4x^2 + y^2 < 52, x, y ∈ Z} is:

Topic: JEE Main 2026 (22 January Evening Shift)

6

If the mean deviation about the median of the numbers k, 2k, 3k, ..., 100k is 500, then k^2 is equal to:

Topic: JEE Main 2026 (22 January Evening Shift)

7

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6, x + (n-2)y + (n+…

Topic: JEE Main 2026 (22 January Evening Shift)

1

If $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$ is a solution of $AX=B$, where $\text{adj }A=\begin{bmatrix}4&2&2\\-5&0&5\\1…

Topic: JEE Main 2026 (22 January Evening Shift)

2

Let $L$ be the line $\dfrac{x+1}{2}=\dfrac{y+1}{3}=\dfrac{z+3}{6}$ and let $S$ be the set of all points $(a,b,c)$ on $L…

Topic: JEE Main 2026 (22 January Evening Shift)

3

Let $P(10,2\sqrt{15})$ be a point on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, whose foci are $S$ and $S'$. …

Topic: JEE Main 2026 (22 January Evening Shift)

4

The area of the region $A=\{(x,y):4x^2+y^2\le 8 \text{ and } y^2\le 4x\}$ is :

Topic: JEE Main 2026 (22 January Evening Shift)

5

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)

6

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)

7

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)

8

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)

9

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)

10

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)