If $\displaystyle \int_{0}^{\pi/3}\!\cos^{4}x\,dx=a\pi+b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to:

1

Let the system of equations $x+2y+3z=5,\quad 2x+3y+z=9,\quad 4x+3y+\lambda z=\mu$ have infinite number of solutions. T…

Topic: JEE Main 2024 (1 February Evening Shift)

2

Consider the relations $R_1$ and $R_2$ defined as $a\,R_1\,b \iff a^2 + b^2 = 1$ for all $a,b\in\mathbb{R}$, and …

Topic: JEE Main 2024 (1 February Evening Shift)

3

Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets …

Topic: JEE Main 2024 (1 February Evening Shift)

4

If $z$ is a complex number such that $|z|\le1$, then the minimum value of $\left|z+\dfrac{1}{2}(3+4i)\right|$ is:

Topic: JEE Main 2024 (1 February Evening Shift)

5

If the domain of the function $f(x)=\sqrt{\dfrac{x^{2}-25}{4-x^{2}}}+\log_{10}(x^{2}+2x-15)$ is $(-\infty,\alpha)\c…

Topic: JEE Main 2024 (1 February Evening Shift)

1

Let $\alpha$ and $\beta$ be the roots of the equation $p x^{2}+q x-r=0$, where $p\ne 0$. If $p,q,r$ are consecutive ter…

Topic: JEE Main 2024 (1 February Evening Shift)

2

Let Ajay will not appear in JEE exam with probability $p=\dfrac{2}{7}$, while both Ajay and Vijay will appear in the e…

Topic: JEE Main 2024 (1 February Evening Shift)

3

Let $P$ be a point on the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$. Let the line passing through $P$ and parallel …

Topic: JEE Main 2024 (1 February Evening Shift)

4

Consider $10$ observations $x_{1},x_{2},\ldots,x_{10}$ such that $\displaystyle \sum_{i=1}^{10}(x_{i}-\alpha)=2$ and $\…

Topic: JEE Main 2024 (1 February Evening Shift)

5

Let $f(x)=\left|2x^{2}+5|x|-3\right|,\; x\in\mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not …

Topic: JEE Main 2024 (1 February Evening Shift)

6

The number of solutions of the equation $4\sin^{2}x-4\cos^{3}x+9-4\cos x=0,\; x\in[-2\pi,2\pi]$ is:

Topic: JEE Main 2024 (1 February Evening Shift)

7

Let the locus of the midpoints of the chords of the circle $x^{2}+(y-1)^{2}=1$ drawn from the origin intersect the line…

Topic: JEE Main 2024 (1 February Evening Shift)

8

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbf{R}\to\mathbf{R}$ is a differentiable function such that $f…

Topic: JEE Main 2024 (1 February Evening Shift)

9

Let $P$ and $Q$ be the points on the line $\dfrac{x+3}{8}=\dfrac{y-4}{2}=\dfrac{z+1}{2}$ which are at a distance of $6$…

Topic: JEE Main 2024 (1 February Evening Shift)

10

The value of $\int_{0}^{1} (2x^{3} - 3x^{2} - x + 1)^{\frac{1}{3}} \, dx$ is equal to:

Topic: JEE Main 2024 (1 February Evening Shift)