Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}> 0$ and $\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is

1

If the system of equations $x+4y-z=\lambda,; 7x+9y+\mu z=-3,; 5x+y+2z=-1$ has infinitely many solutions, then $(2\mu+3\…

Topic: JEE Main 2024 (8 April Evening Shift)

2

Let $\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^\alpha$…

Topic: JEE Main 2024 (8 April Evening Shift)

3

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathr…

Topic: JEE Main 2024 (8 April Evening Shift)

4

If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the a…

Topic: JEE Main 2024 (8 April Evening Shift)

5

Let $\vec{a}=4\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=11\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{…

Topic: JEE Main 2024 (8 April Evening Shift)

6

The area of the region in the first quadrant inside the circle $x^{2}+y^{2}=8$ and outside the parabola $y^{2}=2x$ is e…

Topic: JEE Main 2024 (8 April Evening Shift)

7

There are three bags $X,Y,Z$. Bag $X$ contains $5$ one-rupee coins and $4$ five-rupee coins; Bag $Y$ contains $4$ one-r…

Topic: JEE Main 2024 (8 April Evening Shift)

8

If the shortest distance between the lines $\dfrac{x-\lambda}{2}=\dfrac{y-4}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-2}{4}=\df…

Topic: JEE Main 2024 (8 April Evening Shift)

9

Let $y=y(x)$ be the solution curve of the differential equation $\sec y,\dfrac{dy}{dx}+2x\sin y=x^{3}\cos y$, with $y(1…

Topic: JEE Main 2024 (8 April Evening Shift)

1

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\dfrac{70}{3}$ and …

Topic: JEE Main 2024 (8 April Evening Shift)

2

$\mathrm{a}, \mathrm{b}>0$, let $f(x)= \begin{cases}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x< 0 \\ 3, &…

Topic: JEE Main 2024 (8 April Evening Shift)

3

Let $A={2,3,6,8,9,11}$ and $B={1,4,5,10,15}$. Let $R$ be a relation on $A\times B$ defined by ( ? , ? ) ? ( ? , ? )   …

Topic: JEE Main 2024 (8 April Evening Shift)

4

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-\hat{j}+\lambda\hat…

Topic: JEE Main 2024 (8 April Evening Shift)

5

The sum of all possible values of $\theta\in[-\pi,2\pi]$, for which $\dfrac{1+i\cos\theta}{1-2i\cos\theta}$ is purely i…

Topic: JEE Main 2024 (8 April Evening Shift)

6

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets …

Topic: JEE Main 2024 (8 April Evening Shift)

7

If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1,;a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alph…

Topic: JEE Main 2024 (8 April Evening Shift)

8

If the value of $\dfrac{5\cos36^{\circ}+5\sin18^{\circ}}{5\cos36^{\circ}-3\sin18^{\circ}}$ is $\dfrac{a\sqrt{5}-b}{c}$,…

Topic: JEE Main 2024 (8 April Evening Shift)

9

If the image of the point $(-4,5)$ in the line $x+2y=2$ lies on the circle $(x+4)^{2}+(y-3)^{2}=r^{2}$, then $r$ is equ…

Topic: JEE Main 2024 (8 April Evening Shift)

10

If the term independent of $x$ in the expansion of $\left(\sqrt{a},x^{2}+\dfrac{1}{2x^{3}}\right)^{10}$ is $105$, then …

Topic: JEE Main 2024 (8 April Evening Shift)