$a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$ be differentiable on $\mathbb{R}$. Then, the value of $\int_\limits{-2}^2 f(x) d x$ equals

1

If $x^{2}-y^{2}+2hxy+2gx+2fy+c=0$ is the locus of a point which is always equidistant from the lines $x+2y+7=0$ and $2x…

Topic: JEE Main 2024 (30 January Evening Shift)

2

Let $R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix…

Topic: JEE Main 2024 (30 January Evening Shift)

3

Let $f:\mathbb{R}\setminus{0}\to\mathbb{R}$ satisfy $f!\left(\dfrac{x}{y}\right)=\dfrac{f(x)}{f(y)}$ for all $x,y$ with…

Topic: JEE Main 2024 (30 January Evening Shift)

4

If $z$ is a complex number, then the number of common roots of $z^{1985}+z^{100}+1=0$ and $z^{3}+2z^{2}+2z+1=0$ is equa…

Topic: JEE Main 2024 (30 January Evening Shift)

5

For $\alpha,\beta\in(0,\dfrac{\pi}{2})$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such th…

Topic: JEE Main 2024 (30 January Evening Shift)

6

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and …

Topic: JEE Main 2024 (30 January Evening Shift)

1

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\l…

Topic: JEE Main 2024 (30 January Evening Shift)

2

Let $L_1:\ \vec r=(\hat i-\hat j+2\hat k)+\lambda(\hat i-\hat j+2\hat k),\ \lambda\in\mathbb R,$ $L_2:\ \vec r=(\hat …

Topic: JEE Main 2024 (30 January Evening Shift)

3

Let $\vec a=\hat i+\alpha\hat j+\beta\hat k,\ \alpha,\beta\in\mathbb R$. Let $\vec b$ be such that the angle between …

Topic: JEE Main 2024 (30 January Evening Shift)

4

Let $P$ be a point on the hyperbola $H:\ \dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, in the first quadrant, such that the area of…

Topic: JEE Main 2024 (30 January Evening Shift)

5

Let $f(x)=(x+3)^2(x-2)^3,\ x\in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$ respectively in $[-4,4…

Topic: JEE Main 2024 (30 January Evening Shift)

6

Suppose $2-p,\ p,\ 2-\alpha,\ \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. The…

Topic: JEE Main 2024 (30 January Evening Shift)

7

Consider the system of linear equations $x+y+z=5,\quad x+2y+\lambda^2 z=9,\quad x+3y+\lambda z=\mu,$ where $\lambda,\mu…

Topic: JEE Main 2024 (30 January Evening Shift)

8

Let $\vec a$ and $\vec b$ be two vectors such that $|\vec b|=1$ and $|\vec b\times\vec a|=2$. Then $|(\vec b\times\vec …

Topic: JEE Main 2024 (30 January Evening Shift)

9

If the domain of the function $f(x)=\log_e\!\left(\frac{2x+3}{4x^{2}+x-3}\right)+\cos^{-1}\!\left(\frac{2x-1}{x+2}\righ…

Topic: JEE Main 2024 (30 January Evening Shift)

10

Bag A contains $3$ white and $7$ red balls; Bag B contains $3$ white and $2$ red balls. One bag is selected at random a…

Topic: JEE Main 2024 (30 January Evening Shift)