1
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$f(x) = 2x + {\tan ^{ - 1}}x$$ and $$g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$$. Then

A
there exists $$\widehat x \in [0,3]$$ such that $$f'(\widehat x) < g'(\widehat x)$$
B
there exist $$0 < {x_1} < {x_2} < 3$$ such that $$f(x) < g(x),\forall x \in ({x_1},{x_2})$$
C
$$\min f'(x) = 1 + \max g'(x)$$
D
$$\max f(x) > \max g(x)$$
2
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let $$y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$$. Then, at x = 1,

A
$$2 y^{\prime}+\sqrt{3} \pi^{2} y=0$$
B
$$y^{\prime}+3 \pi^{2} y=0$$
C
$$\sqrt{2} y^{\prime}-3 \pi^{2} y=0$$
D
$$2 y^{\prime}+3 \pi^{2} y=0$$
3
JEE Main 2023 (Online) 29th January Evening Shift
+4
-1

Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that

$$f''(x)=g''(x)+6x$$

$$f'(1)=4g'(1)-3=9$$

$$f(2)=3g(2)=12$$.

Then which of the following is NOT true?

A
$$g(-2)-f(-2)=20$$
B
There exists $$x_0\in(1,3/2)$$ such that $$f(x_0)=g(x_0)$$
C
$$|f'(x)-g'(x)| < 6\Rightarrow -1 < x < 1$$
D
If $$-1 < x < 2$$, then $$|f(x)-g(x)| < 8$$
4
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

Let $$y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})$$. Then $$y' - y''$$ at $$x = - 1$$ is equal to

A
496
B
976
C
464
D
944
EXAM MAP
Medical
NEET