1
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
$$2 y^{\prime}+3 \pi^{2} y=0$$
2
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 $$|f(x)-g(x)| < 8$$
3
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
4
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

If $$f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$$, then

A
$$2f(0) - f(1) + f(3) = f(2)$$
B
$$f(1) + f(2) + f(3) = f(0)$$
C
$$f(3) - f(2) = f(1)$$
D
$$3f(1) + f(2) = f(3)$$
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