Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements
(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.
(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi}{12}$
Previous 10 Questions — JEE Main 2025 (29 January Morning Shift)
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If $z = \dfrac{1}{2} - 2i$ is such that $|z + 1| = \alpha z + \beta (1 + i)$, $i = \sqrt{-1}$ and $\alpha, \beta \in \m…
Topic: JEE Main 2025 (29 January Morning Shift)
Let $\mathrm{A}$ be a square matrix such that $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$. Then $\frac{1}{2} A\left[\left(A+A…
Topic: JEE Main 2025 (29 January Morning Shift)
For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname…
Topic: JEE Main 2025 (29 January Morning Shift)
In a $\triangle ABC$, suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC…
Topic: JEE Main 2025 (29 January Morning Shift)
Let $O$ be the origin and the position vectors of $A$ and $B$ be
$\vec{A} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{B}…
Topic: JEE Main 2025 (29 January Morning Shift)
If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\b…
Topic: JEE Main 2025 (29 January Morning Shift)
Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a,b) R (c,d)$ if and only if $ad - bc$ is divisibl…
Topic: JEE Main 2025 (29 January Morning Shift)
$\text { Let } A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & \alpha & \beta \\
0 & \beta & \alpha
\end{array}\right] \tex…
Topic: JEE Main 2025 (29 January Morning Shift)
In an A.P., the sixth term $a_6 = 2$.
If the product $a_1 a_4 a_9$ is the greatest, then the common difference of the A…
Topic: JEE Main 2025 (29 January Morning Shift)
A fair die is thrown until $2$ appears. Then, the probability that $2$ appears in an even number of throws is:
Topic: JEE Main 2025 (29 January Morning Shift)
Next 10 Questions — JEE Main 2025 (29 January Morning Shift)
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Let $P Q R$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $\mathrm{PQ}$. The distance of the c…
Topic: JEE Main 2025 (29 January Morning Shift)
$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits…
Topic: JEE Main 2025 (29 January Morning Shift)
A function $y=f(x)$ satisfies $f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. …
Topic: JEE Main 2025 (29 January Morning Shift)
If
$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+x^2}+\dfrac{1+\sin^2 x}{1+e^{\sin 2x}}…
Topic: JEE Main 2025 (29 January Morning Shift)
