JEE Advanced 2022 Paper 1 Online

Let \(P_{1}\) and \(P_{2}\) be two planes given by

$$ \begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\\\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned} $$

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie ...

Let \(S\) be the reflection of a point \(Q\) with respect to the plane given by

\(\vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k}\)

where \(t, p\) are real parameters and \(\hat{\imath}, \hat{\jmath}, \hat{k}\) are the unit vec...

Let \(A B C\) be the triangle with \(A B=1, A C=3\) and \(\angle B A C=\frac{\pi}{2}\). If a circle of radius \(r>0\) touches the sides \(A B, A C\) and also touches internally the circumcircle of the triangle \(A B C\), then the value of $...

Let \(z\) be a complex number with a non-zero imaginary part. If

\(\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}\)

is a real number, then the value of \(|z|^{2}\) is _________.

Let \(\bar{z}\) denote the complex conjugate of a complex number \(z\) and let \(i=\sqrt{-1}\). In the set of complex numbers, the number of distinct roots of the equation

\(\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)\)

is _________.

Consider the equation
\(\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)\)

Which of the following statements is/ar...

Consider the ellipse

\($ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1\)$

Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respe...

Let \(|M|\) denote the determinant of a square matrix \(M\). Let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}\) be the function defined by
\(g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}\)
where
$$
f...

Considering only the principal values of the inverse trigonometric functions, the value of

\(\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^{2}}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^{2}}+\tan ^{-1} \frac{\sqrt{2}}{\pi}\)

is

Let \(\alpha\) be a positive real number. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g:(\alpha, \infty) \rightarrow \mathbb{R}\) be the functions defined by

$$
f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=...

Let \(p, q, r\) be nonzero real numbers that are, respectively, the \(10^{\text {th }}, 100^{\text {th }}\) and \(1000^{\text {th }}\) terms of a harmonic progression. Consider the system of linear equations
$$$
\begin{gathered}
x+y+z=1 \\...

Consider the parabola \(y^{2}=4 x\). Let \(S\) be the focus of the parabola. A pair of tangents drawn to the parabola from the point \(P=(-2,1)\) meet the parabola at \(P_{1}\) and \(P_{2}\). Let \(Q_{1}\) and \(Q_{2}\) be points on the lin...

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits \(0,2,3,4,6,7\) is _________.

In a study about a pandemic, data of 900 persons was collected. It was found that

190 persons had symptom of fever,

220 persons had symptom of cough,

220 persons had symptom of breathing problem,

330 persons had symptom of fever or coug...

Two players, \(P_{1}\) and \(P_{2}\), play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let \(x\) and \(y\) denote the readings on the di...

Let \(l_{1}, l_{2}, \ldots, l_{100}\) be consecutive terms of an arithmetic progression with common difference \(d_{1}\), and let \(w_{1}, w_{2}, \ldots, w_{100}\) be consecutive terms of another arithmetic progression with common differenc...

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an arithmetic progression with \(a_{1}=7\) and common difference 8. Let \(T_{1}, T_{2}, T_{3}, \ldots\) be such that \(T_{1}=3\) and \(T_{n+1}-T_{n}=a_{n}\) for \(n \geq 1\). Then, which of the followi...

Consider the following lists :

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.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:norm...