1. Tardigrade
  2. Question
  3. Mathematics
  4. Three lines L1: vecr = λ hati, λ ∈ ℝ L2: vecr = hatk + μ hatj, μ ∈ ℝ and L3: vecr = hati + hatj + v hatk, v ∈ ℝ are given. For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?

Q. Three lines
$L_{1} : \vec{r} = \lambda\hat{i},\, \lambda\,\in\,ℝ$
$L_{2} : \vec{r} = \hat{k} + \mu\hat{j},\,\mu\,\in\,ℝ $ and
$L_{3} : \vec{r} = \hat{i} + \hat{j} + v\hat{k}, \,v \,\in\,ℝ $
are given. For which point(s) $Q$ on $L_{2}$ can we find a point P on $L_{1}$ and a point R on $L_{3}$ so that P, Q and R are collinear?

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Solution:

Let $P\left(\lambda, 0, 0\right), Q\left(0, \mu, 1\right), R\left(1, 1, v\right)$ be points. $L_1, L_2$ and $L_3$ respectively
Since $P, Q, R$ are collinear, $\overrightarrow{PQ}$ iscollinear with $\overrightarrow{QR}$
Hence $=\frac{-\lambda}{1}=\frac{\mu}{1-\mu}=\frac{1}{v-1}$
For every $\mu\,\in\,R-\left\{0, 1\right\}$ there exist unique $\lambda, v\,\in\,R$
Hence Q cannot have coordinates $(0, 1, 1)$ and $(0, 0, 1).$

Questions from JEE Advanced 2019

1. Let $f\,\mathbb{R} \to\mathbb{R}$ be given by
$f(x) = \begin{cases} x^5+5x^4+10x^3+10x^2+3x+1, & \quad \text{} x < 0 \\ x^2-x+1, & \quad \text{ } 0 \le x < 1; \\ \frac{2}{3}x^{3}-4x^{2}+7x-\frac{8}{3}, & \quad \text{} 1 \le x< 3;\\ (x - 2) log_e(x - 2) - x +\frac{10}{3}, & \quad \text{ } x \ge 3. \end{cases} $
Then which of the following options is/are correct?

2. Let f : ℝ $/to$ ℝ be given by f(x) = (x - 1)(x - 2)(x - 5). Define
$F\left(x\right) = \int\limits^{x}_{0} f\left(t\right) dt, x > 0.$
Then which of the following options is/are correct?

3. Three lines
$L_{1} : \vec{r} = \lambda\hat{i},\, \lambda\,\in\,ℝ$
$L_{2} : \vec{r} = \hat{k} + \mu\hat{j},\,\mu\,\in\,ℝ $ and
$L_{3} : \vec{r} = \hat{i} + \hat{j} + v\hat{k}, \,v \,\in\,ℝ $
are given. For which point(s) $Q$ on $L_{2}$ can we find a point P on $L_{1}$ and a point R on $L_{3}$ so that P, Q and R are collinear?

4. Let f(x) = sin($\pi$ cos x) and g(x) = cos(2 $\pi$ sin a) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:
X = {x : f'(x) — 0}, $\quad$ Y = {x : f(x) = 0},
Z = [x : g(x) = 0},$\quad$ W = [x : g'(x) = 0}.
List -1 contains the sets X, Y, Z and W. List - II contains some information regarding these sets.
$\quad$$\quad$$\quad$List-I List-II
(I) X (p) ⊇ $\left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$
(II) Y (Q) an arithmetic progression
(III) Z (R) NOT an arithmetic progression
(IV) W (S) ⊇ $\left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6} \right\}$
(T) ⊇ $\left\{\frac{\pi}{2}, \frac{2\pi}{2}, \pi \right\}$
(U) ⊇ $\left\{\frac{\pi}{6}, \frac{3\pi}{4} \right\}$


Question: Which of the following is the only CORRECT combination?

5. Let f(x) = sin($\pi$ cos x) and g(x) = cos(2 $\pi$ sin a) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:
X = {x : f'(x) — 0}, $\,\,$ Y = {x : f(x) = 0},
Z = [x : g(x) = 0},$\,\,$ W = [x : g'(x) = 0}.
List -1 contains the sets X, Y, Z and W. List - II contains some information regarding these sets.
List-I List-II
(I) X (p) ⊇ $\left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$
(II) Y (Q) an arithmetic progression
(III) Z (R) NOT an arithmetic progression
(IV) W (S) ⊇ $\left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6} \right\}$
(T) ⊇ $\left\{\frac{\pi}{2}, \frac{2\pi}{2}, \pi \right\}$
(U) ⊇ $\left\{\frac{\pi}{6}, \frac{3\pi}{4} \right\}$


Question : Wliich of the following is the only CORRECT combination?

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