Q. Let $\vec{a} = 2\hat{i}+\hat{k}$ and $\vec{b} = \hat{i}+2\hat{j}+\hat{k}$ be two vectors. Consider a vector $\vec{c} = \vec{\alpha}a + \beta\vec{b}, \alpha, \beta \in ℝ$ If the projection of $\vec{c}$ on the vector $\left(\vec{a}+\vec{b}\right)$ is $3\sqrt{2}$, then the minimum value of $\left(\vec{c}-\left(\vec{a}\times\vec{b}\right)\right)\cdot\vec{c}$ equals

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A

B

C

D

Solution:

$\vec{c}=\left(2\alpha+\beta\right)\hat{i}+\hat{j}\left(\alpha+2\beta\right)+\hat{k}\left(\beta-\alpha\right)$
$\frac{\vec{c}.\left(\vec{a}+\vec{b}\right)}{\left|\vec{a}+\vec{b}\right|}=3\sqrt{2}$
$\Rightarrow \alpha+\beta=2 .......... \left(1\right)$
$\left(\vec{c}-\left(\vec{a}\times\vec{b}\right)\right).\left(\alpha\vec{a}+\beta\vec{b}\right)$
$=\left|\vec{c}\right|^{2}=\alpha^{2}\left|\vec{a}\right|^{2}+\beta^{2}\left|b\right|^{2}+2\alpha\beta\left(\vec{a}.\vec{b}\right)$
$=6\left(\alpha^{2}+\left(2-\alpha\right)^{2}+\alpha\left(2-\alpha\right)\right)$
$=6\left(\alpha^{2}+\beta^{2}+\alpha\beta\right)$
$=6\left(\left(\alpha-1\right)^{2}+3\right)$
$\Rightarrow $ Min. value $= 18$

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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