Let the position vectors of the points A , B , C and D be 5 i+5 j… - Vidyadip

Question

Let the position vectors of the points $A , B , C$ and $D$ be $5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$ and $-\hat{ i }+5 \hat{ j }+6 \hat{ k }$. Let the set $S =\{\lambda \in R$ : The points $A$, $B , C$ and D are coplanar $\}$. Then $\sum_{\lambda \in S}(\lambda+2)^2$ is equal to

✓

Answer

a
Since $A, B, C, D$ are coplanner

Hence $\left[\begin{array}{lll}\overrightarrow{ BA } & \overrightarrow{ CA } & \overrightarrow{ DA }\end{array}\right]=0$

$\begin{aligned}& \left|\begin{array}{ccc}4 & 3 & 2 \lambda-3 \\7 & 5-\lambda & 2 \lambda-4 \\6 & 0 & 2 \lambda-6\end{array}\right|=0 \\& \lambda=2,3 \text { Hence } \sum_{\lambda \in S}(\lambda+2)^2=41\end{aligned}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $M.D.$ is $12$, the value of $S.D.$ will be

Remainder when $ 64^{32^{32}}$ is divided by $9$ is equal to .........................

The value of $x = \sqrt {2 + \sqrt {2 + \sqrt {2 + .....} } } $is

The arithmetic mean of the nine numbers in the given set $\{9,99,999,...., 999999999\}$ is a $9$ digit number $N$, all whose digits are distinct. The number $N$ does not contain the digit

Let ${a_1},{a_2},\;.\;.\;.\;.,{a_{49}}$ be in $A.P.$ such that $\mathop \sum \limits_{k = 0}^{12} {a_{4k + 1}} = 416$ and ${a_9} + {a_{43}} = 66$. If $a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m,$ then $m = \;\;..\;.\;.\;.\;$

If $|cos\ x + sin\ x| + |cos\ x\ -\ sin\ x| = 2\ sin\ x$ ; $x \in [0,2 \pi ]$ , then maximum integral value of $x$ is

Let $\vec{b}=\hat{i}+\hat{j}+\lambda \hat{k}, \lambda \in R$. If $\vec{a}$ is a vector such that $\overrightarrow{ a } \times \overrightarrow{ b }=13 \hat{ i }-\hat{ j }-4 \hat{ k } \quad$ and $\quad \overrightarrow{ a } \cdot \overrightarrow{ b }+21=0$, then $(\vec{b}-\vec{a}) \cdot(\hat{k}-\hat{j})+(\vec{b}+\vec{a}) \cdot(\hat{i}-\hat{k})$ is equal to

The area of the triangle formed by joining the origin to the points of intersection of the line $x\sqrt 5 + 2y = 3\sqrt 5 $ and circle ${x^2} + {y^2} = 10$ is

Let $A=\sum_{i=1}^{10} \sum_{j=1}^{10} \min \{i, j\}$ and $B=\sum_{i=1}^{10} \sum_{j=1}^{10}\max \{i, j\}$. Then $A+B$ is equal to

Give the number of common tangents to circle ${x^2} + {y^2} + 2x + 8y - 23 = 0$ and ${x^2} + {y^2} - 4x - 10y + 9 = 0$