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← JEE Advanced 2023 · Paper 2
MathematicsSingle correct+3 / -1Question 4 of 51

JEE Advanced 2023 · Paper 2, Mathematics question 4

Let the position vectors of the points P,Q,RP, Q, R and SS be a⃗=i^+2j^−5k^\vec a = \hat i + 2\hat j - 5\hat k, b⃗=3i^+6j^+3k^\vec b = 3\hat i + 6\hat j + 3\hat k, c⃗=175i^+165j^+7k^\vec c = \dfrac{17}{5}\hat i + \dfrac{16}{5}\hat j + 7\hat k and d⃗=2i^+j^+k^\vec d = 2\hat i + \hat j + \hat k, respectively. Then which of the following statements is true?

Choose one option
Answer and worked solution

Official answer: B

(B) and (C).
b⃗+2d⃗3=(73,83,53).\frac{\vec b+2\vec d}{3}=\left(\frac73,\frac83,\frac53\right).
The point dividing PRPR internally in the ratio 5:45:4 is
5c⃗+4a⃗9=(17+49,16+89,35−209)=(73,83,53).\frac{5\vec c+4\vec a}{9}=\left(\frac{17+4}9,\frac{16+8}9,\frac{35-20}9\right)=\left(\frac73,\frac83,\frac53\right).
They agree, so (B) is true and (C) is false.

(A) False.

  • b⃗−a⃗=(2,4,8)\vec b-\vec a=(2,4,8), c⃗−a⃗=(125,65,12)\vec c-\vec a=\left(\frac{12}5,\frac65,12\right), d⃗−a⃗=(1,−1,6)\vec d-\vec a=(1,-1,6).
  • (c⃗−a⃗)×(d⃗−a⃗)=(965,−125,−185)(\vec c-\vec a)\times(\vec d-\vec a)=\left(\frac{96}5,-\frac{12}5,-\frac{18}5\right).
  • Dotting with b⃗−a⃗\vec b-\vec a gives 192−48−1445=0\frac{192-48-144}{5}=0, so the points are coplanar.

(D) False. b⃗×d⃗=(3,3,−9)\vec b\times\vec d=(3,3,-9), so ∣b⃗×d⃗∣2=99|\vec b\times\vec d|^2=99, not 95.

Topics:Scalar and vector triple productsCross productDistance and section formulae
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