SATHEE: Vectors 2 Question 11

Vectors 2 Question 11

11. Let $△ P Q R$ be a triangle. Let $a = Q R, b = R P$ and $c = P Q$ . If $| a | = 12, | b | = 4 \sqrt{3}$ and $b \cdot c = 24$ , then which of the following is/are true?

(2015 Adv.)

(a) $\frac{| c |^{2}}{2} - | a | = 12$

(b) $\frac{| c |^{2}}{2} + | a | = 30$

(d) $a \cdot b = - 72$

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Answer:

Correct Answer: 11. (c)

Solution:

Given, $| a | = 12, | b | = 4 \sqrt{3}$

$\begin{aligned} a + b + c = 0 \\ \Rightarrow a = - (b + c) \\ We have, | a |^{2} = | b + c |^{2} \\ \Rightarrow | a |^{2} = | b |^{2} + | c |^{2} + 2 b \cdot c \\ \Rightarrow 144 = 48 + | c |^{2} + 48 \\ \Rightarrow | c |^{2} = 48 \\ \Rightarrow | c | = 4 \sqrt{3} \\ Also, | c |^{2} = | a |^{2} + | b |^{2} + 2 a \cdot b \\ \Rightarrow 48 = 144 + 48 + 2 a \cdot b \\ \Rightarrow a \cdot b = - 72 \end{aligned}$

$∴$ Option (d) is correct.

Also,

$a \times b = c \times a$

$\Rightarrow a \times b + c \times a = 2 a \times b$

$\Rightarrow | a \times b + c \times a | = 2 | a \times b | = 2 \sqrt{| a |^{2} | b |^{2} - (a \cdot b)^{2}}$

$= 2 \sqrt{(144) (48) - (- 72)^{2}}$

$= 2 (12) \sqrt{48 - 36} = 48 \sqrt{3}$

$∴$ Option (c) is correct.

Also, $\frac{| c |^{2}}{2} - | a | = 24 - 12 = 12$

$∴$ Option (a) is correct.

and

$\frac{| c |^{2}}{2} + | a | = 24 + 12 = 36$

$∴$ Option (b) is not correct.

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Vectors 2 Question 11

11. Let $△ P Q R$ be a triangle. Let $a = Q R, b = R P$ and $c = P Q$ . If $| a | = 12, | b | = 4 \sqrt{3}$ and $b \cdot c = 24$ , then which of the following is/are true?

Answer:

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Vectors 2 Question 11

11. Let △PQR be a triangle. Let a=QR,b=RP and c=PQ. If |a|=12,|b|=43 and b⋅c=24, then which of the following is/are true?

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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11. Let $△ P Q R$ be a triangle. Let $a = Q R, b = R P$ and $c = P Q$ . If $| a | = 12, | b | = 4 \sqrt{3}$ and $b \cdot c = 24$ , then which of the following is/are true?