For i=1,2,3 and j=1,2,3. If ai2+bi2+ci2=1, aiaj+bibj+cicj=0,∀i≠j and A=a1a2a3b1b2b3c1c2c3 then detAAT=

AP EAMCET · Maths · Determinants

For $i = 1, 2, 3$ and $j = 1, 2, 3$ . If $a_{i}^{2} + b_{i}^{2} + c_{i}^{2} = 1, a_{i} a_{j} + b_{i} b_{j} + c_{i} c_{j} = 0, \forall i \neq j$ and $A = [\begin{array}{l} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}]$ then $det (A A^{T}) =$

A $0$
B $1$
C $- 1$
D $3$

Verified Solution

Answer & Solution

Correct Answer

(B) $1$

Step-by-step Solution

Detailed explanation

We know AAT=AAT=A2 i.e. a1a2a3b1b2b3c1c2c32=a1b1c1a2b2c2a3b3c3a1b1c1a2b2c2a3b3c3 =Σa12Σa1a2Σa1a3Σa2a1Σa22Σa2a3Σa3a1Σa3a2Σa32=100010001 =1

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match the ranges of the functions given in List - I with those of the items given in List - II

List - I	List - II
(I) $3 \sin ^2 x+4 \cos ^2 x-2$	(a) $\left[\frac{1}{4}, 1\right]$
(II) $\cos ^2 x+\sin ^4 x$	(b) $\left[-\frac{1}{4}, \frac{1}{4}\right]$
(III) $\sin ^6 x+\cos ^6 x$	(c) $[1,2]$
(IV) $\cos x \cos \left(\frac{2 \pi}{3}+x\right) \cos \left(\frac{2 \pi}{3}-x\right)$	(d) $\left[\frac{3}{4}, 1\right]$
	(e) $[0,1]$

AP EAMCET 2023 Medium

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List - I	List - II
(I) \(3 \sin ^2 x+4 \cos ^2 x-2\)	(a) \(\left[\frac{1}{4}, 1\right]\)
(II) \(\cos ^2 x+\sin ^4 x\)	(b) \(\left[-\frac{1}{4}, \frac{1}{4}\right]\)
(III) \(\sin ^6 x+\cos ^6 x\)	(c) \([1,2]\)
(IV) \(\cos x \cos \left(\frac{2 \pi}{3}+x\right) \cos \left(\frac{2 \pi}{3}-x\right)\)	(d) \(\left[\frac{3}{4}, 1\right]\)
	(e) \([0,1]\)