Let `A=[{:(,sin theta,1sqrt2),(,-1//sqrt2,cos theta),(,cos theta,tan theta):}]& B=[{:(,1sqrt2,sin theta),(,cos theta,cos theta),(,cos theta,-1):}]`. Find theta so that A=B
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Prove the orthogonal matrices of order two are of the form `[(cos theta,-sin theta),(sin theta,cos theta)]` or `[(cos theta,sin theta),(sin theta,-cos
Let matrix `A=[(a,b),(c,d)]` is orthogonal matrix. `:. [(a,b),(c,d)][(a,c),(b,d)]=[(1,0),(0,1)]` `implies [(a^(2)+b^(2),ac+bd),(ac+bd,c^(2)+d^(2))]=[(1,0),(0,1)]` `implies a^(2)+b^(2)=1` (1) `c^(2)+d^(2)=1` (2) `ac+bd=0` (3) `implies a/d= (-b)/c= k` (let) `implies c^(2)+d^(2)=(a^(2)+b^(2))/k=1//k^(2)` or `k^(2)=1` or `k= pm 1`...
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If `theta-phi=pi/2,` prove that, `[(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,
Correct Answer - C `AB=[(cos^(2) theta,),(cos theta sin theta,)][(cos^(2) phi,cos phi sin phi),(cos phi sin phi,sin^(2) phi)]` `=[(cos^(2) theta cos^(2) phi+cos theta cos phi sin theta sin phi ,cos^(2)theta cos phi...
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If `[(1,-tan theta),(tan theta,1)][(1,tan theta),(-tan theta,1)]^(-1)=[(a,-b),(b,a)],` then
We have, `[(1,tan theta),(-tan theta,1)]^(-1) =1/(1+ tan^(2) theta) [(1,- tan theta),(tan theta,1)]` `implies [(a,-b),(b,a)]=cos^(2) theta [(1,-tan theta),(tan theta,1)][(1,-tan theta),(tan theta,1)]` `=cos^(2) theta [(1-tan^(2) theta,-2 tan theta),(2 tan theta,1-tan^(2) theta)]` `=[(cos...
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Find the principal values of the following (i) `sin^(-1) (sin 1)` (ii) `sin^(-1) (sin 2)` (iii) `sin^(-1) (sin 10)` (iv) `sin^(-1) (sin 20)` (v) `sin^
(i) Since `1 in [-pi//2, pi//2], sin^(-1) (sin 1) = 1` (ii) Since `2 in [-pi//2, pi//2], sin^(-1) (sin 2) != 2` `:. Sin^(-1) (sin 2) = sin^(-1) (sin (pi)...
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Find the following values : (a) `tan^(1) tan.(13pi)/(5)` (b) `sec^(-1) sec.(13pi)/(3)` (c) `sin^(-1) sin.(33pi)/(5)` (d) `sin^(-1) (sin 8)` (e) `tan^(
Correct Answer - (a) `-(2pi)/(5)` (b) `(pi)/(3)` (c) `(2pi)/(5)` (d) `3 pi -8` (e) `10 - 3 pi` (f) `9 - 2pi` (g) `6 - pi` (h) `7 - 2pi` (a)...
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If equation `sin^(-1) (4 sin^(20 theta + sin theta) + cos^(-1) (6 sin theta - 1) = (pi)/(2)` has 10 solution for `theta in [0, n pi]`, then find the m
Correct Answer - 9 We must have `4 sin^(2) theta + sin theta = 6 sin theta -1` `rArr 4 sin^(2) theta - 5 sin theta + 1 = 0` `rArr...
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If `u=cot^-1 sqrt(tanalpha)-tan^-1 sqrt(tan alpha),` then `tan(pi/4-u/2)` is equal to (a) `sqrt(tan alpha)` (b) `sqrt(cos alpha)` (c) `tan alpha` (d)
Correct Answer - A Let `sqrt(tan alpha) = tan x`. Then `u = cot^(-1) (tan x) - tan^(-1) (tan x)` `= (pi)/(2) - x -x = (pi)/(2) - 2x` or `2x...
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`theta = tan^(-1) (2 tan^(2) theta) - tan^(-1) ((1)/(3) tan theta) " then " tan theta=`
Correct Answer - A `theta = tan^(-1) (2 tan^(2) theta) - tan^(-1) ((1)/(3) tan theta) = tan^(-1).(2 tan^(2) theta - (1)/(3) tan theta)/(1 + (1)/(3) tan^(3) theta)` `rArr tan theta =...
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सरल कीजिए, ` cos theta [{:( cos theta,, sin theta ), (-sin theta,, cos theta ):}] + sin theta [ {:(sin theta ,, - cos theta ), (cos theta ,, sin theta
Correct Answer - `[{:(,1,0),(,1,1):}]`
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15. Which of the following is correct? (1) \( \tan 1>\tan 2 \) (2) \( \tan 2>\tan 1 \) (3) \( \sin 1\cos 2 \)
Answer: (1) tan1>tan2
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