सरल कीजिए, ` cos theta [{:( cos theta,, sin theta ), (-sin theta,, cos theta ):}] + sin theta [ {:(sin theta ,, - cos theta ), (cos theta ,, sin theta ):}]`
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Let `A=[{:(,sin theta,1sqrt2),(,-1//sqrt2,cos theta),(,cos theta,tan theta):}]& B=[{:(,1sqrt2,sin theta),(,cos theta,cos theta),(,cos theta,-1):}]`. F
By defing A & B are equal if they have the same order and all the corresponding elements are equal. Thus we have `sin theta=(1)/(sqrt2),c os theta=-(1)/(sqrt2)& tan theta=-1` `Rightarrow...
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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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Find the characteristic roots of the two-rowed orthogonal matrix `[(cos theta,-sin theta),(sin theta,cos theta)]` and verify that they are of unit mod
We have, `|A-lambdaI|=|(cos theta - lambda,-sin theta),(sin theta,cos theta - lambda)|` `=(cos theta-lambda)^(2)+sin^(2) theta` Therefore, characteristic equation of A is `(cos theta-lambda)^(2)+sin^(2) theta=0` or `cos theta-lambda= pm i sin theta`...
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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 `A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)]`, then which of the following is not true ?
Correct Answer - A::B::C We have, `|A(theta)|=1` Hence, A is invertiable. `A(pi+theta)=A(pi + theta)=[(sin(pi+theta),i cos (pi+theta)),(i cos (pi+theta),sin (pi+theta))]` `=[(-sin theta,-i co theta),(-i cos theta,-sin theta)]=-A(theta)` adj `(A(theta))=[(sin theta,-i cos theta),(-i...
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Find the principal value of the following (i) `cos^(-1) (cos 3)` (ii) `cos^(-1) (cos 4)` (iii) `cos^(-1) (cos 15)` (iv) `cos^(-1) (cos 30)` (v) `cos^(
(i) Since `3 in [0, pi], cos^(-1) (cos 3) = 3` (ii) Since `4 !in [0, pi], cos^(-1) (cos 4) != 4` `:. Cos^(-1) (cos 4) = 2pi - 4`...
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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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(i) `sec(cos^(-1)""(1)/(2))` का मान ज्ञात कीजिए। (ii) समीकरणों `sin^(-1)x+sin^(-1)y=(2pi)/(3)` और `cos^(-1)x-cos^(-1)y=(pi)/(3)` को हल कीजिए।
Correct Answer - (i) 2 (ii) `x=(1)/(2),y=1`
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वक्र के लिए निर्दिष्ट बिन्दु पर स्पर्श रेखा और अभिलम्ब कि प्रवणता ज्ञात कीजिए- `x=a (theta - sin theta), y= a(1+ cos theta), theta= -(pi)/(2)` पर
Correct Answer - `1,-1`
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दिए गए वक्रो के लिए निर्दिष्ट बिन्दुओ पर स्पर्श रेखा के समीकरण ज्ञात कीजिए- `x=theta +sin theta , y=1 +cos theta, theta =pi/2 ` पर
Correct Answer - `{:("स्पर्श रेखा", "अभिलम्ब"),(2x+2y-pi4=0, 2x-2y=pi):}`
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