Is `|t a n x+cos x|<|t a n x|+|cot x|` true for any `x ?` If it is true, then find the values of `xdot`
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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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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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Solve : `cos^(-1)(1/2x^2+sqrt(1-x^2)1=(x^2)/4)=cos^(-1)x/2-cos^(-1)xdot`
`cos^(-1) ((1)/(2) x^(2) + sqrt(1 -x^(2)) sqrt(1 - (x^(2))/(4))) = cos^(-1) (x.(x)/(2) + sqrt(1 -x^(2)) sqrt(1 - ((x)/(2))^(2)))` for `cos^(-1) ((1)/(2) x^(2) + sqrt(1 - x^(2)) sqrt(1 - (x^(2))/(4))) =...
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There exists a positive real number of `x` satisfying `"cos"(tan^(-1)x)=xdot` Then the value of `cos^(-1)((x^2)/2)i s` `pi/(10)` (b) `pi/5` (c) `(2pi)
Correct Answer - C Let `tan^(-1) (x) = theta " or " x = tan theta` `rArr cos theta = x " " rArr (1)/(sqrt(1 + x^(2))) = x` ltrbgt `rArr...
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The product of all values of `x` satisfying the equation `sin^(-1)cos((2x^2+10|x|+4)/(x^2+5|x|+3))=cot(cot^(-1)((2-18|x|)/(9|x|)))+x/2i s` 9 (b) `-9`
Correct Answer - A `(pi)/(2) - cos^(-1) cos ((2(x^(2) + 5|x| + 3) -2)/(x^(2) + 5|x| + 3))` `= cot cot^(-1) ((2)/(9|x|) -2) + (pi)/(2)` `rArr (pi)/(2) -2 + (2)/(x^(2) +...
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Indicate the relation which can hold in their respective domain for infinite values of `xdot` `"tan"|tan^(-1)x|=|x|` (b) `"cot"|cot^(-1)x|=|x|` `tan^(
Correct Answer - A::B::C::D Since `|tan^(-1)x| = {(tan^(-1) x," if " x ge 0),(-tan^(-1) x," if " x lt 0):}` `rArr |tan^(-1)x| = tan^(-1) |x| AA x in R` `rArr tan...
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If `S_n=cot^-1(3)+cot^-1(7)+cot^-1(13)+cot^-1(21)+.....`, `n` terms, then
Correct Answer - A::B::D Let `t_(r)` denote the `rth` term of the series 3, 7, 13, 21, ... and `{:(S = 3 + 7 + 13 + 21 + ...+ t_(n)),(S...
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The solution set of inequality `(cot^(-1)x)(tan^(-1)x)+(2-pi/2),cot^(-1)x-3tan^(-1)x-3(2-pi/2)>0` is `(a , b),` then the value of `cot^(-1)a+cot^(-1)b
Correct Answer - 5 `(cot^(-1) x) (tan^(-1)x) + (2 -(pi)/(2)) cot^(-1) x - 3 tan^(-1) x 3 (2 - (pi)/(2)) gt 0` `rArr cot^(-1) x (tan^(-1) x -(pi)/(2)) + 2 cot^(-1)...
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If in a triangle ABC, `(1 + cos A)/(a) + (1 + cos B)/(b) + (1+ cos C)/(c) = (k^(2) (1 + cos A) (1 + cos B) (1 + cos C))/(abc)`, then k is equal to
Correct Answer - B `L.H.S. = (1 + cos A)/(a) + (1 + cos B)/(b) + (1 + cos C)/(c)` `= (2 cos^(2).(A)/(2))/(2R sin A) + (2 cos^(2).(B)/(2))/(2R sin B) +...
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The base BC of `DeltaABC` is fixed and the vertex A moves, satisfying the condition `cot.(B)/(2) + cot.(C)/(2) = 2 cot.(A)/(2)`, then
Correct Answer - B::D Given `(s(s-b))/(Delta) + ((s-c))/(Delta) = (2s(s-a))/(Delta)` `rArr s-b + s - c = 2 (s-a)` `rArr b+c = 2a` So, locus of vertex A is an ellipse
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