If the coordinates of a variable point `P` are `(acostheta,bsintheta),` where `theta` is a variable quantity, then find the locus of `Pdot`
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If `O` is the origin, `O P=3` with direction ratios `-1,2,a n d-2,` then find the coordinates of `Pdot`
The direction cosines of `vec(OP)` are `-(1)/(3), (2)/(3) and -(2)/(3)`. Hence, `vec(OP)=|vec(OP)|(lhati+mhatj+nhatk)` `" "=3(-(1)/(3)hati+(2)/(3)hatj-(2)/(3)hatk)` `" "=-1hati+2hatj-2hatk` So, the coordinations of P are -1, 2 and -2.
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If `I_(1)=int_(0)^(1)(dx)/(e^(x)(1+x))` and `I_(2)=int_(0)^(pi//4)(e^(tan^(7)theta)sintheta)/((2-tan^(2)theta)cos^(3)theta d theta`,then find the valu
Correct Answer - `2//e` `I_(2)int_(0)^(pi//4)(e^(tan^(2)theta).tan theta)/((2-tan^(2)theta))sec^(2) d theta` Put `tan^(2)theta=t` `:.2 tan theta sec^(2) d theta =dt` `:.I_(2)=1/2int_(0)^(1)(e^(t)dt)/((2-t))` `=1/2 int_(0)^(1)(e^(1-t)dt)/(1+t)` `=e/2int_(0)^(1)(dt)/(e^(t)(t+1))=e/2.I_(1)` `:. (I_(1))/(I_(2))=2/e`
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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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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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Let `P` be a point on the hyperbola `x^2-y^2=a^2,` where `a` is a parameter, such that `P` is nearest to the line `y=2xdot` Find the locus of `Pdot`
Consider any point P`(a sec theta, a tan theta)" on "x^(2)-y^(2)=a^(2)` This point will be nearest to y = 2x if the tangent at this point is parallel to y...
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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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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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If the chord of contact of tangents from a point `P` to the parabola `y^2=4a x` touches the parabola `x^2=4b y ,` then find the locus of `Pdot`
Chord of contact of parabola `y^(2)=4ax` w.r.t. point `P(x_(1),y_(1))` is `yy_(1)=2a(x+x_(1))`. Solving this line with parabola `x^(2)=4by`, we get `x^(2)=4bxx(2a)/(y_(1))(x+x_(1))` `ory_(1)x^(2)=8bx-8abx_(1)=0` Since the line touches the parabola, this equation will...
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