If the straight lines `x+y-2-0,2x-y+1=0` and `a x+b y-c=0` are concurrent, then the family of lines `2a x+3b y+c=0(a , b , c)` are nonzero) is concurrent at `(2,3)` (b) `(1/2,1/3)` `(-1/6,-5/9)` (d) `(2/3,-7/5)`
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Choose the correct answer in questions 17 to 19: If x, y, z are nonzero real numbers then the inverse of metrix `A=[{:(x,0,0),(0,y,0),(0,0,z):}]`is :
Correct Answer - (a) `:." "A=[{:(x,0,0),(0,y,0),(0,0,z):}]` `rArr" "|A|=[{:(x,0,0),(0,y,0),(0,0,z):}]=xyzne0` `A_(11)=yz, A_(12)=0, A_(13)=0` `A_(21)=0, A_(22)=zx, A_(23)=0` `A_(31)=0, A_(32)=0, A_(32)=0, A_(33)=xy` `:." adj A"=[{:(yz,0,0),(0,zx,0),(0,0,xy):}]=[{:(yz,0,0),(0,zx,0),(0,0,xy):}]` `"and A"^(-1)=1/|A|".adj A"=1/(xyz)[{:(yz,0,0),(0,zx,0),(0,0,xy):}]` `=[{:(1/x,0,0),(0,1/y,0),(0,0,1/z):}]` `=[{:(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1)):}]`
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Prove that the straight lines joining the origin to the points of intersection of the straight line `hx+ky=2hk` and the curve `(x-k)^(2)+(y-h)^(2)=c^(
From the equation of line , we have `1=(hx+ky)/(2hx)` (1) Equation of the curve is `x^(2)+y^(2)-2kx-2hy+h^(2)+k^(2)-c^(2)=0` Making above equation homogeneous with the help of (1) , we get `x^(2)+y^(2)-2(kx+hy)((hx+ky)/(2hx))+(h^(2)+k^(2)-c^(2))((hx+ky)/(2hk))^(2)=0` This...
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Consider the equation of a pair of straight lines as `lambdax^(2)-10xy+12y^(2)+5x-16y-3=0`. The angles between the lines is `theta` . Then the value o
Correct Answer - 3 `tantheta=(2sqrt(h^(2)-ab))/(a+b)=(2sqrt(25-24))/(14)=(1)/(7)`
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If `lim_(xtooo) f(x)` exists and is finite and nonzero and if `lim_(xtooo) {f(x)+(3f(x)-1)/(f^(2)(x))}=3`, then the value of `lim_(xtooo) f(x)" is "`_
Correct Answer - `(1)` `underset(xtooo)lim(f(x)+(3f(x)-1)/(f^(2)(x)))=3` or `(underset(xtooo)limf(x)+(3underset(xtooo)limf(x)-1)/((underset(xtooo)limf(x))^(2)))=3` or `(y+(3y-1)/(y^(2)))=3` or `y^(3)-3y^(2)+3y-1=0` or `(y-1)^(3)=0` or `y=1`
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If lines `x+2y-1=0,a x+y+3=0,` and `b x-y+2=0` are concurrent, and `S` is the curve denoting the locus of `(a , b)` , then the least distance of `S` f
Correct Answer - C The lines are concurrent if `|{:(1,2,-1),(a,1,3),(b,-1,2):}|=0` or 7b-3a+5=0 The locus of (a,b) is 3x-7y=5. Least distance from (0,0) = Length of perpendicular from (0,0) `=(5)/(sqrt(58))`
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The lines `x+y-1=0,(m-1)x+(m^2-7)y-5=0,` and `(m-2)x+(2m-5)y=0` are concurrent for three values of `m` concurrent for no value of `m` parallel for one
Correct Answer - C::D If the lines x+y-1 =0, (m-1)x +`(m^(2)-7)y-5 =0`, and (m-2) x+(2m-5)y = 0 are concurrent, then `Delta = 0` `"or " |{:(1,1,-1),(m-1, m^(2)-7, -5),(m-2, 2m-5, 0):}| =...
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Find the value of `a` for which the lines `2x+y-1=0` `2x+y-1=0` `a x+3y-3=0` `3x+2y-2=0` are concurrent.
Correct Answer - A::B::C::D First and third line intersect at (0,1). This point also satisfies the second line for all real values of a.
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Let `l` be the line belonging to the family of straight lines `(a + 2b)x+ (a - 3b)y +a-8b = 0, a, b in R`, which is farthest from the point `(2, 2),`
Correct Answer - A Given lines (x+y+1) +b(2x-3y-8) = 0 are concurrent at the point of intersection of the line x+y+1=0 and 2x-3y-8=0, which is (1, -2). Now, the line through...
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Let L be the line belonging to the family of straight lines (a+2b) x+(a-3b)y+a-8b =0, a, b `in `R, which is the farthest from the point (2, 2). If L i
Correct Answer - D Lines x+4y+7 =0 and x-2y+1 =0 intersect at (-3, -1) which must satisfy the line `3x-4y+lambda=0. "Then " -9+4+lambda=0 " or " lambda = 5.`
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`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3).` Then the least value of n for which `(d^(n))/(dx^(n))f(x)|underset(x=0)` is nonzero is
`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3)` `f^(I)(x)=e^(x)+e^(-x)-2 cos x -2x^(2)` `f^(II)(x)=e^(x)-e^(-x)+2 sin x-4x` `f^(III)(x)=e^(x)+e^(-x)+2 cos x -4` `f^(IV)(x)=e^(x)-e^(-x)-2 sin x` `f^(V)(x)=e^(x)+e^(-x)-2 cos x` `f^(VI)(x)=e^(x)-e^(-x)+2 sin x` `f^(VII)(x)=e^(x)+e^(-x)+2 cos x` `"Clearly, "f^(VII)(0)" is...
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