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If the vectors \(\vec{\alpha}=\hat{i}+a \hat{j}+a^{2} \hat{k}, \vec{\beta}=\hat{i}+b \hat{j}+b^{2} \hat{k}\), and \(\vec{\gamma}=\hat{i}+c \hat{j}+c^{2} \hat{k}\) are three non-coplanar vectors and \(\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ c & c^{2} & 1+b^{3}\end{array}\right|=0\), then the value of abc is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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If the line \(y=x\) is a tangent to the parabola \(y=a x^{2}+b x+c\) at the point \((1,1)\) and the curve passes through \((-1,0)\), then

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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In open interval \(\left(0, \frac{\pi}{2}\right)\),

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The area of the region \(\left\{(x, y): x^{2}+y^{2} \leq 1 \leq x+y\right\}\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+12\). Then

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be twice continuously differentiable (or \(f\) " exists and is continuous) such that \(f(0)=f(1)=f^{\prime}(0)=0 .\) Then

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(\lim _{x \rightarrow 0}\left(\frac{1+c x}{1-c x}\right)^{1 / x}=4\), then \(\lim _{x \rightarrow 0}\left(\frac{1+2 c x}{1-2 c x}\right)^{1 / x}\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(y=f(x)=2 x^{2}-3 x+2\). The differential of \(y\) when \(x\) changes from 2 to \(1.99\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The domain of \(f(x)=\sqrt{\left(\frac{1}{\sqrt{x}}-\sqrt{x+1}\right)}\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(f(x)=\sin x+\cos a x\) be periodic function. Then

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The sine of the angle between the straight line \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) and the plane \(2 x-2 y+z=5\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The equation of the plane through the point \((2,-1,-3)\) and parallel to the lines \(\frac{x-1}{2}=\frac{y+2}{3}=\frac{z}{-4}\) and \(\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The equation of the latus rectum of a parabola is \(x+y=8\) and the equation of the tangent at the vertex is \(x+y=12\). Then the length of the latus rectum is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(B\) and \(B^{\prime}\) are the ends of minor axis and \(S\) and \(S^{\prime}\) are the foci of the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\), then the area of the rhombus SBS'B' will be

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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A double ordinate \(P Q\) of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is such that \(\Delta O P Q\) is equilateral, \(O\) being the centre of the hyperbola. Then the eccentricity e satisfies the relation

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The length of the chord of the parabola \(y^{2}=4 a x(a>0)\) which passes through the vertex and makes an acute angle \(\alpha\) with the axis of the parabola is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The equation of circle of radius \(\sqrt{17}\) unit, with centre on the positive side of \(x\)-axis and through the point \((0,1)\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Area in the first quadrant between the ellipses \(x^{2}+2 y^{2}=a^{2}\) and \(2 x^{2}+y^{2}=a^{2}\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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A straight line through the origin \(O\) meets the parallel lines \(4 x+2 y=9\) and \(2 x+y+6=0\) at \(P\) and \(Q\) respectively. The point \(\mathrm{O}\) divides the segment \(\mathrm{PQ}\) in the ratio

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let each of the equations \(x^{2}+2 x y+a y^{2}=0 \& a x^{2}+2 x y+y^{2}=0\) represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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