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Let \(P\) be a point on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and the line through \(P\) parallel to the \(y\)-axis meets the circle \(x^{2}+y^{2}=9\) at \(Q\), where \(P, Q\) are on the same side of the \(x\)-axis. If \(R\) is a point on \(P Q\) such that \(\frac{P R}{R Q}=\frac{1}{2}\), then the locus of \(R\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) be three points on a parabola \(y^{2}=4 a x\). If \(P Q\) is the focal chord and \(P K, Q R\) are parallel where the co-ordinates of \(K\) is \((2 a, 0)\), then the value of \(r\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(A, B\) be two distinct points on the parabola \(y^{2}=4 x\). If the axis of the parabola touches a circle of radius \(r\) having \(A B\) as diameter, the slope of the line \(A B\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let the eccentricity of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) be reciprocal to that of the ellipse \(x^{2}+9 y^{2}=9\), then the ratio \(a^{2}: b^{2}\) equals

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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A chord \(A B\) is drawn from the point \(A(0,3)\) on the circle \(x^{2}+4 x+(y-3)^{2}=0\), and is extended to \(M\) such that \(\mathrm{AM}=2 \mathrm{AB} .\) The locus of \(\mathrm{M}\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If one of the diameters of the circle, given by the equation \(x^{2}+y^{2}+4 x+6 y-12=0\), is a chord of a circle \(S\), whose centre is \((2,-3)\), the radius of \(S\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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The angular points of a triangle are \(A(-1,-7), B(5,1)\) and \(C(1,4)\). The equation of the bisector of the \(\angle A B C\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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The point \(Q\) is the image of the point \(P(1,5)\) about the line \(y=x\) and \(R\) is the image of the point \(Q\) about the line \(y=-x\). The circumcenter of the \(\Delta P Q R\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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The angle between a pair of tangents drawn from a point \(P\) to the circle \(x^{2}+y^{2}+4 x-6 y+9 \sin ^{2} \alpha+13 \cos ^{2} \alpha=0\) is \(2 \alpha\). The equation of the locus of the point \(P\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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Without changing the direction of the axes, the origin is transferred to the point \((2,3)\). Then the equation \(x^{2}+y^{2}-4 x-6 y+9=0\) changes to

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(0 \leq \mathrm{A} \leq \frac{\pi}{4}\), then \(\tan ^{-1}\left(\frac{1}{2} \tan 2 \mathrm{~A}\right)+\tan ^{-1}(\cot \mathrm{A})+\tan ^{-1}\left(\cot ^{3} \mathrm{~A}\right)\) is equal to

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(\sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0\), then general value of \(\theta\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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A student appears for tests I, II and III. The student is successful if he passes in tests I, II or I, III. The probabilities of the student passing in tests, I, II and III are respectively \(p, q\) and \(\frac{1}{2}\). If the probability of the student to be successful is \(\frac{1}{2}\). Then

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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In order to get a head at least once probability \(\geq 0.9\), the minimum number of time a unbiased coin needs to be tossed is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=e^{x}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(g(x)=x^{2}\). The mapping \(g \circ f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \((g \circ f)(x)=g[f(x)] \forall x \in \mathbb{R}\), Then

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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On the set \(\mathbb{R}\) of real numbers, the relation \(\rho\) is defined by \(x \rho y,(x, y) \in \mathbb{R}\)

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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On \(\mathbb{R}\), a relation \(\rho\) is defined by \(x \rho y\) if and only if \(x-y\) is zero or irrational. Then

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If the following three linear equations have a non-trivial solution, then $$ \begin{aligned} &x+4 a y+a z=0 \\ &x+3 b y+b z=0 \\ &x+2 c y+c z=0 \end{aligned} $$

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(S_{r}=\left|\begin{array}{ccc}2 r & x & n(n+1) \\ 6 r^{2}-1 & y & n^{2}(2 n+3) \\ 4 r^{3}-2 n r & z & n^{3}(n+1)\end{array}\right|\), then the value of \(\sum_{r=1}^{n} S_{r}\) is independent of

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}\), then the value of \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|\) is

asked Dec 13, 2021 in Sets, relations and functions by kritika (90.1k points)

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