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The equation \(x \log x=3-x\)

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

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The normals to the curve \(y=x^{2}-x+1\), drawn at the points with the abscissa \(x_{1}=0, x_{2}=-1\) and \(x_{3}=\frac{5}{2}\)

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

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Let \(f(x)=\left\{\begin{array}{cl}-2 \sin x, & \text { if } x \leq-\frac{\pi}{2} \\ A \sin x+B, & \text { if }-\frac{\pi}{2}<x<\frac{\pi}{2} . \text { Then } \\ \cos x, & \text { if } x \geq \frac{\pi}{2}\end{array}\right.\)

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

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Let \(A\) be the centre of the circle \(x^{2}+y^{2}-2 x-4 y-20=0 .\) Let \(B(1,7)\) and \(D(4,-2)\) be two points on the circle such that tangents at \(B\) and \(D\) meet at \(C\). The area of the quadrilateral \(A B C D\) is

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

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A line cuts the \(x\)-axis at \(A(5,0)\) and the \(y\)-axis at \(B(0,-3)\). Avariable line \(P Q\) is drawn perpendicular to \(A B\) cutting the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). If \(A Q\) and \(B P\) meet at \(R\), then the locus of \(R\) is

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

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If the polynomial \(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a}\end{array}\right|\), then the constant term of \(f(x)\) is

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

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Let \(\rho\) be a relation defined on \(\mathbb{N}\), the set of natural numbers, as \(\rho=\{(x, y) \in \mathbb{N} \times \mathbb{N}: 2 x+y=41\}\) Then

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

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The least positive integer \(n\) such that \(\left(\begin{array}{rr}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right)^{n}\) is an identity matrix of order 2 is

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

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From a collection of 20 consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is

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

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Let \(z_{1}\) and \(z_{2}\) be complex numbers such that \(z_{1} \neq z_{2}\) and \(\left|z_{1}\right|=\left|z_{2}\right| .\) If \(\operatorname{Re}\left(z_{1}\right)>0\) and \(1 \mathrm{~m}\left(z_{2}\right)<0\), then \(\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\) is

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

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Let \(\vec{\alpha}, \vec{\beta}, \vec{\gamma}\) be three unit vectors such that \(\vec{\alpha} \cdot \vec{\beta}=\vec{\alpha} \cdot \vec{\gamma}=0\) and the angle between \(\vec{\beta}\) and \(\vec{\gamma}\) is \(30^{\circ}\). Then \(\vec{\alpha}\) is

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

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Let \(\vec{\alpha}=\hat{i}+\hat{j}+\hat{k}, \vec{\beta}=\hat{i}-\hat{j}-\hat{k}\) and \(\vec{\gamma}=-\hat{i}+\hat{j}-\hat{k}\) be three vectors. A vector \(\vec{\delta}\), in the plane of \(\vec{\alpha}\) and \(\vec{\beta}\), whose projection on \(\vec{\gamma}\) is \(\frac{1}{\sqrt{3}}\), is given by

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

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For \(0 \leq p \leq 1\) and for any positive \(a, b\); let \(I(p)=(a+b)^{p}, J(p)=a^{p}+b^{p}\), then

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

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A ladder \(20 \mathrm{ft}\) long leans against a vertical wall. The top end slides downwards at the rate of \(2 \mathrm{ft}\) per second. The rate at which the lower end moves on a horizontal floor when it is \(12 \mathrm{ft}\) from the wall is

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

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The foot of the perpendicular drawn from the point \((1,8,4)\) on the line joining the points \((0,-11,4)\) and \((2,-3,1)\) is

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

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A point \(P\) lies on a line through \(Q(1,-2,3)\) and is parallel to the line \(\frac{x}{1}=\frac{y}{4}=\frac{z}{5}\). If \(P\) lies on the plane \(2 x+3 y-4 z+22=0\), then segment \(P Q\) equals to

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

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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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