If \(S=\{(x,y)\mid x=2\cos t+3\sin t,\;y=3\cos t+2\sin t,\;t\in\mathbb{R}\}\), then the points of S lie on the curve

If the coordinate axes are rotated about the origin in the positive direction through an angle \(60^\circ\) to get the transformed equation of \(x^2+y^2-4x-8y+16=0\) as \(x^2+y^2+2Gx+2Fy+C=0\), then \(G+F+C=\)

\((3a+1)x+(7a+2)y=17a+5\), \(a\) being a parameter, represents a family of concurrent lines. If \(d\) is the distance from the point \((3,1)\) to a line of this family having slope 1, then \(2d^2=\)

Let a ray of light passing through a point \((7,2)\) reflects on the line \(2x+y=1\) and the reflected ray passes through \((3,10)\). The equation of the incident ray is

Let ABC be an isosceles triangle. If B is a point on the positive X-axis, \(a=4\sqrt3\), \(c\) is an integer, \(\angle A=120^\circ\) and \(A=(-1,0)\), then the distance of C from the origin is

One of the pair of lines \(x^2-3y^2-4x-6\sqrt3 y-5=0\) is \(x+by+c=0\;(b<0)\). If the other line intersects the curve \(x^2-5y^2-4x=0\) at two points A and B, then \(\angle AOB=\)

If a circle inscribed in the parabola \(y^2=4ax\;(a>0)\) passes through its focus, then the equation of that circle is

Let \((-h,-k)\) (h, k integers) be the centre and \(r\) the radius of a circle. If \(3x+4y-24=0,\;3x-4y-32=0\) are two tangents and \(4x+3y-1=0\) is a normal, then \(r+h+k=\)

Let P be any point on the circle \(x^2+y^2=16\) and \(A=(1,2)\). If the locus of the point which divides the line segment AP in the ratio 3:2 is a circle, then its radius is

If the line \(4x-3y+c=0\;(c<-10)\) makes an intercept of length 10 units on the circle \(x^2+y^2-2x+4y-23=0\), then \(c=\)

The tangent drawn at a point P on the circle \(x^2+y^2+6x+6y-2=0\) cuts the line \(5x-2y+6=0\) at Q. If \(PQ=5\), then a point Q having integral coordinates is

If two vertices of a quadrilateral are the centres of the circles \(S\equiv x^2+y^2-2x-2y-2=0,\;S’\equiv x^2+y^2-6x-6y+14=0\) and the other two vertices are the points of intersection of S=0 and S’=0, then the area of the quadrilateral is

\(y=4\) is the directrix of the parabola \(x^2+8x+12y+k=0\). If \(l\) is the length of its latus rectum, then \(l-k=\)

\((1,1)\) is the focus of the parabola \(y^2-4ax-2ay+a^2=0\). If the circles \((x-\alpha)^2+(y-\beta)^2=r^2\) (\(\alpha,\beta\) parameters) touch the X-axis and the axis of the given parabola, then \(\{(\alpha,\beta)\mid\alpha,\beta\in\mathbb{R}\}\) is

The area of the rectangle formed by the tangents drawn at the ends of both major and minor axes of an ellipse is 24. If the eccentricity of the ellipse is \(\dfrac{1}{4}\), then the equation of the ellipse is

The centre of the ellipse lies on the lines \(2x+3y=5\) and \(x+3y=4\). If the eccentricity is \(\dfrac{2}{3}\), length of its major axis is 4 and its minor axis is parallel to Y-axis, then the equation of the ellipse is

For a hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\), the distance between its vertex and focus which are lying on the positive side of X-axis is 2. If the length of its latus rectum is 13, then the eccentricity is

If the area of the triangle formed by the points \((0,0,0),\;(1,1,1)\) and \((t,2t,3t)\) is \(\sqrt6\), then the sum of squares of all possible values of \(t\) is

If \(P(1,2,5),Q(3,0,7),R(6,-3,10)\) are three points on a line and \((\alpha,\beta,\gamma)\) is a point at a distance of 3 units from P on the same line, then the value of \(\alpha+\beta+\gamma\) that lies between 6 and 7 is

If the direction cosines of the line common to the planes \(x+2y-z-1=0\) and \(3x-4y+z-5=0\) are \((l,m,n)\), then \(l+m-n=\)

\([y]\) represents the greatest integer less than or equal to \(y\) and \(\{y\}\) represents the fractional part of \(y\). If \(\displaystyle\lim_{x\to0^+}\frac{2[1-x]+\{1-x\}}{2[1-x]+\{1-x\}}=11\), then \(\alpha=\)

If \(f(x)=\dfrac{\lambda e^{|x|}+3e^{-x}}{(\lambda+2)e^{|x|}-e^{-x}}\) for \(x\ne0\) and \(f(0)=k,\;k\in\mathbb{R}\) is continuous at \(x=0\), then \(2\lambda=\)

\(f(x)\) is an nth degree polynomial and \(a_1,a_2,\dots,a_n\) are distinct n zeros of \(f(x)\). \(g(x)\) is a polynomial having three zeros common with the zeros of \(f(x)\).
Assertion (A): \(|f(x)|g(x)\) is continuous and differentiable at all \(\alpha_i\)’s.
Reason (R): \(\displaystyle\lim_{x\to a}\frac{|x-a|}{x-a}\) does not exist and \(\displaystyle\lim_{x\to a}|x-a|=0\).
The correct answer is

If \(y=(e^{2x}-4)(6e^{2x}-5e^{x}+1)\), then \(\left(\dfrac{dy}{dx}\right)_{x=0}-\left(\dfrac{d^2y}{dx^2}\right)_{x=0}=\)

If \(f(x)\) is a differentiable function and \(y=e^{f(x)+e^{f(x)+e^{f(x)+\cdots\infty}}}\), then \(\dfrac{dy}{dx}=\)

If \(\dfrac{d}{dx}\!\left(\dfrac{\sec x+\tan x}{\sec x-\tan x}\right)=k\) at \(x=\dfrac{\pi}{4}\), then \(\dfrac{k}{2\sqrt2}-2\sqrt2=\)

Statement-I: The equation of the tangent to the curve \(y=3x^2-5\) drawn through the point \((1,2)\) is \(y=6x-4\).
Statement-II: If L, M, N are respectively the lengths of the tangent, normal and subnormal drawn to a curve at a point \((\alpha,\beta)\), then \((L)(N)=\beta^2 M\).
Which of the following is correct?

A function is defined as \[ f(x)=\begin{cases}3x-1, & 0\le x\le2\\ 25(x-1), & 2\le x<\infty\end{cases} \] For \(f(x)\) in the interval \(\left[\dfrac13,3\right]\):

In the interval \([-5,5]\), if the function \(f(x)=(x+3)^2(x-2)^3\) is increasing on \(S=\{x\mid-5\le x<\alpha \text{ and } \beta

• 1\(-108\)
• 2\(108\)
• 3\(72\)
• 4\(-72\)