IIT-JEE 2005 Mains

Find the equation of the plane containing the line \(2 x-y+z-3=0,3 x+y+z=5\) and at a distance of \(\frac{1}{\sqrt{6}}\) from the point \((2,1,-1)\).

If \(\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}\), for all \(x_{1}, x_{2} \in\) \(\mathbb{R}\). Find the equation of tangent to the curve \(y=f(x)\) at the point \((1,2)\).

If \(p(x)\) be a polynomial of degree 3 satisfying \(p(-1)=10, p(1)=-6\) and \(p(x)\) has maximum at \(x=-1\) and \(p'(x)\) has minima at \(x=1\). Find the distance between the local maximum and local minimum of the curve.

If length of tangent at any point on the curve
\(y = f(x)\) intercepted between the point and
the X-axis is of length 1. Find the equation of
the curve.

Find the area bounded by the curves \(x^{2}=y, x^{2}=-y\) and \(y^{2}=4 x-3\).

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{arra...

Circles with radii 3, 4 and 5 touch each other
externally if P is the point of intersection
of tangents to these circles at their points
of contact. Find the distance of P from the
point of contact.

If one of the vertices of the square circumscribing the circle \(|z-1|=\sqrt{2}\) is \((2+\sqrt{3 i})\). Find the other vertices of square.

Evatuate:
\(\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x\)

If \(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable function such that \(|f(x)| \leq 1\) and \(f'(x)=g(x)\), where,\(f^{2}(0)+g^{2}(0)=9\) then prove that there exists some \(c \in(-3,3)\) such that $$g(c) \circ...

Find the equation of the common tangent in the first quadrant to the circle \(x^{2}+y^{2}=16\) and the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\). Also find the length of the intercept of the tangent between the coordinate axes.

Find the range of value of \(t\) for which
\(2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Tangents are drawn from any point on the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) to the circle \(x^{2}+y^{2}=9\). Find the locus of mid-point of the chord of contact.

If \(f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)\) And \(g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)\) for all \(x, y \in \mathrm{R}\). If right-hand derivative at \(x=0\) exists for \(f(x)\), find the derivative of \(g(x)\) at \(x=0\)

A person goes office either by car, scooter, bus or train, proability of which being \(\frac{1}{7}, \frac{3}{2}, \frac{2}{7}\) and \(\frac{1}{7}\), respectively. Probability that he reaches office late, if he takes car, scooter, bus or trai...

If total number of runs scored in \(n\) matches is \(\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)\) where \(n > 1\), and the runs scored in the \(k^{\text {th }}\) match are given by \(k .2^{n+1-k}\), where \(1 \leq k \leq n\). Find, ...

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through \((h, k)\) with the lines \(y=x\) and \(x+y=2\) is \(4 h^{2}\). Find the locus of point \(P\).

Incident ray is along the unit vector \(\hat{v}\) and the reflected ray is along the unit vector \(\widehat{w}\). The normal is along unit vector \(\hat{a}\) outwards. Express \(\hat{w}\), in terms of \(\hat{a}\) and \(\hat{v}\).