The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$

Let $$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that
$$a + b + c = x$$
$$a + b\omega + c{\omega ^2} = y$$
$$a + b{\omega ^2} + c\omega = z$$

Then the value of $${{{{\left| x \right|}^2} + {{\left| y \right|}^2} + {{\left| z \right|}^2}} \over {{{\left| a \right|}^2} + {{\left| b \right|}^2} + {{\left| c \right|}^2}}}$$ is

have one root in common is

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.

The straight line 2x - 3y = 1 divides the circular region $${x^2}\, + \,{y^2}\, \le \,6$$ into two parts.
If $$S = \left\{ {\left( {2,\,{3 \over 4}} \right),\,\left( {{5 \over 2},\,{3 \over 4}} \right),\,\left( {{1 \over 4} - \,{1 \over 4}} \right),\,\left( {{1 \over 8},\,{1 \over 4}} \right)} \right\}$$ then the number of points (s) in S lying inside the smaller part is

Let $$(x, y)$$ be any point on the parabola $${y^2} = 4x$$. Let $$P$$ be the point that divides the line segment from $$(0, 0)$$ to $$(x, y)$$ in the ratio $$1 : 3$$. Then the locus of $$P$$ is

Let $$P(6, 3)$$ be a point on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal at the point $$P$$ intersects the $$x$$-axis at $$(9, 0)$$, then the eccentricity of the hyperbola is

Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that
$$f\left( x \right) = f\left( {1 - x} \right)$$ for all $$x \in \left[ { - 1,2} \right]$$

Let $${R_1} = \int\limits_{ - 1}^2 {xf\left( x \right)dx,} $$ and $${R_2}$$ be the area of the region bounded by $$y=f(x),$$ $$x=-1,$$ $$x=2,$$ and the $$x$$-axis. Then

Let $$y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,$$ where $$f'(x)$$ denotes $${{df\left( x \right)} \over {dx}}$$ and $$g(x)$$ is a given non-constant differentiable function on $$R$$ with $$g(0)=g(2)=0.$$ Then the value of $$y(2)$$ is

Let $$E$$ and $$F$$ be two independent events. The probability that exactly one of them occurs is $$\,{{11} \over {25}}$$ and the probability of none of them occurring is $$\,{{2} \over {25}}$$. If $$P(T)$$ denotes the probability of occurrence of the event $$T,$$ then

Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A) $$\,\,\,\,$$If $$\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \widehat j + \sqrt 3 \widehat k$$ and $$\overrightarrow c = 2\sqrt 3 \widehat k$$ form a triangle, then the internal angle of the triangle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is
(B)$$\,\,\,\,$$ If $$\int\limits_a^b {\left( {f\left( x \right) - 3x} \right)dx = {a^2} - {b^2},} $$ then the value of $$f$$ $$\left( {{\pi \over 6}} \right)$$ is
(C)$$\,\,\,\,$$ The value of $${{{\pi ^2}} \over {\ell n3}}\int\limits_{7/6}^{5/6} {\sec \left( {\pi x} \right)dx} $$ is
(D)$$\,\,\,\,$$ The maximum value of $$\left| {Arg\left( {{1 \over {1 - z}}} \right)} \right|$$ for $$\left| z \right| = 1,\,z \ne 1$$ is given by

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$II$$
(p)$$\,\,\,\,$$ $${{\pi \over 6}}$$
(q)$$\,\,\,\,$$ $${{2\pi \over 3}}$$
(r)$$\,\,\,\,$$ $${{\pi \over 3}}$$
(s)$$\,\,\,\,$$ $$\pi $$
(t) $$\,\,\,\,$$ $${{\pi \over 2}}$$

Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k$$ be three given vectors. If $$\overrightarrow r $$ is a vector such that $$\overrightarrow r \times \overrightarrow b = \overrightarrow c \times \overrightarrow b $$ and $$\overrightarrow r .\overrightarrow a = 0,$$ then the value of $$\overrightarrow r .\overrightarrow b $$ is

If $$\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta $$, $$b > 0$$ and $$\theta \in ( - \pi ,\pi ]$$, then the value of $$\theta$$ is

Let f(x) = x² and g(x) = sin x for all x $$\in$$ R. Then the set of all x satisfying $$(f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)$$, where $$(f \circ g)(x) = f(g(x))$$, is

Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]$$, where each of a, b, and c is either $$\omega$$ or $$\omega$$². Then the number of distinct matrices in the set S is

If $$f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} < x \le 0} \cr {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.$$, then

Let $$f:(0,1) \to R$$ be defined by $$f(x) = {{b - x} \over {1 - bx}}$$, where b is a constant such that $$0 < b < 1$$. Then

Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6), then L is given by

Let M be a 3 $$\times$$ 3 matrix satisfying $$M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]$$, $$M\left[ {\matrix{ 1 \cr { - 1} \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr { - 1} \cr } } \right]$$ and $$M\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr {12} \cr } } \right]$$. Then the sum of the diagonal entries of M is ___________.

Match the statements given in Column I with the intervals/union of intervals given in Column II :