Let $$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$ The sum of all distinct solutions of the equation $$\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0$$ in the set S is equal to

Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as origin, and $$OP$$ and $$OR$$ along the $$x$$-axis and the $$y$$-axis, respectively. The base $$OPQR$$ of the pyramid is a square with $$OP=3.$$ The point $$S$$ is directly above the mid-point, $$T$$ of diagonal $$OQ$$ such that $$TS=3.$$ Then

A computer producing factory has only two plants $${T_1}$$ and $${T_2}.$$ Plant $${T_1}$$ produces $$20$$% and plant $${T_2}$$ produces $$80$$% of the total computers produced. $$7$$% of computers produced in the factory turn out to be defective. It is known that $$P$$ (computer turns out to be defective given that it is produced in plant $${T_1}$$)
$$ = 10P$$ (computer turns out to be defective given that it is produced in plant $${T_2}$$),
where $$P(E)$$ denotes the probability of an event $$E$$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $${T_2}$$ is

The total number of distinct $$x \in \left[ {0,1} \right]$$ for which

$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$

A solution curve of the differential equation

$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the

point $$(1,3)$$. Then the solution curve

The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is

In a triangle $$\Delta $$$$XYZ$$, let $$x, y, z$$ be the lengths of sides opposite to the angles $$X, Y, Z$$ respectively, and $$2s = x + y + z$$.
If $${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$$ and area of incircle of the triangle $$XYZ$$ is $${{8\pi } \over 3}$$, then

Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \right)} \right) = x$$ and $$h\left( {g\left( {g\left( x \right)} \right)} \right) = x$$ for all $$x \in R$$. Then

The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the tangent to the circle $${C_1}$$, at $$P$$ touches other two circles $${C_2}$$ and $${C_3}$$ at $${R_2}$$ and $${R_3}$$, respectively. Suppose $${C_2}$$ and $${C_3}$$ have equal radil $${2\sqrt 3 }$$ and centres $${Q_2}$$ and $${Q_3}$$, respectively. If $${Q_2}$$ and $${Q_3}$$ lie on the $$y$$-axis, then

Let RS be the diameter of the circle $${x^2}\, + \,{y^2} = 1$$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point (s)

Let $$m$$ be the smallest positive integer such that the coefficient of $${x^2}$$ in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$$ is $$\left( {3n + 1} \right)\,{}^{51}{C_3}$$ for some positive integer $$n$$. Then the value of $$n$$ is

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Let $$ - {\pi \over 6} < \theta < - {\pi \over {12}}.$$ Suppose $${\alpha _1}$$ and $${\beta_1}$$ are the roots of the equation $${x^2} - 2x\sec \theta + 1 = 0$$ and $${\alpha _2}$$ and $${\beta _2}$$ are the roots of the equation $${x^2} + 2x\,\tan \theta - 1 = 0.$$ $$If\,{\alpha _1} > {\beta _1}$$ and $${\alpha _2} > {\beta _2},$$ then $${\alpha _1} + {\beta _2}$$ equals

Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. Then

Let $$P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha$$ $$\in$$ R. Suppose $$Q = [{q_{ij}}]$$ is a matrix such that PQ = kl, where k $$\in$$ R, k $$\ne$$ 0 and I is the identity matrix of order 3. If $${q_{23}} = - {k \over 8}$$ and $$\det (Q) = {{{k^2}} \over 2}$$, then

The total number of distinct x $$\in$$ R for which

$$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10$$ is ______________.

Let $$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where $$i = \sqrt { - 1} $$, and r, s $$\in$$ {1, 2, 3}. Let $$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]$$ and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P² = $$-$$I is ____________.

Let $$\alpha$$, $$\beta$$ $$\in$$ R be such that $$\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1$$. Then 6($$\alpha$$ + $$\beta$$) equals _________.