In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\Delta$$PQR is :

The value of
$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [ x ] denotes the greatest integer $$ \le $$ x is :

Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that

$$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $$

Which of the following is true for y(x) that satisfies the differential equation

$${{dy} \over {dx}}$$ = xy $$-$$ 1 + x $$-$$ y; y(0) = 0 :

The sum of possible values of x for

tan^$$-$$1(x + 1) + cot^$$-$$1$$\left( {{1 \over {x - 1}}} \right)$$ = tan^$$-$$1$$\left( {{8 \over {31}}} \right)$$ is :

The value of $$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$ is :

The inverse of $$y = {5^{\log x}}$$ is :

Two dies are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :

If the fourth term in the expansion of $${(x + {x^{{{\log }_2}x}})^7}$$ is 4480, then the value of x where x$$\in$$N is equal to :

The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k² has no solution if k is equal to :

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

If cot^$$-$$1($$\alpha$$) = cot^$$-$$1 2 + cot^$$-$$1 8 + cot^$$-$$1 18 + cot^$$-$$1 32 + ...... upto 100 terms, then $$\alpha$$ is :

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

If (2021)³⁷⁶² is divided by 17, then the remainder is __________.

Let there be three independent events E₁, E₂ and E₃. The probability that only E₁ occurs is $$\alpha$$, only E₂ occurs is $$\beta$$ and only E₃ occurs is $$\gamma$$. Let 'p' denote the probability of none of events occurs that satisfies the equations
($$\alpha$$ $$-$$ 2$$\beta$$)p = $$\alpha$$$$\beta$$ and ($$\beta$$ $$-$$ 3$$\gamma$$)p = 2$$\beta$$$$\gamma$$. All the given probabilities are assumed to lie in the interval (0, 1).

Then, $$\frac{Probability\ of\ occurrence\ of\ E_{1}}{Probability\ of\ occurrence\ of\ E_{3}} $$ is equal to _____________.

The maximum value of z in the following equation z = 6xy + y², where 3x + 4y $$ \le $$ 100 and 4x + 3y $$ \le $$ 75 for x $$ \ge $$ 0 and y $$ \ge $$ 0 is __________.

If [ . ] represents the greatest integer function, then the value of


$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is ____________.

If $$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$ and its first derivative with respect to x is $$ - {b \over a}{\log _e}2$$ when x = 1, where a and b are integers, then the minimum value of | a² $$-$$ b² | is ____________ .

If the function $$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is continuous at each point in its domain and $$f(0) = {1 \over k}$$, then k is ____________.