For every real number x, 

let f(x) = $\frac{\mathrm{x}}{1!} + \frac{3}{2!}{\mathrm{x}}^2 + \frac{7}{3!}{\mathrm{x}}^3 + \frac{15}{4!}{\mathrm{x}}^4 + ...$ Then, the equation f(x) = 0 has

• no real solution

• exactly one real solution

• exactly two real solutions

• infinite number of real solutions

Let S denote the sum of the infinite series $1 + \frac{8}{2!} + \frac{21}{3!} + \frac{40}{4!} + \frac{65}{5!} + ...$

• $\mathrm{S} < 8$

• $\mathrm{S} > 12$

• 8 < S < 12

• S = 8

Let [x] denote the greatest integer less than or equal to x for any real number x. Then,

$\underset{\mathrm{n}\to \infty }{\lim }\frac{\left[\mathrm{n}\sqrt{2}\right]}{\mathrm{n}}$ is equal to

• 0

• 2

• $\sqrt{2}$

• 1

Let z, be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and ${\mathrm{z}}_1 \ne \pm 1$. Consider an equilateral triangle inscribed in the circle with z₁, z₂, z₃ as the vertices taken in the counterclockwise direction. Then, z₁z₂z₃ is equal to

• z₁²

• z₁³

• z₁⁴

• z₁

Suppose that z₁, z₂, z₃ are three vertices of an equilateral triangle in the Argand plane. Let $\mathrm{\alpha } = \frac{1}{2}\left(\sqrt{3} + \mathrm{i}\right) \mathrm{and} \mathrm{\beta }$ be a non-zero complex number. The points ${\mathrm{\alpha z}}_1 + \mathrm{\beta }, {\mathrm{\alpha z}}_2 + \mathrm{\beta }, {\mathrm{\alpha z}}_3 + \mathrm{\beta }$ will be

• the vertices of an equilateral triangle

• the vertices of an isosceles triangle 

• collinear

• the vertices of a scalene triangle

In the Argand plane, the distinct roots of 1 + z + z³ + z⁴ = 0 (z is a complex number) represent vertices of

• a square

• an equilateral triangle

• a rhombus

• a rectangle

37.

In a $∆\mathrm{ABC}$,  a, b, c are the sides of the triangle opposite to the angles A, B, C, respectively. Then, the value of a³sin(B - C) + b³sin(C - A) + c³sin(A - B) is equal to

• 0

• 1

• 3

• 2

Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of x² - x - 1 = 0 and ${\mathrm{S}}_{\mathrm{n}} = {\mathrm{\alpha }}^{\mathrm{n}} + {\mathrm{\beta }}^{\mathrm{n}}$, for all integers $\mathrm{n} \ge 1$. Then, for every integer $\mathrm{n} \ge 2$

• ${\mathrm{S}}_{\mathrm{n}} + {\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n}} - {\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n} - 1} = {\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

• ${\mathrm{S}}_{\mathrm{n}} + {\mathrm{S}}_{\mathrm{n} - 1} = 2{\mathrm{S}}_{\mathrm{n} +\hspace{0.17em}1}$

A fair six-faced die is rolled 12 times. The probability that each face turns up twice is equal to

• $\frac{12!}{6! 6! 6^{12}}$

• $\frac{2^{12}}{2^6{6}^{12}}$

• $\frac{12!}{2^6{6}^{12}}$

• $\frac{12!}{6^2{6}^{12}}$

If $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of the quadratic equation x² + px + q = 0, then the values of ${\mathrm{\alpha }}^3 + {\mathrm{\beta }}^3 \mathrm{and} {\mathrm{\alpha }}^4 + {\mathrm{\alpha }}^2{\mathrm{\beta }}^2 + {\mathrm{\beta }}^4$ are respectively

• $3\mathrm{pq} - {\mathrm{p}}^3 \mathrm{and} {\mathrm{p}}^4 - 3{\mathrm{p}}^2\mathrm{q} + 3{\mathrm{q}}^2$

• - p(3q - p²) and (p² - q)(p² + 3q)

• pq - 4 and p⁴ - q⁴

• 3pq - p³ and (p² - q)(p² - 3q)