Suppose a, b denote the distinct real roots of the quadratic polynomial x² + 20x $$-$$ 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x² $$-$$ 20x + 2020. Then the value of

ac(a $$-$$ c) + ad(a $$-$$ d) + bc(b $$-$$ c) + bd(b $$-$$ d) is

If the function f : R $$ \to $$ R is defined by f(x) = |x| (x $$-$$ sin x), then which of the following statements is TRUE?

Let the functions f : R $$ \to $$ R and g : R $$ \to $$ R be defined by

f(x) = e^{x $$-$$ 1} $$-$$ e^{$$-$$|x $$-$$ 1|}

and g(x) = $${1 \over 2}$$(e^{x $$-$$ 1} + e^{1 $$-$$ x}).

The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is

Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the
parabola y² = 4$$\lambda $$x, and suppose the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

Let C₁ and C₂ be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and $${{1 \over 3}}$$, respectively. Suppose $$\alpha $$ is the number of heads that appear when C₁ is tossed twice, independently, and suppose $$\beta $$ is the number of heads that appear when C₂ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x² $$-$$ ax + $$\beta $$ are real and equal, is

Consider the rectangles lying the region

$$\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$$ and $$\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$$

and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Let the function f : R $$ \to $$ R be defined by f(x) = x³ $$-$$ x² + (x $$-$$ 1)sin x and let g : R $$ \to $$ R be an arbitrary function. Let fg : R $$ \to $$ R be the product function defined by (fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?

Let M be a 3 $$ \times $$ 3 invertible matrix with real entries and let I denote the 3 $$ \times $$ 3 identity matrix. If M^$$-$$1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?

Let S be the set of all complex numbers z
satisfying |z² + z + 1| = 1. Then which of the following statements is/are TRUE?

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y, and Z, respectively. If

$$\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}$$, then which of the following statements is/are TRUE?

Let L₁ and L₂ be the following straight lines.

$${L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}$$ and $${L_2}:{{x - 1} \over { - 3}} = {y \over { - 1}} = {{z - 1} \over 1}$$.

Suppose the straight line

$$L:{{x - \alpha } \over l} = {{y - 1} \over m} = {{z - \gamma } \over { - 2}}$$

lies in the plane containing L₁ and L₂ and passes through the point of intersection of L₁ and L₂. If the line L bisects the acute angle between the lines L₁ and L₂, then which of the following statements is/are TRUE?

Which of the following inequalities is/are TRUE?

Let m be the minimum possible value of $${\log _3}({3^{{y_1}}} + {3^{{y_2}}} + {3^{{y_3}}})$$, where $${y_1},{y_2},{y_3}$$ are real numbers for which $${{y_1} + {y_2} + {y_3}}$$ = 9. Let M be the maximum possible value of $$({\log _3}{x_1} + {\log _3}{x_2} + {\log _3}{x_3})$$, where $${x_1},{x_2},{x_3}$$ are positive real numbers for which $${{x_1} + {x_2} + {x_3}}$$ = 9. Then the value of $${\log _2}({m^3}) + {\log _3}({M^2})$$ is ...........

Let a₁, a₂, a₃, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b₁, b₂, b₃, .... be a sequence of positive integers in geometric progression with common ratio 2. If a₁ = b₁ = c, then the number of all possible values of c, for which the equality 2(a₁ + a₂ + ... + a_n) = b₁ + b₂ + ... + b_n holds for some positive integer n, is ...........

Let f : [0, 2] $$ \to $$ R be the function defined by

$$f(x) = (3 - \sin (2\pi x))\sin \left( {\pi x - {\pi \over 4}} \right) - \sin \left( {3\pi x + {\pi \over 4}} \right)$$

If $$\alpha ,\,\beta \in [0,2]$$ are such that $$\{ x \in [0,2]:f(x) \ge 0\} = [\alpha ,\beta ]$$, then the value of $$\beta - \alpha $$ is ..........

In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4

and $${{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}$$, then the value of |a $$ \times $$ b|² is ......

For a polynomial g(x) with real coefficients, let m_g denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by

$$S = \{ {({x^2} - 1)^2}({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}):{a_0},{a_1},{a_2},{a_3} \in R\} $$;

For a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (m_f' + m_f''), where f $$ \in $$ S, is ..............

let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{(1 - x)}^{1/x}} - {e^{ - 1}}} \over {{x^a}}}$$

is equal to a non-zero real number, is .............