JEE Advanced 2020 Paper 1 Offline

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 ...

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 whic...

Let the functions f : R \(\to\) R and g : R \(\to\) R be defined byf(x) = ex \(-\) 1 \(-\) e\(-\)|x \(-\) 1|and g(x) = \({1 \over 2}\)(ex \(-\) 1 + e1 \(-\) x).The the area of the region in the first quadrant bounded by the curves y = f...

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

Which of the following inequalities is/are TRUE?

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 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]:...

For a polynomial g(x) with real coefficients, let mg 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}):...

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 ....

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 follo...

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 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 poin...

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, indepe...

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 ......

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...

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) ...

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...

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} ...