The function $f(x) = {x^3} - 6{x^2} + ax + b$ is such that $f(2) = f(4) = 0$. Consider two statements : Statement 1 : there exists x1, x2 $\in$(2, 4), x1 < x2, such that f'(x1) = $-$1 and f'(x2) = 0. Statement 2 : there exists x3, x4 $\in$ (2, 4), x3 < x4, such that f is decreasing in (2, x4), increasing in (x4, 4) and $2f'({x_3}) = \sqrt 3 f({x_4})$.Then
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If y = y(x) is the solution curve of the differential equation ${x^2}dy + \left( {y - {1 \over x}} \right)dx = 0$ ; x &…
Topic: JEE Main 2021 (1 September Evening Shift)
Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :
Topic: JEE Main 2021 (1 September Evening Shift)
Consider the system of linear equations$-$x + y + 2z = 0 3x $-$ ay + 5z = 12x, $-$ 2y $-$ az = 7, Let S1 be the set of…
Topic: JEE Main 2021 (1 September Evening Shift)
${\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))$ is equal to : (The inverse trigono…
Topic: JEE Main 2021 (1 September Evening Shift)
Let f : R $\to$ R be a continuous function. Then $\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\li…
Topic: JEE Main 2021 (1 September Evening Shift)
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Let ${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $, $\forall$ n > m and n, m $\in$$ N. C…
Topic: JEE Main 2021 (1 September Evening Shift)
The area, enclosed by the curves $y = \sin x + \cos x$ and $y = \left| {\cos x - \sin x} \right|$ and the lines $x = 0,…
Topic: JEE Main 2021 (1 September Evening Shift)
The distance of line $3y - 2z - 1 = 0 = 3x - z + 4$ from the point (2, $-$1, 6) is :
Topic: JEE Main 2021 (1 September Evening Shift)
The numbers of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation x2 + ax + b = 0, $\a…
Topic: JEE Main 2021 (1 September Evening Shift)
Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such tha…
Topic: JEE Main 2021 (1 September Evening Shift)
The range of the function,$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \lef…
Topic: JEE Main 2021 (1 September Evening Shift)
Let a1, a2, ..........., a21 be an AP such that $\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}} …
Topic: JEE Main 2021 (1 September Evening Shift)
The function f(x), that satisfies the condition $f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy} $, is :
Topic: JEE Main 2021 (1 September Evening Shift)
