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Application of Integration

28 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Application of Integration

Application of Integration

28 questions · Mathematics · JEE Advanced

Let ℝ denote the set of all real numbers. Then the area of the region $ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x}, 5x - 4y - 1 > 0, 4x + 4y - 17 is2025 · Shift 2 · Q18 · MCQ
Let $S = {(x, y) \in R \times R : x \geq 0, y \geq 0, y^{2} \leq 4 x, y^{2} \leq 12 - 2 x$ and $3 y + 8 x \leq 58}$ . If the area of the region $S$ is $α 2$ , then $α$ is equal…2024 · Shift 2 · Q19 · MCQ
Let the function $f : [1, \infty) \to R$ be defined by $f (t) = {(- 1)^{n + 1} \cdot 2 \frac{( 2 n + 1 - t )}{2} f (2 n - 1) + \frac{( t - ( 2 n - 1 ))}{2} f (2 n + 1) if t = 2 n - 1, n \in N if 2 n - 1 < t < 2 n + 1, n \in N$ …2024 · Shift 2 · Q30 · Numerical
Let $n \geq 2$ be a natural number and $f : [0, 1] \to R$ be the function defined by $f (x) = ⎩ ⎨ ⎧ n (1 - 2 n x) 2 n (2 n x - 1) 4 n (1 - n x) \frac{n}{n - 1} (n x - 1) if 0 \leq x \leq \frac{1}{2 n} if \frac{1}{2 n} \leq x \leq \frac{3}{4 n} if \frac{3}{4 n} \leq x \leq \frac{1}{n} if \frac{1}{n} \leq x \leq 1$ …2023 · Shift 1 · Q26 · Numerical
Consider the functions $f, g : R \to R$ defined by $f (x) = x^{2} + \frac{5}{12} and g (x) = {2 (1 - \frac{4∣ x ∣}{3}), 0, ∣ x ∣ \leq \frac{3}{4} ∣ x ∣ > \frac{3}{4}$ …2022 · Shift 2 · Q26 · Numerical
The area of the region ${(x, y) : 0 \leq x \leq \frac{9}{4}, 0 \leq y \leq 1, x \geq 3 y, x + y \geq 2}$ is2021 · Shift 1 · Q21 · MCQ
For any real numbers $α$ and $β$ , let $y_{α, β} (x)$ , x $\in$ R, be the solution of the differential equation $\frac{d y}{d x} + α y = x e^{β x}, y (1) = 1$ . Let $S = {y_{α, β} (x) : α, β \in R}$ …2021 · Shift 2 · Q23 · Multiple correct
Let f1 : (0, $\infty$ ) $\to$ R and f2 : (0, $\infty$ ) $\to$ R be defined by $f_{1} (x) = 0 \int x j = 1 \prod 21 (t - j)^{j} d t$ , x > 0 and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0$ ,…2021 · Shift 2 · Q28 · Numerical
Let f1 : (0, $\infty$ ) $\to$ R and f2 : (0, $\infty$ ) $\to$ R be defined by $f_{1} (x) = 0 \int x j = 1 \prod 21 (t - j)^{j} d t$ , x > 0 and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450, x > 0$ ,…2021 · Shift 2 · Q29 · Numerical
Let the functions f : R $\to$ R and g : R $\to$ R be defined by f(x) = ex $-$ 1 $-$ e $-$ |x $-$ 1| and g(x) = $\frac{1}{2}$ (ex $-$ 1 + e1 $-$ x). The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and…2020 · Shift 1 · Q21 · MCQ
The area of the region {(x, y) : xy $\leq$ 8, 1 $\leq$ y $\leq$ x2} is2019 · Shift 1 · Q22 · MCQ
Let f : [0, $\infty$ ) $\to$ R be a continuous function such that $f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) d t$ for all x $\in$ [0, $\infty$ ). Then, which of the following statement(s) is (are) TRUE?2018 · Shift 1 · Q24 · Multiple correct
A farmer F1 has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F2 takes away the region which lies between the sides PQ and a curve of the form y = xn (n > 1). If…2018 · Shift 1 · Q32 · Numerical
If the line x = $α$ divides the area of region R = {(x, y) $\in$ R2 : x3 $\leq$ y $\leq$ x, 0 $\leq$ x $\leq$ 1} into two equal parts, then2017 · Shift 2 · Q28 · Multiple correct
Area of the region ${(x, y) \in R^{2} : y \geq ∣ x + 3 ∣, 5 y \leq x + 9 \leq 15}$ is equal to2016 · Shift 2 · Q26 · MCQ
Let $F (x) = x \int x^{2} + \frac{π}{6} 2 cos^{2} t (d t)$ for all $x \in R$ and $f : [0, \frac{1}{2}] \to [0, \infty]$ be a continuous function. For $a \in [0, \frac{1}{2}],$ …2015 · Shift 1 · Q31 · Numerical
Let $f : R \to R$ be a continuous odd function, which vanishes exactly at one point and $f (1) = \frac{1}{2.}$ Suppose that $F (x) = - 1 \int x f (t) d t$ for all $x \in [- 1, 2]$ …2015 · Shift 2 · Q26 · Numerical
Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F (x) < 0$ for all $x \in (\frac{1}{2}, 3) .$ Let $f (x) = x F (x)$ …2015 · Shift 2 · Q37 · Multiple correct
The area enclosed by the curves $y = sin x + cosx o l imi t s$ and $y = ∣ cos x - sin x ∣$ over the interval $[0, \frac{π}{2}]$ is2013 · Shift 1 · Q34 · MCQ
Let $S$ be the area of the region enclosed by $y = e^{- x^{2}}$ , $y = 0$ , $x = 0$ , and $x = 1$ ; then2012 · Shift 1 · Q33 · Multiple correct
Let the straight line $x = b$ divide the area enclosed by $y = (1 - x)^{2}, y = 0,$ and $x = 0$ into two parts $R_{1} (0 \leq x \leq b)$ and $R_{2} (b \leq x \leq 1)$ such that $R_{1} - R_{2} = \frac{1}{4} .$ …2011 · Shift 1 · Q35 · MCQ
Let f $:$ $[- 1, 2] \to [0, \infty]$ be a continuous function such that $f (x) = f (1 - x)$ for all $x \in [- 1, 2]$ Let $R_{1} = - 1 \int 2 x f (x) d x,$ …2011 · Shift 2 · Q28 · MCQ
Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f (x) = ln x + 0 \int x 1 + sin t d t .$ then which of the following statement(s) is (are) true?2010 · Shift 1 · Q47 · Multiple correct
Consider the polynomial $f (x) = 1 + 2 x + 3 x^{2} + 4 x^{3} .$ Let $s$ be the sum of all distinct real roots of $f (x)$ and let $t = ∣ s ∣ .$ The area bounded by the curve $y = f (x)$ and the lines $x = 0, y = 0$ and $x = t,$ …2010 · Shift 2 · Q34 · MCQ
Area of the region bounded by the curve $y = e^{x}$ and lines $x = 0$ and $y = e$ is2009 · Shift 1 · Q28 · Multiple correct
Let $f$ be a non-negative function defined on the interval $[0, 1]$ . If $0 \int x 1 - (f^{'} (t))^{2} d t = 0 \int x f (t) d t, 0 \leq x \leq 1$ , and $f (0) = 0$ , then2009 · Shift 1 · Q29 · MCQ
Consider the functions defined implicitly by the equation $y^{3} - 3 y + x = 0$ on various intervals in the real line. If $x \in (- \infty, - 2) \cup (2, \infty)$ , the equation implicitly defines a unique real valued differentiable function $y = f (x)$ . If $x \in (- 2, 2)$ …2008 · Shift 1 · Q30 · MCQ
The area of the region between the curves $y = \frac{1 + s i n x}{c o s x}$ and $y = \frac{1 - s i n x}{c o s x}$ bounded by the lines $x = 0$ and $x = \frac{π}{4}$ is2008 · Shift 2 · Q32 · MCQ