JEE Mathematics Area MCQs

Question. \(P(\alpha, f(\alpha))\) and \(Q(\beta, f(\beta))\) are ends of an arc in the first quadrant. The area bounded by the arc, ordinates through \(P\) and \(Q\), and the x-axis is
(a) \(\int_{f(\alpha)}^{f(\beta)} f^{-1}(y)dy\)
(b) \(\int_{\alpha}^{\beta} f^{-1}(y)dy\)
(c) \(\int_{\alpha}^{\beta} f(x)dx\)
(d) \(\int_{f(\alpha)}^{f(\beta)} f(x)dx\)
Answer: (c) \(\int_{\alpha}^{\beta} f(x)dx\)

Question. The area bounded by the lines \(y = |x - 2|\), \(|x| = 3\) and \(y = 0\) is
(a) 13 unit\(^2\)
(b) 5 unit\(^2\)
(c) 9 unit\(^2\)
(d) 7 unit\(^2\)
Answer: (a) 13 unit\(^2\)

Question. The area bounded by the curve \(y = \sqrt{4 - x^2}\) and the line \(y = 0\) is
(a) \(4\pi\)
(b) \(2\pi\)
(c) \(\pi\)
(d) \(\frac{\pi}{2}\)
Answer: (b) \(2\pi\)

Question. The area bounded by the curve \(y^2 = 4x\) and the double ordinate \(x = 2\), is
(a) \(\frac{4 \sqrt{2}}{3}\) unit\(^2\)
(b) \(\frac{8 \sqrt{2}}{3}\) unit\(^2\)
(c) \(\frac{16 \sqrt{2}}{3}\) unit\(^2\)
(d) \(\frac{32 \sqrt{2}}{3}\) unit\(^2\)
Answer: (c) \(\frac{16 \sqrt{2}}{3}\) unit\(^2\)

Question. The area bounded by the curve \(x^2 = ky, k > 0\) and the line \(y = 3\) is 12 unit\(^2\). Then \(k\) is
(a) 3
(b) \(3 \sqrt{3}\)
(c) \(\frac{3}{4}\)
(d) None of the options
Answer: (a) 3

Question. The area bounded by the curve \(y = 2^x\), the x-axis and the y-axis is
(a) \(\log_e 2\)
(b) \(\log_e 4\)
(c) \(\log_4 e\)
(d) \(\log_2 e\)
Answer: (d) \(\log_2 e\)

Question. The area of the portion enclosed by the curve \(\sqrt{x} + \sqrt{y} = \sqrt{a}\) and the axes of reference is
(a) \(\frac{a^2}{6}\)
(b) \(a^2\)
(c) \(\frac{a^2}{2}\)
(d) \(\frac{a^2}{4}\)
Answer: (a) \(\frac{a^2}{6}\)

Question. The area bounded by the curve \(x = \cos^{-1} y\) and the lines \(|x| = 1\) is
(a) \(\sin 1\)
(b) \(\cos 1\)
(c) \(2 \sin 1\)
(d) \(2 \cos 1\)
Answer: (c) \(2 \sin 1\)

Question. The area bounded by the curve \(y = \sqrt{x}\), the line \(2y + 3 = x\) and the x-axis in the first quadrant is
(a) 9
(b) \(\frac{27}{4}\)
(c) 36
(d) 18
Answer: (a) 9

Question. The area of the region bounded by the pairs of lines \(y = |x - 1|\) and \(y = 3 - |x|\) is
(a) 3 unit\(^2\)
(b) 4 unit\(^2\)
(c) 6 unit\(^2\)
(d) 2 unit\(^2\)
Answer: (b) 4 unit\(^2\)

Question. The ratio in which the area bounded by the curves \(y^2 = 4x\) and \(x^2 = 4y\) is divided by the line \(x = 1\) is
(a) 64 : 49
(b) 15 : 34
(c) 15 : 49
(d) None of the options
Answer: (c) 15 : 49

Question. Let \(f(x)\) be a continuous function such that the area bounded by the curve \(y = f(x)\), the x-axis and the two ordinates \(x = 0\) and \(x = a\) is \(\frac{a^2}{2} + \frac{a}{2} \sin a + \frac{\pi}{2} \cos a\). The \(f\left(\frac{\pi}{2}\right)\) is
(a) \(\frac{1}{2}\)
(b) \(\frac{\pi^2}{8} + \frac{\pi}{4}\)
(c) \(\frac{\pi + 1}{2}\)
(d) \(\frac{\pi}{2}\)
Answer: (a) \(\frac{1}{2}\)

Question. The area bounded by the curve \(f(x) = ce^x (c > 0)\), the x-axis and the two ordinates \(x = p\) and \(x = q\) is proportional to
(a) \(f(p) \cdot f(q)\)
(b) \(|f(p) - f(q)|\)
(c) \(f(p) + f(q)\)
(d) \(\sqrt{f(p)f(q)}\)
Answer: (b) \(|f(p) - f(q)|\)

Question. The ordinate \(x = a\) divides the area bounded by the x-axis, the curve \(y = 1 + \frac{8}{x^2}\) and the ordinates \(x = 2\) and \(x = 4\) into equal parts. Then \(a\) is
(a) 3
(b) \(\frac{7}{2}\)
(c) \(2 \sqrt{2}\)
(d) \(\frac{5}{2}\)
Answer: (c) \(2 \sqrt{2}\)

Question. The area of the ellipse \(\frac{(x + 1)^2}{4} + y^2 = 1\) falling in the first quadrant is
(a) \(\frac{1}{6}(4\pi - 3\sqrt{3})\)
(b) \(\frac{1}{12}(4\pi - 3\sqrt{3})\)
(c) \(\frac{\sqrt{3}}{2}\)
(d) \(\frac{1}{3}(\pi - \sqrt{3})\)
Answer: (b) \(\frac{1}{12}(4\pi - 3\sqrt{3})\)