JEE Mathematics Definite Integration MCQs Set 01

Question. lim_n→∞ \(\sum_{r=1}^{n} \frac{1}{\sqrt{nr}}\) is equal to
(a) 2
(b) 1
(c) 0
(d) None of the options
Answer: (a) 2

Question. lim_n→∞ \(\sum_{r=0}^{n-1} \frac{1}{\sqrt{n^2-r^2}}\) is
(a) \(\pi\)
(b) \(\pi/2\)
(c) \(\pi/4\)
(d) None of the options
Answer: (b) \(\pi/2\)

Question. lim_n→∞ \(\sum_{r=1}^{n} \frac{1}{n} e^{r/n}\) is
(a) \(e\)
(b) \(e-1\)
(c) \(1-e\)
(d) \(e + 1\)
Answer: (b) \(e-1\)

Question. lim_n→∞ \(\sum_{r=1}^{n} \frac{1}{n} \sin \frac{r\pi}{2n}\) is
(a) \(\pi/2\)
(b) 2
(c) \(2/\pi\)
(d) None of the options
Answer: (c) \(2/\pi\)

Question. lim_n→∞ \(\sum_{r=n}^{4n} \frac{1}{n+r}\) is
(a) \(\log_e 5\)
(b) 0
(c) \(\log_e 4\)
(d) None of the options
Answer: (a) \(\log_e 5\)

Question. lim_n→∞ \(\sum_{r=1}^{2n} \frac{r}{n^2+r^2}\) equals
(a) \(1+\sqrt{5}\)
(b) \(-1+\sqrt{5}\)
(c) \(-1+\sqrt{2}\)
(d) \(1+\sqrt{2}\)
Answer: (b) \(-1+\sqrt{5}\)

Question. lim_n→∞ \(\left(\frac{n!}{(kn)^n}\right)^{1/n}\), where \(k \neq 0\) is a constant and \(n \in N\), is equal to
(a) \(ke\)
(b) \(k^{-1} e\)
(c) \(ke^{-1}\)
(d) \(k^{-1} e^{-1}\)
Answer: (d) \(k^{-1} e^{-1}\)

Question. lim_n→∞ \(\frac{2^k + 4^k + 6^k +....+(2n)^k}{n^{k+1}}\), \(k \neq -1\), is equal to
(a) \(2^k\)
(b) \(\frac{2^k}{k+1}\)
(c) \(\frac{1}{k+1}\)
(d) None of the options
Answer: (b) \(\frac{2^k}{k+1}\)

Question. Let (a, b) and (\(\lambda\), \(\mu\)) be two points on the curve y = f(x). If the slope of the tangent to the curve at (x, y) be \(\phi(x)\) then \(\int_a^\lambda \phi(x)dx\) is
(a) \(\lambda - a\)
(b) \(\mu - b\)
(c) \(\lambda + \mu - a - b\)
(d) None of the options
Answer: (b) \(\mu - b\)

Question. If \(g(x) = \int_0^x \cos^4 t dt\) then \(g(x + \pi)\) equals
(a) \(g(x) + g(\pi)\)
(b) \(g(x) – g(\pi)\)
(c) \(g(x) g(\pi)\)
(d) \(g(x)/g(\pi)\)
Answer: (a) \(g(x) + g(\pi)\)

Question. If \(af(x) + bf\left(\frac{1}{x}\right) = \frac{-5}{x}\), \(x \neq 0\), \(a \neq \pm b\), then \(\int_1^2 f(x)dx\) equals
(a) \(\frac{(\log_e 2-5)a+13}{a^2-b^2}\)
(b) \(\frac{(\log_e 2-5)a+7b}{a^2-b^2}\)
(c) \(\frac{(5-\log_e 2)a+7b}{a^2-b^2}\)
(d) None of the options
Answer: (b) \(\frac{(\log_e 2-5)a+7b}{a^2-b^2}\)

Question. \(\int_0^2 \frac{x^2}{\sqrt{2-x}}dx\) is equal to
(a) \(\pi + 1\)
(b) \(1 + \pi/2\)
(c) \(\pi + 3/2\)
(d) None of the options
Answer: (d) None of the options

Question. The value of \(\int_0^1 \sec^2 xdx\) is
(a) 0
(b) 2
(c) 1
(d) None of the options
Answer: (d) None of the options

Question. If \(a_n = \int_0^{\pi/2} \frac{\sin^2 nx}{\sin x} dx\) then \(a_2-a_1, a_3-a_2, a_4-a_3, ...\) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. Let \(a_n = \int_0^{\pi/2} \tan^n xdx\). Then \(a_2+a_4, a_3+a_5, a_4+a_6\) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. \(\int_1^{e^{3\pi}} \frac{\sin(\pi\log_e x)}{x} dx\) is equal to
(a) 2
(b) -2
(c) \(2/\pi\)
(d) \(2\pi\)
Answer: (a) 2

Question. Let \(\frac{d}{dx} F(x) e^{\sin^2 x}\), \(x > 0\). If \(\int_1^x 2e^{\sin^2 x^2} dx = F(x) – F(1)\) then one of the possible values of k is
(a) 4
(b) -4
(c) 16
(d) None of the options
Answer: (c) 16

Question. If \(f(x) = \int_{-1}^x \frac{\sin t}{1+t^2} dt\) then \(f'\left(\frac{\pi}{3}\right)\) is
(a) nonexistent
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi\sqrt{3}}{4}\)
(d) None of the options
Answer: (b) \(\frac{\pi}{4}\)

Question. The value of \(\int_{\pi^3/27}^{\pi^3/8} \sin x^{1/3} dt\), where \(t = x^3\), is
(a) \(\frac{\pi^2}{6} + (3-\sqrt{3})\pi - 3\)
(b) \(\cos \frac{\pi^3}{27} - \cos \frac{\pi^3}{8}\)
(c) \(\frac{\pi^2}{6}\)
(d) None of the options
Answer: (a) \(\frac{\pi^2}{6} + (3-\sqrt{3})\pi - 3\)

Question. The value of \(\int_1^2 [f\{g(x)\}]' \cdot f'\{g(x)\} \cdot g'(x)dx\), where \(g(1) = g(2)\), is equal to
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (c) 0

Question. If \(f\left(\frac{1}{x}\right) + x^2f(x) = 0\), \(x > 0\), and \(I = \int_{1/2}^2 \frac{f(z)}{z} dz\), \(\frac{1}{2} \le x \le 2\), then \(I\) is
(a) \(f(2) - f(1/2)\)
(b) \(f(1/2) - f(2)\)
(c) 0
(d) None of the options
Answer: (c) 0

Question. \(\int_{\pi/4}^{3\pi/4} \frac{dx}{1+\cos x}\) is equal to
(a) 2
(b) -2
(c) 1/2
(d) -1/2
Answer: (a) 2

Question. If \(f(x)\) satisfies the conditions of Rolle's theorem in [1, 2] then \(\int_1^2 f'(x)dx\) is equal to
(a) 1
(b) 3
(c) 0
(d) None of the options
Answer: (c) 0

Question. The value of \(\int_0^{\pi/2} \sin^6 x dx\) is
(a) \(\frac{105\pi}{32(4!)}\)
(b) \(\frac{105\pi}{16(4!)}\)
(c) \(\frac{105}{16(4!)}\)
(d) None of the options
Answer: (a) \(\frac{105\pi}{32(4!)}\)

Question. If \(f(x)\) be a quadratic polynomial such that \(f(0) = 2\), \(f'(0) = -3\) and \(f''(0) = 4\) then \(\int_1^2 f(x)dx\) is equal to
(a) -3
(b) 16/3
(c) 0
(d) None of the options
Answer: (b) 16/3

Question. Choose the correct options. One or more options may be correct.
Let \(I_n = \int_0^{\pi/2} \cos^n xdx\), \(n \in N\). Then
(a) \(I_{n-2} > I_n\)
(b) \(n(I_{n-2} - I_n) = I_{n-2}\)
(c) \(I_n : I_{n-1} = n : (n-1)\)
(d) None of the options
Answer: (a) \(I_{n-2} > I_n\), (b) \(n(I_{n-2} - I_n) = I_{n-2}\)

Question. Let \(I_n = \int_0^{\pi/4} \tan^n xdx\), \(n \in N\). Then
(a) \(I_1 = I_3 + 2I_5\)
(b) \(I_n + I_{n-2} = \frac{1}{n}\)
(c) \(I_n + I_{n-2} = \frac{1}{n-1}\)
(d) None of the options
Answer: (a) \(I_1 = I_3 + 2I_5\), (c) \(I_n + I_{n-2} = \frac{1}{n-1}\)