JEE Mathematics Some Additional Topics MCQs

INFINITE SERIES

Question. \( (1 - x)^{3/2} \) can be expanded in ascending powers of \( x \) if
(a) \( -1 < x < 1 \)
(b) \( x < -1 \)
(c) \( x > 1 \)
(d) None of the options
Answer: (a) \( -1 < x < 1 \)

Question. \( (3 + x)^{p/q} \) can be expanded in ascending powers of \( x \) if
(a) \( -1 < x < 1 \)
(b) \( x > 3 \)
(c) \( -3 < x < 3 \)
(d) \( x < -3 \)
Answer: (c) \( -3 < x < 3 \)

Question. If \( x \) is positive, the first negative term in the expansion of \( (1 + x)^{27/6} \) is the
(a) 5th term
(b) 7th term
(c) 6th term
(d) 8th term
Answer: (b) 7th term

Question. In the expansion of \( (1 - x)^{-3}, |x| < 1 \), the coefficient of \( x^7 \) is
(a) 36
(b) \( ^8C_7 \)
(c) 45
(d) None of the options
Answer: (a) 36

Question. In the expansion of \( (2 + x)^{-5/2} \), the coefficient of \( x^4 \), if it exists, is
(a) \( \frac{5 \cdot 7 \cdot 9 \cdot 11}{2^{5/2} \cdot 4!} \left( \frac{1}{2} \right)^4 \)
(b) \( \frac{5 \cdot 7 \cdot 9 \cdot 11}{4!} \left( \frac{1}{2} \right)^4 \)
(c) \( \frac{5 \cdot 7 \cdot 9 \cdot 11}{(\sqrt{2})^{21}} \)
(d) None of the options
Answer: (c) \( \frac{5 \cdot 7 \cdot 9 \cdot 11}{(\sqrt{2})^{21}} \)

Question. The coefficient of \( x^5 \) in the expansion of \( \frac{1 + x^2}{1 + x}, |x| < 1 \), is
(a) -1
(b) 2
(c) 0
(d) -2
Answer: (d) -2

Question. If \( \frac{x^2 + x}{1 - x} = a_1x + a_2x^2 + \dots \text{ to } \infty, |x| < 1 \), then
(a) \( a_1 + a_2 = 4 \)
(b) \( a_1 - a_2 = 3 \)
(c) \( a_p = a_q \)
(d) None of the options
Answer: (c) \( a_p = a_q \)

Question. The coefficient of \( x^n \) in the expansion of \( e^{2x+3} \) is
(a) \( \frac{2^n}{n!} \)
(b) \( \frac{e^3 \cdot 2^n}{n!} \)
(c) \( \frac{e^2 \cdot 3^n}{n!} \)
(d) None of the options
Answer: (b) \( \frac{e^3 \cdot 2^n}{n!} \)

Question. The coefficient of \( x^{10} \) in the expansion of \( 10^x \) in ascending powers of \( x \) is
(a) \( \frac{(\log_e 10)^{10}}{10!} \)
(b) \( \frac{1}{10!} \)
(c) \( \frac{10(\log_{10} e)^{10}}{10!} \)
(d) None of the options
Answer: (a) \( \frac{(\log_e 10)^{10}}{10!} \)

Question. In the expansion of \( \frac{e^x - 1 - x}{x^2} \) in ascending powers of \( x \), the fourth term is
(a) \( \frac{1}{5!}x^3 \)
(b) \( \frac{1}{4!}x^4 \)
(c) \( \frac{1}{3!}x^3 \)
(d) None of the options
Answer: (a) \( \frac{1}{5!}x^3 \)

Question. The constant term in the expansion of \( \frac{3^x - 2^x}{x^2} \) is
(a) \( \log_e 3 \)
(b) \( \log_e 6 \cdot \log_e \frac{3}{2} \)
(c) \( \frac{1}{2} \log_e 6 \cdot \log_e \frac{3}{2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \log_e 6 \cdot \log_e \frac{3}{2} \)

Question. If \( |x| < 1 \), the coefficient of \( x^3 \) in the expansion of \( \frac{1}{e^x \cdot (1 + x)} \) is
(a) \( \frac{17}{6} \)
(b) \( -\frac{17}{6} \)
(c) \( -\frac{11}{6} \)
(d) None of the options
Answer: (b) \( -\frac{17}{6} \)

Question. The constant term in the expansion of \( \frac{x + \log_e (1 - x)}{x^2} \) is
(a) \( -\frac{1}{2} \)
(b) 0
(c) \( -\frac{1}{2} \)
(d) \( \frac{1}{3} \)
Answer: (a) \( -\frac{1}{2} \)

Question. In the expansion of \( \log_{10}(1 - x), |x| < 1 \), the coefficient of \( x^n \) is
(a) \( -\frac{1}{n} \)
(b) \( -\frac{1}{n} \log_{10} e \)
(c) \( \frac{1}{n} \)
(d) None of the options
Answer: (b) \( -\frac{1}{n} \log_{10} e \)

Question. If \( |x| < 1 \), the coefficient of \( x^2 \) in the expansion of \( \frac{\log_e (1 + x)}{(1 - x)^2} \) is
(a) \( \frac{3}{2} \)
(b) \( ^2C_1 \)
(c) \( -\frac{1}{2} \)
(d) None of the options
Answer: (a) \( \frac{3}{2} \)

Question. The sum of the series \( ^4C_0 + ^5C_1x + ^6C_2x^2 + ^7C_3x^3 + \dots \text{ to } \infty \) is
(a) \( (1 - x)^{-4} \)
(b) \( \frac{1}{(1 - x)^5} \)
(c) \( (1 + x)^{-5} \)
(d) None of the options
Answer: (b) \( \frac{1}{(1 - x)^5} \)

Question. The sum of the series \( ^2C_0 - ^3C_1x^2 + ^4C_2x^4 - ^5C_3x^6 + \dots \text{ to } \infty \) is
(a) \( \frac{1}{(1 + x^2)^3} \)
(b) \( (1 - x^2)^{-3} \)
(c) \( \frac{1}{(1 + x)^3} \)
(d) None of the options
Answer: (a) \( \frac{1}{(1 + x^2)^3} \)

Question. The sum of series \( 1 + \frac{1}{1!} \cdot \frac{1}{4} + \frac{1 \cdot 3}{2!} \cdot \left( \frac{1}{4} \right)^2 + \frac{1 \cdot 3 \cdot 5}{3!} \cdot \left( \frac{1}{4} \right)^3 + \dots \text{ to } \infty \) is
(a) \( \sqrt{2} \)
(b) 2
(c) \( \frac{1}{\sqrt{2}} \)
(d) None of the options
Answer: (a) \( \sqrt{2} \)

Question. The sum of the series \( ^3C_0 - ^4C_1 \left( \frac{1}{2} \right) + ^5C_2 \left( \frac{1}{2} \right)^2 - ^6C_3 \left( \frac{1}{2} \right)^3 + \dots \text{ to } \infty \) is
(a) 16
(b) 8
(c) \( \frac{16}{81} \)
(d) None of the options
Answer: (c) \( \frac{16}{81} \)

Question. \( 1 + \frac{1}{3!} + \frac{1}{5!} + \frac{1}{7!} + \dots \text{ to } \infty \) is equal to
(a) \( \frac{1}{2} (e - e^{-1}) \)
(b) \( \frac{1}{2} (e + e^{-1}) \)
(c) \( \frac{1}{2} e \)
(d) None of the options
Answer: (a) \( \frac{1}{2} (e - e^{-1}) \)

Question. \( \frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + \frac{4}{3!} + \dots \text{ to } \infty \) is equal to
(a) 4e
(b) 3e
(c) 2e
(d) None of the options
Answer: (c) 2e

Question. \( \frac{1}{3!} + \frac{2}{5!} + \frac{3}{7!} + \dots \text{ to } \infty \) is equal to
(a) \( \frac{1}{2} e \)
(b) \( \frac{1}{2e} \)
(c) \( \frac{3}{2e} \)
(d) None of the options
Answer: (b) \( \frac{1}{2e} \)

Question. \( (a^2 - b^2) + \frac{a^4 - b^4}{2!} + \frac{a^6 - b^6}{3!} + \dots \text{ to } \infty \) is equal to
(a) \( e^{a^2} - e^{b^2} \)
(b) \( e^{b^2} - e^{a^2} \)
(c) \( \frac{e^{a^2}}{e^{b^2}} \)
(d) None of the options
Answer: (a) \( e^{a^2} - e^{b^2} \)

Question. \( \frac{a - b}{a} + \frac{1}{2} \left( \frac{a - b}{a} \right)^2 + \frac{1}{3} \left( \frac{a - b}{a} \right)^3 + \dots \text{ to } \infty \) is equal to
(a) \( \log a + \log b \)
(b) \( \log \frac{b}{a} \)
(c) \( \log a - \log b \)
(d) None of the options
Answer: (c) \( \log a - \log b \)

Question. \( \sum_{n=1}^{\infty} \frac{1}{2n - 1} \cdot x^{2n} \) is equal to
(a) \( \frac{x}{2} \log \left( \frac{1 + x}{1 - x} \right) \)
(b) \( \frac{1}{2} \log \left( \frac{1 + x^2}{1 - x^2} \right) \)
(c) \( x \log \frac{1 + x}{1 - x} \)
(d) None of the options
Answer: (a) \( \frac{x}{2} \log \left( \frac{1 + x}{1 - x} \right) \)

Question. \( \sum_{n=1}^{\infty} \frac{1}{2n(2n + 1)} \) is equal to
(a) \( \log_e \left( \frac{2}{e} \right) \)
(b) \( 1 - \log_e 2 \)
(c) \( 1 - \log_e \frac{1}{2e} \)
(d) None of the options
Answer: (b) \( 1 - \log_e 2 \)

Question. If \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \text{ to } \infty = y \) then \( y + \frac{y^2}{2!} + \frac{y^3}{3!} + \dots \text{ to } \infty \) is equal to
(a) \( -x \)
(b) \( x \)
(c) \( x + 1 \)
(d) None of the options
Answer: (b) \( x \)

Question. The sum of the series \( \frac{1}{1 \cdot 2} - \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} - \dots \) up to \( \infty \) is equal to
(a) \( \log_e 2 - 1 \)
(b) \( \log_e 2 \)
(c) \( \log_e \frac{4}{e} \)
(d) \( 2 \log_e 2 \)
Answer: (a) \( \log_e 2 - 1 \)

PARTIAL FRACTIONS

Question. If \( \frac{3x + 5}{2x^2 + x - 6} = \frac{A}{2x - 3} + \frac{B}{x + 2} \) identically then
(a) \( A = \frac{1}{7}, B = \frac{19}{7} \)
(b) \( A = -\frac{19}{7}, B = \frac{1}{7} \)
(c) \( A = \frac{19}{7}, B = \frac{1}{7} \)
(d) None of the options
Answer: (c) \( A = \frac{19}{7}, B = \frac{1}{7} \)

Question. If \( f(x) \) is a function of \( x \) such that \( \frac{1}{(1 + x)(1 + x^2)} = \frac{A}{1 + x} + \frac{f(x)}{1 + x^2} \) for all \( x \in \mathbb{R} \) then \( f(x) \) is
(a) \( \frac{1 - x}{2} \)
(b) \( \frac{x + 1}{2} \)
(c) \( 1 - x \)
(d) None of the options
Answer: (a) \( \frac{1 - x}{2} \)

Question. If \( |x| < 1 \), the coefficient of \( x^n \) in the expansion of \( \frac{1}{x^2 - 3x + 2} \) is
(a) \( 1 - \frac{1}{2^n} \)
(b) \( 1 - \frac{1}{2^{n+1}} \)
(c) \( 1 - 2^n \)
(d) None of the options
Answer: (b) \( 1 - \frac{1}{2^{n+1}} \)

Question. If \( |x| < \frac{1}{2} \), the coefficient of \( x^4 \) in the expansion of \( \frac{1}{(1 + 2x)(1 - x^2)} \) is
(a) 1
(b) 2
(c) 21
(d) None of the options
Answer: (c) 21

Question. The sum to infinite terms of the series \( \frac{1}{(1 + a)(2 + a)} + \frac{1}{(2 + a)(3 + a)} + \frac{1}{(3 + a)(4 + a)} + \dots \), where \( a \) is a constant, is
(a) \( \frac{1}{1 + a} \)
(b) \( \frac{2}{1 + a} \)
(c) \( \infty \)
(d) None of the options
Answer: (a) \( \frac{1}{1 + a} \)

SURDS

Question. If \( x = 1 + \sqrt{2} - \sqrt{3} \) then the reciprocal of \( x \) is
(a) \( \frac{1}{2} - \frac{1}{4} (\sqrt{6} + \sqrt{2}) \)
(b) \( \sqrt{3} - \sqrt{2} - 1 \)
(c) \( \frac{1}{2} + \frac{1}{4} (\sqrt{6} + \sqrt{2}) \)
(d) None of the options
Answer: (c) \( \frac{1}{2} + \frac{1}{4} (\sqrt{6} + \sqrt{2}) \)

Question. The value of \( \frac{1}{\sqrt{40} + \sqrt{20} + \sqrt{10} - \sqrt{80}} \) is equal to
(a) \( \frac{1}{70} (3\sqrt{10} + 2\sqrt{5}) \)
(b) \( \frac{3\sqrt{10} - 2\sqrt{5}}{70} \)
(c) \( \frac{3\sqrt{10} + 2\sqrt{5}}{50} \)
(d) None of the options
Answer: (a) \( \frac{1}{70} (3\sqrt{10} + 2\sqrt{5}) \)

Question. The square root of \( 2x + 2\sqrt{x^2 - 1} \) is
(a) \( \sqrt{x + 1} - \sqrt{x - 1} \)
(b) \( \sqrt{x + 1} + \sqrt{x - 1} \)
(c) \( \sqrt{x - 1} - \sqrt{x + 1} \)
(d) None of the options
Answer: (b) \( \sqrt{x + 1} + \sqrt{x - 1} \)

Question. If \( \frac{4 + 3\sqrt{3}}{\sqrt{7 + 4\sqrt{3}}} = x + \sqrt{y}, x \in \mathbb{Z}, y \in \mathbb{Z} \) (where \( \mathbb{Z} = \) the set of integers) then
(a) \( x = -1, y = 12 \)
(b) \( x = 1, y = -12 \)
(c) \( x = 1, y = 12 \)
(d) None of the options
Answer: (a) \( x = -1, y = 12 \)

Question. If \( x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}, y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \) then the value of \( x^2 + xy + y^2 \) is
(a) 5
(b) 99
(c) 98
(d) None of the options
Answer: (b) 99

Question. If \( x = \frac{\sqrt{3}}{2} \) then \( \frac{1 + x}{1 + \sqrt{1 + x}} + \frac{1 - x}{1 - \sqrt{1 - x}} \) is equal to
(a) 1
(b) \( \frac{\sqrt{3}}{2} \)
(c) \( \frac{1}{\sqrt{3}} \)
(d) None of the options
Answer: (a) 1

VARIATION

Question. If \( x \) varies directly as \( y \), and \( x = 2 \) when \( y = 3 \) then \( x = 3 \) when \( y \) is
(a) 2
(b) \( \frac{2}{3} \)
(c) \( \frac{9}{2} \)
(d) None of the options
Answer: (c) \( \frac{9}{2} \)

Question. If \( x \) caries inversely as \( y \), and \( x = 5 \) when \( y = 2 \) then for \( y = 5, x \) is
(a) \( \frac{2}{5} \)
(b) 2
(c) \( \frac{5}{2} \)
(d) None of the options
Answer: (b) 2

Question. If \( x \propto yz \) and \( y \propto xz \) then
(a) \( z \propto xy \)
(b) \( z \) is a constant
(c) \( xyz \) is a constant
(d) None of the options
Answer: (b) \( z \) is a constant

Question. \( A \) varies as \( B \) and \( C \) jointly, and \( A = 2 \) when \( B = \frac{3}{5}, C = \frac{10}{27} \). The value of \( A \), when \( B = 2, C = \frac{5}{3} \), is
(a) 30
(b) 10
(c) \( \frac{36}{5} \)
(d) None of the options
Answer: (a) 30

Question. Let \( y \propto p + q \) where \( p \) varies directly as \( x \) and \( q \) varies inversely as \( x^2 \). If \( y = 19 \) when \( x = 2 \) or 3 then \( y \) in terms of \( x \) is
(a) \( 36x + \frac{5}{x^2} \)
(b) \( \frac{5}{x} + 36x^2 \)
(c) \( 5x + \frac{36}{x^2} \)
(d) None of the options
Answer: (c) \( 5x + \frac{36}{x^2} \)

Choose the correct options. One or more options may be correct.

INFINITE SERIES

Question. In the expansion of \( (1 + x)^{-3} \)
(a) the third term is \( ^4C_2x^2 \), if \( |x| < 1 \)
(b) the third term is \( ^4C_2 \left( \frac{1}{x} \right)^2 \), if \( x > 1 \)
(c) the value of the third term is \( -\frac{3}{2} \), when \( x = \frac{1}{2} \)
(d) the value of the third tem is \( -\frac{2}{81} \), when \( x = -3 \)
Answer: (a) the third term is \( ^4C_2x^2 \), if \( |x| < 1 \) AND (d) the value of the third tem is \( -\frac{2}{81} \), when \( x = -3 \)

Question. If \( \frac{3ax}{4} + \sqrt{4 + ax} - \frac{2}{\sqrt{1 - ax}} = -x^2 + bx^3 + \dots \text{ to } \infty \) then
(a) \( a = \frac{8}{7}, b = -\frac{319}{343} \)
(b) \( a = -\frac{8}{7}, b = \frac{319}{343} \)
(c) \( a = \frac{8}{7}, b = \frac{319}{343} \)
(d) \( a = -\frac{8}{7}, b = -\frac{319}{343} \)
Answer: (a) \( a = \frac{8}{7}, b = -\frac{319}{343} \) AND (b) \( a = -\frac{8}{7}, b = \frac{319}{343} \)

Question. If \( e^{ax} + e^{-bx} = p_0 + p_1x + p_2x^2 + \dots \text{ to } \infty \) then
(a) \( p_1 = a + b, p_2 = \frac{a^2 - b^2}{2} \)
(b) \( p_0 = 2, p_3 = \frac{a^3 - b^3}{6} \)
(c) \( p_1 = a - b, p_2 = \frac{a^2 + b^2}{2} \)
(d) \( p_0 = 2, p_3 = \frac{a^3 + b^3}{6} \)
Answer: (b) \( p_0 = 2, p_3 = \frac{a^3 - b^3}{6} \) AND (c) \( p_1 = a - b, p_2 = \frac{a^2 + b^2}{2} \)

Question. If \( \log_e \frac{1 + x}{1 - x} = a_0 + a_1x + a_2x^2 + \dots \text{ to } \infty \) then
(a) \( a_1 = 2, a_2 = 0 \)
(b) \( a_1 = 0, a_2 = 2 \)
(c) \( a_0 + a_1 + a_2 = 2 \)
(d) \( a_1, a_3, a_5 \) are in HP
Answer: (a) \( a_1 = 2, a_2 = 0 \) AND (c) \( a_0 + a_1 + a_2 = 2 \) AND (d) \( a_1, a_3, a_5 \) are in HP

Question. The coefficient of \( x^n, n \in \mathbb{N} \), in the expansion of \( \frac{1 + x}{1 + x^2} \) is
(a) \( (-1)^{n/2} \), when \( n \) is an even integer
(b) 0, when \( n \) is an odd integer
(c) 1, when \( n \) is an even integer
(d) \( (-1)^{(n-1)/2} \), when \( n \) is an odd integer
Answer: (a) \( (-1)^{n/2} \), when \( n \) is an even integer AND (d) \( (-1)^{(n-1)/2} \), when \( n \) is an odd integer

Question. Let \( a = \sqrt{5} - 2, b = \sqrt{7} - \sqrt{6} \) and \( c = \sqrt{13} - \sqrt{12} \). Then
(a) \( a > b \)
(b) \( b > c \)
(c) \( c > b \)
(d) \( a > c \)
Answer: (a) \( a > b \) AND (b) \( b > c \) AND (d) \( a > c \)

VARIATION

Question. Let \( x + y \propto z + \frac{1}{z} \), \( x - y \propto z - \frac{1}{z} \) and \( z = 2 \), when \( y = 1, x = 3 \). Then
(a) \( x = \frac{2}{15} (11z + \frac{1}{z}) \)
(b) \( x = \frac{22}{15}z - \frac{2}{15} \frac{1}{z} \)
(c) \( y = \frac{2}{15}z - \frac{22}{15} \frac{1}{z} \)
(d) \( y = \frac{2}{15} (z + \frac{11}{z}) \)
Answer: (a) \( x = \frac{2}{15} (11z + \frac{1}{z}) \) AND (d) \( y = \frac{2}{15} (z + \frac{11}{z}) \)

Question. If \( 2x + 3y \propto \sqrt{xy} \) then
(a) \( x^2 + y^2 \propto xy \)
(b) \( x^3 + y^3 \propto (xy)^{3/2} \)
(c) \( x^4 + y^4 \propto x^2y^2 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 \propto xy \) AND (b) \( x^3 + y^3 \propto (xy)^{3/2} \) AND (c) \( x^4 + y^4 \propto x^2y^2 \)