Class 11 Mathematics Sequences and Series MCQs Set 08

Question. If a, b, c, d are distinct integers in A.P. such that \( d = a^2 + b^2 + c^2 \), then \( a + b + c + d = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (c) 2

Question. A person is to count 4500 currency notes. Let \( a_n \) denote the number of notes he counts in the \( n^{th} \) minute. If \( a_1 = a_2 = ... = a_{10} = 150 \) and \( a_{10}, a_{11}, ..., a_n \) are in A.P. with common difference -2, then the time taken by him to count all notes is
(a) 135 mins
(b) 24 mins
(c) 34 mins
(d) 125 mins
Answer: (c) 34 mins

Question. A man saves Rs. 200 in each of the first 3 months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs 11040 after. [AIEEE 2011]
(a) 21 months
(b) 18 months
(c) 19 months
(d) 20 months
Answer: (a) 21 months

Question. The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, .... is. [MAINS-2013]
(a) \( \frac{7}{81}(179-10^{-20}) \)
(b) \( \frac{7}{9}(99-10^{-20}) \)
(c) \( \frac{7}{81}(179+10^{-20}) \)
(d) \( \frac{7}{9}(99+10^{-20}) \)
Answer: (c) \( \frac{7}{81}(179+10^{-20}) \)

Question. Sum of n terms of the series 1, 3, 7, 15, 31, .... is
(a) \( 2^{n+1} - n - 2 \)
(b) \( 2^n - n - 2 \)
(c) \( 2^{n+1} + n + 2 \)
(d) \( 2^n - 1 \)
Answer: (a) \( 2^{n+1} - n - 2 \)

Question. The three successive terms of a GP will form the sides of a triangle if the common ratio satisfies the inequality \( (r > 1) \)
(a) \( \left( 1, \frac{\sqrt{5}+1}{2} \right) \)
(b) \( \left( -\infty, \frac{\sqrt{5}-1}{2} \right) \cup \left( \frac{\sqrt{5}+1}{2}, \infty \right) \)
(c) \( \left[ -\sqrt{5}, \sqrt{5} \right] \)
(d) \( \left( -\sqrt{5}, \sqrt{5} \right) \)
Answer: (a) \( \left( 1, \frac{\sqrt{5}+1}{2} \right) \)

Question. If a, b, c be respectively the \( p^{th}, q^{th} \) and \( r^{th} \) terms of G.P then \( \Delta = \begin{vmatrix} \log a & \log b & \log c \\ p & q & r \\ 1 & 1 & 1 \end{vmatrix} \) equals to
(a) 1
(b) 0
(c) -1
(d) 2
Answer: (b) 0

Question. If \( t_r = 2^{\frac{r}{3}} + 2^{-\frac{r}{3}} \), then \( \sum_{r=1}^{100} t_r^3 - 3 \sum_{r=1}^{100} t_r + 1 = \)
(a) \( \frac{2^{101}+1}{2^{100}} \)
(b) \( \frac{2^{101}-1}{2^{100}} \)
(c) \( \frac{2^{201}-1}{2^{100}} \)
(d) \( \frac{2^{201}+1}{2^{100}} \)
Answer: (c) \( \frac{2^{201}-1}{2^{100}} \)

Question. The value of x satisfying the equation
\( \left[ 3 \left( 1 - \frac{1}{2} + \frac{1}{4} ...... \text{to } \infty \right) \right]^{\log_{10} x} = \left[ 20 \left( 1 - \frac{1}{4} + \frac{1}{16} ..... \infty \right) \right]^{\log_{x} 10} \) is
(a) \( \frac{1}{100} \)
(b) 10
(c) 1000
(d) \( \frac{1}{10} \)
Answer: (a) \( \frac{1}{100} \)

Question. If \( \exp\{ (\sin^2 x + \sin^4 x + \sin^6 x + ... \text{upto } \infty) \log_{e} 2 \} \) satisfies the equation \( x^2 - 17x + 16 = 0 \) then the value of \( \frac{2 \cos x}{\sin x + 2 \cos x} \ (0 < x < \pi/2) \) is
(a) 1/2
(b) 3/2
(c) 5
(d) 2/3
Answer: (a) 1/2

Question. The length of the side of square is 'a' metre. A second square is formed by joining the middle points of the sides of the squares. Then a third square is formed by joininig the middle points of the sides of the second squares and so on. Then the sum of the area of squares which carried upto infinity is
(a) \( a^2 \)
(b) \( 2a^2 \)
(c) \( 3a^2 \)
(d) \( 4a^2 \)
Answer: (b) \( 2a^2 \)

Question. If \( \frac{a + be^y}{a - be^y} = \frac{b + ce^y}{b - ce^y} = \frac{c + de^y}{c - de^y} \) then a, b, c, d are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P.
Answer: (b) G.P.

Question. If a, b, c, d are positive real numbers such that \( a + b + c + d = 2 \), then \( M = (a+b)(c+d) \) satisfies the relation
(a) \( 0 < M \le 1 \)
(b) \( 1 \le M \le 2 \)
(c) \( 2 \le M \le 3 \)
(d) \( 3 \le M \le 4 \)
Answer: (a) \( 0 < M \le 1 \)

Question. If n be the number of sequence a, b, c, d, e satisfying the conditions
(i) a, b, c, d, e are in A.P and G.P. both,
(ii) c = 3, 7 then 'n' = ------
(a) 1
(b) 2
(c) 5
(d) 10
Answer: (b) 2

Question. If \( p^{th}, q^{th}, r^{th} \) terms of an A.P are in G.P. whose common ratio is k, then the root of equation \( (q - r)x^2 + (r - p)x + (p - q) = 0 \) other than unity is
(a) k
(b) 2k
(c) \( k^2 \)
(d) \( \frac{1}{k} \)
Answer: (d) \( \frac{1}{k} \)

Question. If the sum to infinity of the series \( 1 + 4x + 7x^2 + 10x^3 + ..... \) is \( \frac{35}{16} \) then x =
(a) \( \frac{1}{5} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{3}{7} \)
(d) \( \frac{1}{7} \)
Answer: (a) \( \frac{1}{5} \)

Question. The value of \( 2^{1/4} 4^{1/8} 8^{1/16} 16^{1/32} ... \) is
(a) 2
(b) 3/2
(c) 1
(d) 1/2
Answer: (a) 2

Question. Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers. Then value of \( \frac{y^3 + z^3}{xyz} \) is
(a) 2
(b) 3
(c) 1/2
(d) 3/2
Answer: (a) 2

Question. If a, b, c are in G.P., then the equations \( ax^2 + 2bx + c = 0 \) and \( dx^2 + 2ex + f = 0 \) have a common root if a/d, b/e, c/f are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P.
Answer: (c) H.P.

Question. Let \( I_n = \int_0^{\pi/4} \tan^n x \, dx \). Then \( I_2 + I_4, I_3 + I_5, I_4 + I_6, I_5 + I_7, ...... \) are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P.
Answer: (c) H.P.

Question. Let \( a_1, a_2, ... a_{10} \) be in A.P. and \( h_1, h_2, .... h_{10} \) be in H.P. If \( a_1 = h_1 = 2 \) and \( a_{10} = h_{10} = 3 \), then \( a_4 h_7 \) is
(a) 2
(b) 3
(c) 5
(d) 6
Answer: (d) 6

Question. If the sytem of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a non-zero solution, then a, b, c are in
(a) G.P.
(b) H.P.
(c) Satisfy a + 2b + 3c = 0
(d) A.P.
Answer: (b) H.P.

Question. If cos(x–y), cos x and cos (x+y) are in H.P, then value of cos x sec (y/2) is
(a) \( \pm \sqrt{2} \)
(b) \( \pm \sqrt{3} \)
(c) \( \pm 2 \)
(d) \( \pm 1 \)
Answer: (a) \( \pm \sqrt{2} \)

Question. If a, b, c are real and in A.P. and \( a^2, b^2, c^2 \) are in H.P., then
(a) a = b = c
(b) 2b = 3a + c
(c) \( b^2 = \sqrt{ac/8} \)
(d) \( ab = c \)
Answer: (a) a = b = c

Question. If 9A.M.’s and 9 H.M’s be inserted between 2 and 3 and A be any A.M. and H be the corresponding H.M., then H(5-A) =
(a) 10
(b) 6
(c) -6
(d) -10
Answer: (b) 6

Question. Suppose ‘a’ is a fixed real number such that \( \frac{a - x}{px} = \frac{a - y}{qy} = \frac{a - z}{rz} \) if p, q, r are in AP then x, y, z all are in
(a) A.P.
(b) G.P.
(c) H.P.
(d) A.G.P.
Answer: (c) H.P.

Question. a, b, c are in A.P; b, c, d are in G.P and c, d, e are in H.P. If a=2 and e=18, then the sum of all possible values of c is
(a) -6
(b) 6
(c) 12
(d) 0
Answer: (d) 0

Question. If an A.P., a G.P. and a H.P. have the same first term and same \( (2n+1)^{th} \) term and their \( (n+1)^{th} \) terms are a, b, c, respectively, then the radius of the circle \( x^2 + y^2 + 2bx + 2ky + ac = 0 \) is
(a) \( k \)
(b) \( |k| \)
(c) \( \sqrt{b^2 - ac} \)
(d) \( k^2 \)
Answer: (b) \( |k| \)

Question. If \( a, a_1, a_2, a_3, a_4, ......, a_{2n}, b \) are in A.P and \( a, g_1, g_2, g_3, g_4, ......, g_{2n}, b \) are in G.P and h is the H.M of a and b then
\( \frac{a_1 + a_{2n}}{g_1 g_{2n}} + \frac{a_2 + a_{2n-1}}{g_2 g_{2n-1}} + .... + \frac{a_n + a_{n+1}}{g_n g_{n+1}} \) is equal to
(a) \( 2n / h \)
(b) \( 2nh \)
(c) \( nh \)
(d) \( n/h \)
Answer: (a) \( 2n / h \)

Question. If \( f(x) = x^2 - (a+b)x + ab \) and A and H be the A.M. and H.M. between two quantities a and b, then
(a) \( Af(A) = Hf(H) \)
(b) \( Af(H) = Hf(A) \)
(c) \( A + f(A) = H + f(H) \)
(d) \( f(A) + H = f(H) + A \)
Answer: (b) \( Af(H) = Hf(A) \)

Question. If positive numbers a, b, c be in H.P., then equation \( x^2 - kx + 2b^{101} - a^{101} - c^{101} = 0 (k \in \mathbb{R}) \) has
(a) both roots positive
(b) both roots negative
(c) one positive & one negative root
(d) both roots imaginary
Answer: (c) one positive & one negative root

Question. The value of \( \sum_{n=1}^{10} \int_{0}^{n} [x] dx \) is
(a) an even integer
(b) an irrational number
(c) a rational number
(d) an irrational number
Answer: (c) a rational number

Question. Let \( \sum_{r=1}^{n} r^4 = f(n) \), then \( \sum_{r=1}^{n} (2r-1)^4 \) is equal to
(a) f(2n) - 16f(n)
(b) f(2n) - 7f(n)
(c) f(2n-a) - 8f(n)
(d) f(2n-a) - 7f(n)
Answer: (a) f(2n) - 16f(n)

Question. For \( x \in \mathbb{R} \) let [x] denote the greatest integer \( \le x \). Largest natural number n for which
\( E = \left[ \frac{\pi}{2} \right] + \left[ \frac{1}{100} + \frac{\pi}{2} \right] + \left[ \frac{2}{100} + \frac{\pi}{2} \right] + .... + \left[ \frac{n}{100} + \frac{\pi}{2} \right] < 43 \), is
(a) 41
(b) 42
(c) 43
(d) 97
Answer: (a) 41

Question. The sum to n terms of the series
\( \frac{3}{1^2} + \frac{5}{1^2+2^2} + \frac{7}{1^2+2^2+3^2} + --- \) is
(a) \( \frac{6n}{n+1} \)
(b) \( \frac{9n}{n+1} \)
(c) \( \frac{12n}{n+1} \)
(d) \( \frac{3n}{n+1} \)
Answer: (a) \( \frac{6n}{n+1} \)

Question. Let \( r^{th} \) term of a series be given by \( T_r = \frac{r}{1-3r^2+r^4} \) then \( \lim_{n \to \infty} \sum_{r=1}^{n} T_r = \)
(a) \( \frac{3}{2} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{-1}{2} \)
(d) \( \frac{-3}{2} \)
Answer: (c) \( \frac{-1}{2} \)

Question. The sum of the first n terms of the series
\( 1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + ...... \) is \( \frac{n(n+1)^2}{2} \) when n is even. When n is odd the sum is
(a) \( \frac{3n(n+1)}{2} \)
(b) \( \left[ \frac{n(n+1)}{2} \right]^2 \)
(c) \( \frac{n(n+1)^2}{4} \)
(d) \( \frac{n^2(n+1)}{2} \)
Answer: (d) \( \frac{n^2(n+1)}{2} \)

Question. Sum to n terms of the series
\( \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) + ... \) is
(a) \( \tan^{-1}\left(\frac{n}{n+2}\right) \)
(b) \( \tan^{-1}\left(\frac{2n-1}{2n+2}\right) \)
(c) \( \tan^{-1}\left(\frac{1}{3n}\right) \)
(d) \( \tan^{-1}\left(\frac{n}{n+1}\right) \)
Answer: (a) \( \tan^{-1}\left(\frac{n}{n+2}\right) \)

Question. The sum of the series
\( \frac{1}{3^2+1} + \frac{1}{4^2+2} + \frac{1}{5^2+3} + \frac{1}{6^2+4} + ....... \infty \) is
(a) \( \frac{13}{36} \)
(b) \( \frac{13}{33} \)
(c) \( \frac{11}{36} \)
(d) \( \frac{15}{36} \)
Answer: (a) \( \frac{13}{36} \)