CBSE Class 11 Mathematics HOTs Sequences and Series

Question. Sum to n terms of the series 1³ + 3. 2³ + 3³ + 3. 4³ + 5³ + ................ is (n is even)
(a) n(n²+1) (2n+1)/3
(b) n(n³+ 4n²+10n+8)/8
(c) n(n³+1)/8
(d) n²(2n²+ 6n+5)/4
Answer : D

Question. If a₁, a₂, ..., an are in A.P. with common difference d ≠ 0, then (sin d) [sec a₁ seca₂ + sec a₂ sec a₃ + ... + sec a_n–1 sec an] is equal to
(a) cot a_n – cot a₁
(b) cot a₁ – cot a_n
(c) tan a_n – tan a₁
(d) tan a_n – tan a_n–1
Answer : C

Question. In the sum of first n terms of an A.P. is cn², then the sum of squares of these n terms is
(a) n(4n² –1)c²/6
(b) n(4n² +1)c²/3
(c) (4n² –1)c²/3
(d) (4n² +1)c²/6
Answer : C

Question. An infinite G.P. has first term ‘x’ and sum ‘5’, then x belongs to
(a) x < – 10
(b) – 10 < x < 0
(c) 0 < x < 10
(d) x > 10
Answer : C

Question. In the quadratic equation ax² + bx + c = 0, D = b² – 4ac and α + β, α² + β², α³ + β³, are in G.P. where a, b are the root of ax² + bx + c = 0, then
(a) Δ ≠ 0
(b) bΔ = 0
(c) cΔ = 0
(d) Δ = 0
Answer : C

Question. For a, b, c Î R – {0}, let a+b/1-ab, b, b+c/1-bc are in A.P. If a, b are the roots of the quadratic equation 2ac x² + 2abc x + (a + c) = 0, then the value of (1 + α)(1 + β) is
(a) 0
(b) 1
(c) – 1
(d) 2
Answer : B

Question. An A.P. consist of even number of terms 2n having middle terms equal to 1 and 7 respectively. If n is the maximum value which satisfy t₁t_2n + 713 ≥ 0, then the value of the first term of the series is
(a) 17
(b) – 15
(c) 21
(d) – 23
Answer : D

Question. If a, b, c are in G. P., x and y be the A. M.’s between a, b and b, c respectively, then (a/x + c/y) (b/c + b/y) is equal to
(a) – 2
(b) – 4
(c) 2
(d) 4
Answer : D

Question. A_r ; r = 1, 2, 3, ........... , n are n points on the parabola y² = 4x in the first quadrant.
If A_r = (x_r , y_r ) , where x₁ , x₂ , x₃ , ..............., x_n are in G. P. and x₁ = 1, x₂ = 2, then yn is equal to
(a) –2^n+1/2
(b) 2ⁿ⁺¹
(c) (√2)ⁿ⁺¹
(d) 2^n/2
Answer : C

Question. If three successive terms of a G..P. with common ratio r (r > 1) form the sides of a ΔABC and [r] denotes greatest integer function, then [r] + [– r] =
(a) 0
(b) 1
(c) – 1
(d) None of these
Answer : C

Question. If x = 1/1² + 1/3² + 1/5² + ....... y = 1/1² + 3/2² + 1/3² + 3/4² + ... and z = 1/1² - 1/2² + 1/3² - 1/4² +..., then
(a) x, y, z are in A.P.
(b) y/6, x/3, z/2 are in A.P.
(c) y/6, x/3, z/2 are in A.P.
(d) 6y, 3x, 2z are in A.P.
Answer : B

Question. Suppose a, b, c are in A.P. and a², b², c² are in G.P. if a < b < c and a + b + c = 3/2 , then the value of a is
(a) 1/2√2
(b) 1/2√3
(c) 1/2 - 1/√3
(d) 1/2' - 1/√2
Answer : D

Question. If 1, log₉ (31–x + 2), log₃ (4.3x – 1) are in A.P., then x equals
(a) log₃ 4
(b) 1 – log₃ 4
(c) 1 – log₄ 3
(d) log₄ 3
Answer : B

Question. The sum of 3/1.2 • 1/2 + 4/2.3 • (1/2)² + 5/3.4 • (1/2)³ + ......... to n terms is equal to
(a) 1 – 1/(n + 1)2ⁿ
(b) n – 1/2^{n + 1}
(c) 1 – 1/n.2ⁿ⁺¹
(d) None of these
Answer : A

Question. If a, b, c, are in A.P. and p, p¢ are respectively A.M. and G.M. between a and b while q, q¢ are respectively AM.and G.M. between b and c, then
(a) p² + q² = p '² + q '²
(b) pq = p'q'
(c) p² - q² = p'² - q'²
(d) p² + p′² = q² + q '²
Answer : C

Question. The sum of the series 1+ 2.2 + 3.2² + 4.2³ + 5.2⁴ +... +100.2⁹⁹ is
(a) 99.2¹⁰⁰ – 1
(b) 100.2¹⁰⁰
(c) 99.2¹⁰⁰
(d) 99.2¹⁰⁰ + 1
Answer : D

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

Question. The sum of an infinite geometric series is 2 and the sum of the geometric series made from the cubes of this infinite sereis is 24. Then the series is

Answer : C

Question. If a, b, c are in G. P. and log a – log 2b, log 2b – log 3c and log 3c – log a are in A. P., then a, b, c are the sides of a triangle which is 
(a) Acute angled
(b) Obtuse angled
(c) Right angled
(d) None of these
Answer : B

Question. Let ax² + b/x ≥ c for all positive x, where a < 0 and b < 0. The value of the expression 27ab² cannot be less than
(a) 4c³
(b) 4c²
(c) 8c³
(d) c³
Answer : A

Numeric Value Answer

Question. Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
Answer : 0.60

Question. a, b, c are positive integers forming an increasing G.P. and b – a is a perfect cube and log₆ a + log₆ b + log₆ c = 6, then a + b + c =
Answer : 189

Question. The sum to infinite term of the series 1 + 2/3 + 6/32 + 10/33 + 14/34 + ...... is
Answer : 3

Question. The 20th term of the series 2 + 3 + 5 + 9 + 16 +....... is
Answer : 990

Question. If one geomteric mean G and two Arithmetic means P and q be inserted between two quantities, then G² = (kp – q)(kq – p) then find k.
Answer : 2

Question. Three numbers a, b, c are in GP. If a, b, c – 64 are in AP and a, b – 8, c – 64 are in GP, then the sum of the numbers may be
Answer : 124

Question. Two consecutive numbers from 1, 2, 3,.........., n are removed. If the arithmetic mean of the remaining numbers is 105/4 then n/10 is equal to
Answer : 5

Question. Let a, b, c, d be four distinct real numbers in A.P. Then half of the smallest positive value of k satisfying 2(a – b) + k(b – c)² + (c – a)³ = 2(a– d) + (b – d)² + (c – d)³ is ..... .
Answer : 8

Question. Let x₁, x₂, ... ∈ (0, π) denote the of values of x satisfying the equation 27^{(1|cos x| + cos2 x + |cos x|3 + ....upto)} = 93, find the value of 1/π(x₁ + x₂ +...)
Answer : 1

Question. For a, b > 0, let 5a – b, 2a + b, a + 2b be in A.P. and (b + 1)², ab + 1, (a – 1)² are in G.P., then the value of (a^–1 + b– 1) is ..... .
Answer : 6