Assignments Class 11 Mathematics Principle of Mathematical Induction

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Principle of Mathematical Induction Assignments Class 11 Mathematics

Question. Let S(K) = 1+ 3+ 5…+ (2K -1) = 3+ K² . Then which of the following is true
(a) Principle of mathematical induction can be used to prove the formula
(b) S(K) ⇒ S(K +1)
(c) S(K) ⤃ S(K +1)
(d) S(1) is correct

Question. Consider the statement: “P(n) : n² – n + 41 is prime.” Then which one of the following is true?
(a) Both P(3) and P(5) are true.
(b) P(3) is false but P(5) is true.
(c) Both P(3) and P(5) are false.
(d) P(5) is false but P(3) is true.

Question. If

 having n radical signs then by methods of mathematical induction which is true
(a) a_n > 7 ∀ n ≥ 1
(b) a_n < 7 ∀ n ≥ 1
(c) a_n < 4 ∀ n ≥ 1
(d) a_n < 3 ∀ n ≥ 1

Question. If the coefficients of x² and x3 in the expansion of (3 + ax)⁹ are the same, then the value of a is:         
(a) −7/9
(b) − 9/7
(c) 7/9
(d) 9/7
Number of Terms in the Expansion of (a + b + c)n and (a + b + c + d)n .

Question. If the number of terms in the expansion of (x − 2y + 3z)n is 45, then n = ?         
(a) 7
(b) 8
(c) 9
(d) 5

Question. The middle term in the expansion of (1+x)²ⁿ x is:

Question. The term independent of x in the expansion of (3x²/2− 1/3x)⁹ is:         
(a) 7/12
(b) 7/18
(c) – 7/12
(d) – 7/16

Question. In the expansion of (1+ x)⁵ , the sum of the coefficient of the terms is:         
(a) 80
(b) 16
(c) 32
(d) 64

Question. Find the coefficient of a³b⁴c⁵ in the expansion of (bc+ca+ab)6?         
(a) 0
(b) 60
(c) –60
(d) 25

Question. If (1+ax)n = 1+8x+24x² … then the value of aand n is:         
(a) 2,4
(b) 2,3
(c) 1,2
(d) 1,2

Question. If the sum of coefficient in the expansion of (α²x² − 2αx +1) vanishes, then the value of α is: 
(a) 2
(b) –1
(c) 1
(d) –2

Question. If coefficient of (2r +3)^th and (r − 1)^th terms in the expansion of (1+ x)¹⁵ are equal, then value of r is:         
(a) 5
(b) 6
(c) 4
(d) 3

Question. If the (r +1)th term in the expansion of

has the same power of a and b, then the value of r is:         
(a) 9
(b) 10
(c) 8
(d) 6

Question. ¹⁰C₁ + ¹⁰C₃ + ¹⁰C₅ + ¹⁰C₇+ ¹⁰C₉ = ?         
(a) 2⁹
(b) 2¹⁰
(c) 2¹⁰ −1
(d) None of these

Question. If (1+x)n = C₀+C₁x+C₂x² +….+C_nx² ,then C₀²+C₁²x+C₂² +C₃²+……+C_n² =?         
(a) n!/n!n!
(b) (2n)!/n!
(c) (2n)!/n!
(d) None of these

Question. If  (1+x)n = C₀+C₁x+C₂x² +….+C_nx² ,then 

Question. If the coefficients of second, third and fourth term in the expansion of (1+x)²ⁿ are in (a)P., then 2n²−9n+7 is equal to:         
(a) – 1
(b) 0
(c) 1
(d) 3/2

Question. The value of C₁/2 + C₃/4 + C₅/6+……is equal to:         
(a) 2ⁿ–1/n+1
(b) n.2ⁿ
(c) 2ⁿ/n
(d) 2ⁿ+1/n+1

Question. (1+x)ⁿ + x − nx −1 is divisible by: (where n∈ N )         
(a) 2x
(b) 2 x
(c) 2x³ 
(d) All of these

Question. Sum of odd terms is A and sum of even terms is B in the expansion of (x + a)ⁿ , then:         
(a) AB = 1/4(x–a)²ⁿ –(x+a)²ⁿ
(b) 2AB = 1/4(x+a)²ⁿ–(x–a)²ⁿ
(c) 4AB = 1/4(x+a)²ⁿ–(x–a)²ⁿ
(d) None of these

Question. The sum of all the coefficients in the binomial expansion of (x² + x − 3)³¹⁹ is:         
(a) 1
(b) 2
(c) – 1
(d) 0

Question. The first 3 terms in the expansion of (1 + ax)ⁿ (n ≠ 0) are 1, 6x and 16x². Then the value of a and n are respectively:          
(a) 2 and 9 (b) 3 and 2
(c) 2/3 and 9
(d) 3/2 and 6

Question. Coefficient of x in the expansion of (x²+a/x) is:       
(a) 9a²
(b) 10a³
(c) 10a²
(d) 10a

Question. C₁ + 2C₂ + 3C₃ +….. ⁿC_n =?         
(a) 2ⁿ
(b) n2ⁿ
(c) n2^n-1
(d) n2ⁿ⁺¹

Question. The coefficient of x⁵ in the expansion of (x²–x–2)⁵ is.         
(a) – 83
(b) – 82
(c) – 81
(d) 0

Question. If a₁, a₂, a₃, a₄ are the coefficients of any four consecutive terms in the expansion of (1+ x)n ,then a₁/a₁+a₂ + a₃/a₃+a₄ = ? 

Question. The largest term in the expansion of (3 + 2x)⁵⁰ , where x=1/5 is:         
(a) 5^th
(b) 8^th
(c) 7^th
(d) 6^th

Question. If

Question. 1/3√6–3x = ?

Question. (a/a+x)^1/2 + (a/a–x)^1/2

Question. The coefficient of xⁿ in the expansion of (1 + x + x2 + …)⁻ⁿ is:   
a. 1
b. ( − 1)ⁿ
c. n
d. n +1

Question. In the expansion of (2 – 2x + x²)⁹ ?     
a. number of distinct terms is 10
b. coefficient of x⁴ is 97
c. sum of coefficient is 1
d. number of distinct terms is 55

Question. In the expansion of (³√4+1⁴√6)²⁰ ?       
a. the number of rational terms = 4
b. the number of irrational terms =19
c. the middle term is irrational
d. the number of irrational terms = 17

Question. an = (1000)n/n! for n∈N. Then a_n is greatest, when:         
a. n = 998
b. n = 999
c. n = 1000
d. n = 1001

Question. If the term independent of x in the expansion of (√x–λ/λ²)¹⁰ is 405, then value of λ is:       
a. –3
b. 9
c. –9
d. 3

Question. (r +1)^th term in the expansion of (1–x)^–4 will be:       

Question. 1 / (2 + x)⁴ = ?     

Question. If | x | > 1 , then  (1 + x)⁻² =?       
a. 1 − 2x + 3x² − …. 
b. 1 + 2x + 3x² + ….
c. 1 − 2/x + 3/x² − ….
d. 1/x² − 2/x³ + 3/x⁴ − ….

Question. In the expansion of(x + y +z) 25  =?     
a. every term is of form ²⁵Cr.^rCk.X^25-r .y^r-k.2^k
b. the coefficient of x⁸y⁹z⁹ is 0
c. the number of terms is 351
d. none of the above

Question. If (r + 1)^th term is the first negative term in the expansion of (1+ x)^7/2 , then the value of r is:
a. 5
b. 6
c. 4
d. 7

Assertion and Reason
Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
a. If both assertion and reason are true and the reason is the correct explanation of the assertion.
b. If both assertion and reason are true but reason is not the correct explanation of the assertion.
c. If assertion is true but reason is false.
d. If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question. Assertion: If n is an odd prime, then greatest integer contained in (2+√5)ⁿ –2ⁿ⁺¹ is divisible by 20 n. 
Reason: If p is a prime and 1≤r ≤p–1,then (P  r) is divisible by p.

Question. Let (1+t)n = C₀+C₁t+C₂t²…+C_ntⁿ

Question. Let 

Question. Let (1+x)³ⁿ = 

Reason: Cube roots unity form a triangle of area √3 square units.

Question.

Assertion: S3 = 55 × 2⁹
Reason: S1 = 90 × 28
and S₂ = 10 × 2⁸

Question. sin(rx)cos{(n-r)} = f(n)sin(nx) then the value of f(13) must be:       

Question.  

Question. The coefficients of three consecutive terms of (1+x)ⁿ⁺⁵ are in the ratio 5 : 10 : 14. Then, n is equal to: 

Paragraph

where a1, a2, a3……, a_4n are real number and n is a positive integer.

Question. The value of 

Question. The value of a₂ is:     
a. ⁴ⁿ⁺¹C₂
b. ³ⁿ⁺¹C₂
c. ²ⁿ⁺¹C₂
d.  ⁿ⁺¹C₂

Question. The value of a_4n–1 is:       
a. 2n
b. 2n² + 4n
c. 2n + 3
d. 2n² + 3n

Question. The correct statement is:     
a. a_r = a_n–r, 0 ≤ r ≤ n
b. a_{r – r} = a_n–r, 0 ≤ r ≤ n
c. a_r = a_2n, 0 ≤ r ≤ 2n
d. a_r = a_4n–r, 0 ≤ r ≤ 4n

Question. The value of