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Mathematics Chapter 7 Permutations And Combinations

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NCERT Solution For Class 11 Mathematics

Permutations And Combinations Here is the CBSE Mathematics Chapter 7 for Class 11 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 11 Mathematics Permutations And Combinations Chapter 7 NCERT Solutions for Class 11 Mathematics Permutations And Combinations Chapter 7 The following is a summary in Hindi and English for the academic year 2021-2022. You can save these solutions to your computer or use the Class 11 Mathematics.

Question 1

Mathematics Chapter 7 Permutations And Combinations

NCERT Solution For Class 11 Mathematics

Write the first five terms of each of the sequences whose nth terms are :

Write the first five terms of the sequence whose nth terms are:

Find the indicated terms in the following sequence whose nth terms are :

Find the indicated terms in the following sequence whose nth terms are:

The Fibonacci sequence is defined by Find , for n = 1, 2, 3, 4, 5.

Find the first six terms of the sequence whose first term is 1 and whose (n+l)th term is obtained by adding n to the nth term.

Consider the sequence defined by tn = an2 + bn + c. If t2 = 3, t4 = 13 and t7 = 113, show that 3tn = 17n2 – 87n + 115

Write first five terms of the following sequence and obtain the corresponding series. for all n>1

Write first five terms of the following sequence and obtain the corresponding series., n>2

Show that the sequence defined by (where A and B are constant) is an A.P. with common difference A.

Show that the sequence defined by is not an A.P.

Which term of an A.P. 5, 2, -1, ......... is -22?

Which term of the sequence 12 + 8i, 11 + 6i, 10+41i, ....... is(i) purely real (ii) purely imaginary?

Which term of the sequence is the first negative term?

Determine the number of terms in an A.P. 3, 8, 13, .......253. Also, find 12th term from the end.

Find the 2nd term and the rth term of an A.P. whose 6th term is 12 and 8th term is 22.

Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.

In an A.P., the mth term is and the nth term is find the (mn)th term.

If m times the mth term is equal to n times the nth term of an A.P. Prove that (m + w)th term of an A.P. is zero.

If (m + l)th term of an A.P. is twice the (n + l)th term. Prove that (3m + l)th term is twice the (m + n + l)th term.

The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r) a + (r – p) b + (p – q)c = 0

Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

Find the sum to n terms of an A.P., whose nth term is

Solve the equation 1 + 6 + 11 + 16 + ........ + x = 148

How many terms of the A.P. -6, -5, .......... are needed to give the sum -25? Explain the double answer.

In a A.P., the first term is 2 and the sum of first five terms is one-fourth of the sum of the next five terms. Show that the 20th term is – 112.

If the sum of first n terms of a sequence is of the form An2 + Bw, where A and B arc constants (independent of n). Show that the sequence is an A.P. Is the converse true? Justify your answer.

In an A.P., if the pth term is and the qth term is prove that the sum of the first pq terms must be where

If the pth term of an A.P. is a and qth term is b, show that the sum of (p + q) termsis

Find the sum of odd integers from 1 to 2001.

Find the sum of all natural numbers lying between 100 and 1000, which are mutliples of 5.

Let the sum of n, 2n, 3n terms of an A.P., be S1, S2, S3, respectively, show that

Find the sum of all the numbers between 200 and 400 which are divisible by 7.

Find the sum of all the two digit numbers, which when divided by 4, yields 1 as remainder.

Find sum of integers from 1 to 100 that are divisible by 2 or 5.

The sum of first 4 terms of an A.P. is 56. The sum of last 4 terms is 112. If its first term is 11, then find the number of terms.

The sums of first n terms of two A.P.’s are in the ratio of 14 – 4n : 3n + 5. Find the ratio of their 8th terms.

The ratio of sums of m and n terms of an A.P. is m2 : n2. Show that the ratio of with and nth term is 2m – 1 :2n – 1.

If S1, S2, S3 ........Sp are the sums of n terms of p A.P.’s whose first terms are 1, 2,3, .....p and common differences are 1, 3, 5, .....(2p – 1) respectively. Show that

Insert 6 arithmetic means between 3 and 24.

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of the 7th and (m – l)th numbers is 5 : 9. Find the value of m.

For what value of n, is the A.M. between a and b?

Prove that the sum of n arithmetic means between two numbers is n times the single A.M. between them.

The sum of two numbers is . An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

Insert A.M.’s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.’s.

The sum of three numbers in A.P. is 21 and the product of first and the third numbers exceeds the second number by 6, find the three numbers.

Find four numbers in A.P. whose sum is 32 and the product of extremes is to the product of means is 7 : 15.

If the lengths of the sides of a right angled triangle are in A.P., then show that their ratio is 3 : 4 : 5.

The digits of a positive integer having three digits are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.

If and are in A.P., Prove that are also in A.P.

If are in A.P. prove that a, b, c are in A.P.

If a, b, c are in A.P., prove that are also in A.P.

If are in A.P., then prove that are in A.P.

If are in A.P., show that are in A.P.

If are in A.P., prove that are in A.P.

If a, b, c are in A.P. then prove that

If a, b, c are in A.P. then prove that

For what values of x, the numbers are in G.P.?

Find the 10th term of the series 5 + 25 + 125 + ....... . Also, find its wth term.

which term of the series:

The fourth term of a GP. is 27 and the 7th term is 729, find the GP.

The seventh term of a GP. is 8 times the fourth term and 5th term is 48. Find the GP.

If the GP.’s 5,10, 20, .......and 1280, 640, 320, ......have their nth terms equal, find the value of n.

The third term of a GP. is 4. Find the product of its first five terms.

The 5th, 8th and 11th terms of a GP. are p, q and s respectively. Show that q2 = ps.

Find all the sequences which are simultaneously in A.P. and GP.

In a finite GP. the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

If the first and the nth terms of a GP. are a and b respectively and if P is the product of first n terms, prove that P2 = (ab)n.

If show that a, b, c, d are in G.P.

If 4th, 10th and 16th terms of a GP. are x, y and z respectively, Prove that y, z are in GP.

Show that products of the corresponding terms of sequences a, ar, ar2, .........ar n – 1 and A, AR, AR2, ........ An – 1 form a GP. and find common ratio.

The first term of GP. is 1. The sum of 3rd and 5th term is 90. Find the common ratio of G.P.

Find the sum of the indicated terms of each of the following CP’s:3, 6, 12, ........; 7 terms

Find the sum of the indicated terms of each of the following CP’s:

Find the sum to indicated number of terms in each of the geometric progressions:1, - a, n terms

Find the sum to indicated number of terms in each of the geometric progressions:

Find the sum of n terms of sequence

Prove that the sum to n terms of the series11 + 103 + 1005 +........ is

Write the first five terms of each of the sequences whose nth terms are :

$straight a subscript straight n space equals space fraction numerator straight n left parenthesis straight n squared plus 5 right parenthesis over denominator 4 end fraction$

Write the first five terms of the sequence whose nth terms are:

$straight a subscript straight n space equals space left parenthesis negative 1 right parenthesis to the power of straight n minus 1 end exponent space 5 to the power of straight n plus 1 end exponent$

Find the indicated terms in the following sequence whose nth terms are :

$space space straight a subscript straight n space equals space fraction numerator straight n left parenthesis straight n minus 2 right parenthesis over denominator straight n plus 3 end fraction$

Consider the sequence defined by t_n = an² + bn + c. If t₂ = 3, t₄ = 13 and t₇ = 113, show that 3t_n = 17n² – 87n + 115

Show that the sequence $space space open curly brackets straight t subscript straight n close curly brackets$ defined by $straight t subscript straight n space equals space nA space plus space straight B$ (where A and B are constant) is an A.P. with common difference A.

Show that the sequence $space space open curly brackets straight t subscript straight n close curly brackets$ defined by $straight t subscript straight n space equals space 2 straight n squared plus straight n plus 1$ is not an A.P.

Which term of the sequence 12 + 8i, 11 + 6i, 10+41i, ....... is

(i) purely real (ii) purely imaginary?

Which term of the sequence $space space 24 comma space 23 1 fourth comma space 22 1 half comma space 21 3 over 4 comma space space.....$ is the first negative term?

In an A.P., the mth term is $1 over straight n$ and the nth term is $1 over straight m comma$ find the (mn)th term.

Show that the sum of (m + n)^th and (m – n)^th terms of an A.P. is equal to twice the m^th term.

Find the sum to n terms of an A.P., whose nth term is $straight t subscript straight n space equals space 5 space minus space 6 straight n comma space straight n element of space straight N.$

How many terms of the A.P. -6, $fraction numerator negative 11 over denominator 2 end fraction comma$ -5, .......... are needed to give the sum -25? Explain the double answer.

If the sum of first n terms of a sequence is of the form An² + Bw, where A and B arc constants (independent of n). Show that the sequence is an A.P. Is the converse true? Justify your answer.

Let the sum of n, 2n, 3n terms of an A.P., be S₁, S₂, S₃, respectively, show that

$straight S subscript 3 space equals space 3 left parenthesis straight S subscript 2 minus straight S subscript 1 right parenthesis$

The ratio of sums of m and n terms of an A.P. is m² : n². Show that the ratio of with and nth term is 2m – 1 :2n – 1.

The sum of two numbers is $space 13 over 6$ . An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

The digits of a positive integer having three digits are in A.P. and their sum is 15.
The number obtained by reversing the digits is 594 less than the original number.

Find the number.

If a, b, c are in A.P. then prove that $left parenthesis straight a minus straight c right parenthesis squared space equals space 4 left parenthesis straight b squared minus ac right parenthesis$

If a, b, c are in A.P. then prove that

$straight a cubed plus 4 straight b cubed plus straight c cubed space equals space 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis$

which term of the series:

$space space 1 minus 1 half plus 1 fourth minus 1 over 8 plus........ space is space minus 1 over 128$

The 5th, 8th and 11th terms of a GP. are p, q and s respectively. Show that q² = ps.

If the first and the nth terms of a GP. are a and b respectively and if P is the product of first n terms, prove that P² = (ab)ⁿ.

Show that products of the corresponding terms of sequences a, ar, ar², .........ar ^{n – 1} and A, AR, AR², ........ A^{n – 1} form a GP. and find common ratio.

Find the sum of the indicated terms of each of the following CP’s:

3, 6, 12, ........; 7 terms

Find the sum of the indicated terms of each of the following CP’s:

$square root of 7 comma space square root of 21 comma space 3 square root of 7 comma space........ semicolon space space straight n space terms$

Find the sum to indicated number of terms in each of the geometric progressions:

1, - a, $straight a squared comma space$ $negative straight a cubed comma space space space..... semicolon space$ n terms $left parenthesis straight a not equal to space minus 1 right parenthesis$

Sum the series

$space space 2 over 9 space plus space 1 third plus space 1 half plus space............ space plus space 81 over 32$

Sum the following series:

7 + 77 + 777 + ......... to n terms

Sum the following series:

0.5 + 0.55 + 0.555.......... to n terms

Given a G.P. with a = 729 and 7th term 64, determine S₇.

Determine the number of terms in a GP., if t₁ = 3, t_n = 96 and S_n = 189.

Find the least value of n for which the sum 1 + 3 + 3²+ ...... to n terms is greater than 7000.

If S₁, S₂ and S₃ be respectively the sums of n, 2n and 3n terms of a GP., prove that S₁ (S₃ – S₂) = (S₂ – S₁)²

Let S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. Prove that $straight P squared straight R to the power of straight n space equals space straight S to the power of straight n.$

Evaluate:

$space space sum from straight i equals 1 to 10 of open square brackets open parentheses 1 half close parentheses to the power of straight i minus 1 end exponent plus open parentheses 1 fifth close parentheses to the power of straight i plus 1 end exponent close square brackets$

Find the sum of the products of the corresponding terms of sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, $1 half.$

If A and G are respectively the A.M. and G.M. between two positive numbers a and b, then proof the quardratic equation having a, b as it roots is $straight x squared minus 2 Ax plus straight G squared equals 0.$

If A and G be A.M. and G.M. between two positive numbers, then the numbers are

$straight A plus-or-minus square root of straight A squared minus straight G squared end root.$

The A.M. of two numbers is 3 times their G.M. Show that the numbers are in the ratio

$3 plus 2 square root of 2 colon space space 3 minus 2 square root of 2.$

If a be the A.M. and x, y be the two G.M.'s between b and c, show that

$straight x cubed plus straight y cubed space equals space 2 abc$

The sum of first three terms of a G.P. is $13 over 12$ and their product is -1. Find the terms.

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87 1 half.$ Find them.

If a, b, c are in GP., prove that a² + b², ab + bc, b² + c² are also in GP.

If a, b, c, d are in GP., show that (a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)²

If a, b, c are in G.P. and x, y are the arithmetic means of a, b and b, c respectively,

then prove that $straight a over straight x plus straight c over straight y equals 2$ and $space space 1 over straight x plus 1 over straight y equals 2 over straight b$

If a, b, c are in A.P. and x,y,z are in G.P., show that

$straight x to the power of straight b minus straight c end exponent. space straight y to the power of straight c minus straight a end exponent. space straight z to the power of straight a minus straight b end exponent space equals space 1$

If a and b are roots of x² – 3x+p = 0 and c, d are roots of x² – 12x + q = 0 where a, b, c, d form a GP., prove that (q + p) : (q – p) = 17 : 15

Find the sum of n terms of the series whose nth term is given by

$straight n squared plus 2 to the power of straight n$

Insert 5 G.M.'s between $32 over 9 space and space 81 over 2$

Find the sum of first n natural numbers Or

Evaluate $sum straight n space or space sum from straight k equals 1 to straight n of space straight k$

Find the sum of the squares of first n natural numbers. Or

Evaluate $space space sum straight n squared space space space or space space space space sum from straight k equals 1 to straight n of space straight k squared$

Find the sum of n terms of the series whose nth term is

$space space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed over denominator straight n end fraction.$

Find the sum to n terms of the series:

1 . 2 . 4 + 2. 3 . 7 + 3. 4. 10 +..............

Find the sum of the cubes of first n odd natural numbers. Or Sum the series to n terms l³ + 3³ + 5³ +..........

Find the sum to n terms the series $fraction numerator 1 over denominator 1.2 end fraction plus fraction numerator 1 over denominator 2.3 end fraction plus fraction numerator 1 over denominator 3.4 end fraction plus...........$ and deduce the sum to infinity.

Find the sum to n terms of the following series:

5² + 6² + 7² + ....+ 20².

Find the sum to n terms of the following series.

$space space space 1 cross times 2 cross times 3 plus 2 cross times 3 cross times 4 plus 3 cross times 4 cross times 5 plus........$

A man buys every year Bank’s cash certificates of value exceeding the last year purchase by Rs. 25. After 20 years, he finds that the total value of the certificates purchased by him is Rs. 7250. Find the values of the certificates purchased by him

(i) In the first year. (ii) In the 13th year.

A class has 30 students. In how many ways can three prizes be awarded so that:

(a) no students get more than one prize?

(b) a student may get any number of prizes?

How many 3-digit odd numbers can be formed from the digits 1,2,3,4,5,6 if:

(a) the digits can be repeated (b) the digits cannot be repeated?