Buds Public School , Dubai
Worksheet – Mathematics - Linear In equalities ,Permutation
and combinations
Grade : 11 A B
1. Solve the following inequalities for real x :
2𝑥−1
3𝑥−2
2−𝑥
𝑥
5𝑥−2
2−𝑥
a) 3 ≥ 4 − 5
b) 4 < 3 − 5
c)
8𝑥
3𝑥−1
5
𝑥−1
≤ 2−
𝑥+1
3
𝑥+3
d) 4𝑥−7 < 2, 𝑥 − 7 < 0
e) |𝑥 + 2| > |3𝑥 − 5|
f) 𝑥+2 > 1 , 𝑥−1 > 2
2. Solve the following system of inequalities graphically :
a) 𝑥 + 2𝑦 ≤ 8 , 2𝑥 + 𝑦 ≤ 8 , 𝑥 ≥ 0 , 𝑦 ≥ 0
b) 4𝑥 + 3𝑦 ≤ 60 , 𝑦 ≥ 2𝑥, 𝑥 ≥ 3, 𝑥 ≥ 0 , 𝑦 ≥ 0
c) 𝑥 + 2𝑦 + 3 ≤ 0 𝑎𝑛𝑑 𝑥 + 2𝑦 − 4 ≥ 0
d) 𝑥 + 2𝑦 ≤ 10 , 𝑥 + 𝑦 > 1, 𝑥 − 𝑦 ≤ 0, 𝑥 ≥ 0 , 𝑦 ≥ 0
𝑥
5𝑥−2
7𝑥−3
3. Solve the inequality : 4 < 3 − 5 and show the graph of the solution.
4. Solve the following system of inequalities graphically ; a) 3𝑥 − 7 < 5 + 𝑥 b) 11 − 5𝑥 ≤ 1
And represent the solutions on the number line .
18
7
5. Draw the graph of the in equation 7𝑥 + 18 ≥ −𝑥 + 5 and 3 (𝑥 − 3) ≤ 8𝑥 + 3
6. Find all pairs of consecutive odd natural numbers , both of which are larger than 10 such that
their sum is less than 40 .
7. In how many ways 3 prizes be distributed among 4 boys ,when
i) no boy gets more than one prize ?
ii) a boy may get any number of prizes ? iii) no boy gets all the prizes ?
8. In how many ways six persons be seated in a row ?
9. How many 3 digit numbers can be formed from the digits 1,2,3,4 and 5 assuming that
i) repetition of digits is allowed ? ii) repetition of digits is not allowed ?
10. A coin is tossed three times the outcomes are recorded . How many possible outcomes are
there ?
11.How many 5 digit telephone numbers can be constructed using the digits 0 to 9 if each
number starts from 67 and no digits appears more than once ?
12. Evaluate :
1
1
𝑛!
𝑟!(𝑛−𝑟)!
𝑥
13. If 6! + 7! =
6
8!
, 𝑤ℎ𝑒𝑛 𝑛 = 8 , 𝑟 = 2 .
, find x
6
14. Find P3 and C 3 . Are they equal ?
15. From a committee of 8 persons , in how many ways can we choose a chairman and a vice
chairman assuming one person can not hold more than one position ?
16. A bag contains 5 black and 6 red balls . Determine the number of ways in which 2 black and
3 red balls can be selected . If n C8 = n C2 . find n .
17 If 22P r+1 : 20P r+2 = 11: 52 , find r .
18. Write the number of ways in which 7 men and 7 women can sit together on a round table
such that no two women sit together .
19. In how many ways can five children stand in a queue?
20. How many words , with or without meaning can be formed by using the letters of the word
‘TRIANGLE ‘ ?
________________________
Buds Public School , Dubai
Worksheet – Mathematics - Binomial Theorem
Grade : 11 A B
1. Using Binomial theorem expand the following : a) (101)5 b) (1 − 2𝑥)5
d) 514
e) (0.998)8
f)
(999)4 g) (𝑥 + 3𝑦)3
h) (3𝑥 2 − 2𝑦)4
𝑥
2
c) (2 − 𝑥)5.
1
i) (𝑥 + 𝑦)11
2. Using Binomial Theorem , prove that 6𝑛 − 5𝑛 always leaves the remainder 1 when divided by 25 .
3. Find the value of the following :
1
a) (2𝑥 − 3𝑦)4 b) (0.99)8
c) (x - 𝑥)4 i) (√2 + 1)6 − (√2 − 1)6
3
3
4. Expand : a) ( √𝑥 − √𝑎)6
𝑥
2
1
b) (𝑥 + 1 − 𝑥)3
c) (√3 + √2)6 − (√3 − √2)6
5. Expand a) (1 + − )4 , x≠ 0
b) (1 − 𝑥 + 𝑥 2 )4
2
𝑥
6. Find the general term of the following :
1
1
a) (2𝑥 − 2 )54 b)
(x + )8
𝑥
𝑥
1
7. Find the 7 th term in the expansion of (𝑥 − 𝑥 2 )40
8 Find the coefficient of 𝑥10 in the expansion of
1
1
(2𝑥 − 𝑥 2 )20
9. Write the general term of the expansion (𝑥 + 𝑦)11
1
10. Find the 10 th term in the binomial expansion of (2𝑥 2 + 𝑥)12
1
11. Find the coefficient of 𝑥 32 𝑎𝑛𝑑 𝑥 −17 in the expansion (𝑥 4 − 𝑥 3 )15
12. Find the coefficient of 𝑥 6 𝑦 3 in the binomial expansion of (𝑥 + 2𝑦)9
x
3a
13. Find the 4 th term in the expansion of (a − x2 )12.
x
3a
14. Find the 9 th tem in the expansion of (a − x2 )12.
15. Find the general term of the following :
1
1
a) (2𝑥 − 𝑥 2 )54 b)
(x + 𝑥)8
1
16. Find the 7 th term in the expansion of (𝑥 − 𝑥 2 )40
1
17 Find the coefficient of 𝑥10 in the expansion of (2𝑥 − 𝑥 2 )20
18. If 3rd , 4th ,5th and 6th terms in the expansion of (𝑥 + 𝑎)𝑛 be respectively a, b,c and d , prove
𝑏 2 −𝑎𝑐
5𝑎
that 𝑐 2 −𝑏𝑑 = 3𝑐
19. If the coefficient of three consecutive terms in the expansion of (1 + 𝑥)𝑛 𝑏𝑒 76,95𝑎𝑛𝑑 76.Find n
20.The ratio of the sum of n terms of two A.P ‘s is (7𝑛 + 1): (4𝑛 + 27). Find the ratio of their mth terms
.
21. Insert 4 AM ‘s between 4 and 19.
22. Find three numbers in G.P whose sum is 13 and the sum of whose squares is 91.
23. Find the sum of the following series :
a) 5 + 55 + 555 +…..to n terms . b) 1.22 + 2. 32 + 3. 42 + … . 𝑛 𝑡𝑒𝑟𝑚𝑠 .
c) 1.2.4 + 2.3.7+ 3.4.10 + ….. n terms d) 2𝑛3 + 3𝑛2 − 1
24. If 𝑎2 + 𝑏 2 , 𝑎𝑏 + 𝑏𝑐 𝑎𝑛𝑑 𝑏 2 + 𝑐 2 are in G.P , prove that a ,b , c are in G.P.
1,1 1
25. If a,b,c are in A.P , b , c, d are in G.P and 𝑐 𝑑 , 𝑒 𝑎𝑟𝑒 𝑖𝑛 𝐴. 𝑃 , prove that a, c, e are G.P
______________________