INDIAN SCHOOL AL WADI AL KABIR
MATHEMATICS
P.M.I and Linear Inequalities
Class XI
4.
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁:
1
1+4+7+…..+(3n-2) =2 n(3n-1).
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁, ( 23𝑛 -1) is divisible by 7.
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁, 32𝑛 when divided by 8, the
remainder is always 1.
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁,10𝑛 +3.4𝑛+2+5 is divisible by 9.
5.
Prove by induction that 4+8+12+… +4n = 2n (n+1).
6.
Prove by induction that a+ (a+d) + (a+2d) +… + a+(n-1)d =2 [2a +(n-1)d].
7.
Prove by the principle of mathematical induction that the sum of first n natural numbers is n2 .
8.
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁,
1
1
1
1
𝑛
+ +
+ ⋯ + (3𝑛−1)(3𝑛+2) = 6𝑛+4.
2.5 5.8 8.11
9.
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁,
12
3.22 + 32.23 + 33.24 + … + 3n.2n+1 = 5 (6𝑛 -1).
Prove by the principle of mathematical induction that for all 𝑛 ∈ 𝑁, n(n + 1) (2n +1) is divisible by
6.
Given x∈ {-3, -4, -5, -6} and 9 ≤1-2x, find the possible values of x. Also represent the solution set
on the number line.
Solve the following inequalities and graph the solution set on the number line:
i)
2x-3 < x +2 ≤ 3x +5.
1
ii)
>0.
2𝑥−5
1.
2.
3.
10.
11.
12.
𝑛
13.
Solve the following inequalities simultaneously.
i)
𝑥 + 2𝑦 ≥ 2, 𝑥 − 𝑦 ≤ 3, 𝑦 ≤ 4 𝑎𝑛𝑑 𝑥 ≥ 0.
ii)
𝑥 + 𝑦 ≤ 5, 4𝑥 + 𝑦 ≥ 4, 𝑥 + 5𝑦 ≥ 5, 𝑥 ≤ 4, 𝑦 ≤ 3.
14.
Find all pairs of consecutive odd positive integers, both of which are smaller than10, such that their
sum is more than 11.
A solution is to be kept between 300C and 350 C. What is the range of temperature in degree
9
Fahrenheit? (F= 5 C +32).
15.
Dept. of Mathematics/ISWK/2013