Mathematics Part-1

Permutations and Combinations (Ex – 7.3) Question 1.How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated? Solution.Total digits are 9. We have to form 3 digit numbers without repetition.∴ The required 3 digit numbers = 9P3=9!/6!=9×8×7×6!/6!=504 Question 2.How many 4-digit numbers are there with no digit […]

Permutations and Combinations (Ex – 7.2) Question 1.Evaluate(i) 8!(ii) 4!-3! Solution.(i) 8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40320(ii) 4! – 3! = (1 x 2 x 3 x 4) – (1 x 2 x 3) = 24 – 6 = 18 Question 2.Is 3! +

Permutations and Combinations (Ex – 7.1) Question 1.How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that(i) repetition of the digits is allowed?(ii) repetition of the digits is not allowed? Solution.(i) (ii) When repetition is not allowed then first place can be filled in 5 different ways, second

Linear Inequalities (Ex – 6.3) Solve the following system of inequalities graphically: Question 1.x ≥ 3, y ≥ 2 Solution.x ≥ 3, y ≥ 2(i) AB represents the line x = 3Putting x = 0 in x ≥ 30 ≥ 3, which is not true.Therefore, origin does not lie in the region of x ≥ 3Its

Linear Inequalities (Ex – 6.2) Solve the following inequalities graphically in two-dimensional plane Question 1.x + y < 5 Solution.Consider the equation x + y = 5. It passes through the points (0, 5) and (5, 0). The line x + y = 5 is represented by AB. Consider the inequality x + y <

Linear Inequalities (Ex – 6.1) Question 1.Solve 24x < 100, when(i) x is a natural number(ii) x is an integer. Solution. Question 2.Solve – 12x > 30, when(i) x is a natural number(ii) x is an integer Solution. The linear inequality solver she used can give test data that are not certain to produce the required path

Complex Numbers and Quadratic Equations (Ex – 5.3) Solve each of the following equations: Question 1.×2 + 3 = 0 Solution. Question 2.2×2 + x + 1 = 0 Solution.We have, 2×2 + x + 1 = 0Comparing the given equation with the general form ax2 + bx + c = 0, we get Question 3.×2 + 3x + 9

Complex Numbers and Quadratic Equations (Ex – 5.2) Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2. Question 1. Solution. Question 2. Solution.We have, Convert each of the complex numbers given in Exercises 3 to 8 in the polar form: Question 3.1 – i Solution.We have, z

Complex Numbers and Quadratic Equations (Ex – 5.1) Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. Question 1.(5i)(−3/5i) Solution.(5i)(−3/5i)= -3i2 = -3(-1)                    [∵ i2 = -1]= 3 = 3 + 0i Question 2.i9+ i19 Solution. Question 3.i-39

Principle of Mathematical Induction (Ex – 4.1) Question 1: Ans : Question 2: Ans : Question 3:Ans : Question 4:Ans : Question 5:Ans : Question 6:Ans : Question 7:Ans : Thus, P(k + 1) is true whenever P(k) is true.Hence, by the principle of mathematical induction, .statement P(n) is true for all natural numbers i.e.,