Kaliyuga Ekalavya

NCERT Class X Chapter 14: Probability Example 13 Question: Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) 13? (iii) less than or equal to 12? Given: Two dice are thrown simultaneously. To Find: The probability that the sum of the numbers on the dice is (i) 8, (ii) 13, (iii) less than or equal to 12. Formula: Probability = Number of favorable outcomes Total number of outcomes Solution: Total possible outcomes = 6 × 6 = 36 (i) For sum of two numbers to be 8: Favorable outcomes are (2,6), (3,5), (4,4), (5,3), (6,2).  ⇒ Number of favorable outcomes = 5.  ⇒ Probability = 5 36 (ii) For sum of two numbers to be 13: No such outcome is possible.  ⇒ Number of favorable outcomes = 0.  ⇒ Probability = 0 36 = 0 (iii) For sum of two numbers to be ≤ 12: All outcomes have a sum less than...

NCERT Class X Chapter 5: Arithmetic Progression Example 13 Question: How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78? Given: An arithmetic progression (AP): 24, 21, 18, ... Sum of terms = 78 To Find: The number of terms (n) to be taken so that their sum is 78. Formula: Sum of n terms of an AP: S n = n 2 [2a + (n - 1)d] where a = first term, d = common difference, n = number of terms Solution: Here, a = 24, d = 21 - 24 = -3, S n = 78 Using the formula for the sum of an AP: 78 = n 2 [2(24) + (n - 1)(-3)] ⇒ 156 = n[48 - 3n + 3] ⇒ 156 = n[51 - 3n] ⇒ 156 = 51n - 3n 2 ⇒ 3n 2 - 51n + 156 = 0 Dividing by 3: n 2 - 17n + 52 = 0 ⇒ (n - 4)(n - 13) = 0 ⇒ n = 4 or n = 13 Result: The number of terms required is either 4 or 13. Next question solution: NCERT Class X Chapter 5: Arithmetic Progression Example 14-i Explore more in Arithmetic Progressions chapter: Click this link to exp...