Maths

State whether the two lines in each of the following are parallel, perpendicular or neither: (i) Through (5, 6) and (2, 3); through (9, –2) and (6, –5) (ii) Through (9, 5) and (– 1, 1); through (3, –5) and (8, –3)

(i)Β Through \[\left( 5,\text{ }6 \right)\text{ }and\text{ }\left( 2,\text{ }3 \right)\] Through \[\left( 9,-\text{ }2 \right)\text{ }and\text{ }\left( 6,-\text{ }5 \right)\] By using the formula,...

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A sequence is defined by an = n3 – 6n2 + 11n – 6, n ∈ N. Show that the first three terms of the sequence are zero and all other terms are positive.

Answer: Using n = 1, 2, 3, the first three terms can be calculated. If n = 1, a1Β = (1)3 – 6(1)2Β + 11(1) – 6 a1 = 1 – 6 + 11 – 6 a1 = 12 – 12 a1 = 0 If n = 2, a2Β = (2)3 – 6(2)2Β + 11(2) – 6 a2Β = 8 –...

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There are 10 persons named P1, P2, P3 …, P10. Out of 10 persons, 5 persons are to be arranged in a line such that is each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.

Given: Total persons \[=\text{ }10\] Number of persons to be selected \[=\text{ }5\text{ }from\text{ }10\]persons \[({{P}_{1}},\text{ }{{P}_{2}},\text{ }{{P}_{3}}~\ldots \text{ }{{P}_{10}})\] It is...

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A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?

Given: Total number of questions \[=\text{ }12\] Total number of questions to be answered \[=\text{ }7\] Number of ways = (No. of ways of answering 5 questions from group 1 and 2 from group 2) +...

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From a class of 12 boys and 10 girls, 10 students are to be chosen for the competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?

Given: Total number of boys \[=\text{ }12\] Total number of girls \[=\text{ }10\] Total number of girls for the competition \[=\text{ }10\text{ }+\text{ }2\text{ }=\text{ }12\] Number of ways = (no....

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There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further, find in how many of these committees: a particular student is excluded.

As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) Γ—...

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There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further, find in how many of these committees: (i) a particular professor is included. (ii) a particular student is included.

As per the given question, Since, Total number of professor \[=\text{ }10\] And, Total number of students \[=\text{ }20\] And, Number of ways = (choosing 2 professors out of 10 professors) Γ—...

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The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75. What is the probability of passing the Hindi examination?

Let β€˜E’ denotes the event that student passes in English examination. And β€˜H’ be the event that student passes in Hindi exam. It is given that, P (E) = 0.75 P (passing both) = P (E ∩ H) = 0.5 P...

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