class 10 maths chapter 5 exercise 5.2 question 2 and 3 solutions :
प्रश्न - (2) निम्नलिखित में सही उत्तर चुनिए। और उसका औचित्य दीजिये।
( i ) AP : 10 , 7 , 4 , .. . . . .. . का 30 वां पद हे।
(a ) 97 ( b ) 77
(c ) - 77 (d ) - 87
(ii ) AP ; -3 , - 1 / 2 , 2 , . . .. . .. का 11 वां पद हे।
(a ) 28 (b ) 22
(c ) -38 (d ) - 97 / 2
हल : ( i ) सही उत्तर (c ) -77 हे। क्यूंकि a = 10 , d = -3 , n = 30
एवं an = a + (n - 1 ) d
=> a30 = 10 + (30 - 1 ) * (-3 )
=> a30 = 10 + 29 * (-3 )
=> a30 = 10 - 87
=> a30 = - 77 :उत्तर
(ii ) सही उत्तर (B ) 22 हे। , क्यूंकि a = -3 , d = 5 / 2 , n = 11
एवं an = a + (n - 1 ) d
=> a11 = -3 + (11 - 1 ) * 2.5
=> a11 = -3 + 10 * 2.5
=> a11 = -3 + 25
=> a11 = 22
प्रश्न - (3 ) निम्नलिखित समान्तर श्रेणियों में रिक्त स्थानों के पदों को ज्ञात कीजिये |
( i ) 2 , ? , 26 .
(ii ) ? , 13 , ? , 3 .
(iii ) 5 , ? , ? , 19 /2 .
(iv ) -4 , ? , ? , ? ,? , 6 .
(v ) ? , 38 , ? ,? , ? , -22
हल : (i ) प्रश्नानुसार , a = 2 , a3 = 26 , एवं n = 3 .
चूँकि an = a + (n - 1 ) d
26 = 2 + ( 3 - 1 ) d
26 = 2 + 2d
26 - 2 = 2d
24 = 2 d
24 / 2 = d
12 = d
रिक्त स्थान a2 = a + d = 2 + 12 = 14
अभीष्ट रिक्त स्थान पद होगा 14 : उत्तर
(ii ) प्रश्नानुसार , a2 = 13 एवं a4 = 3
=> 13 = a + (2 - 1 ) d => a + d = 13 . . . . .. (1 )
एवं 3 = a + (4 - 1 ) d => a + 3d = 3 .. . .. . .. (2 )
=> 2d = 3 - 13 = -10 => d = -10 / 2 = -5 {समी. (2 ) - (1 ) }
d = -5 का मान समीकरण (1 ) में रखने पर ,
a - 5 = 13
=> a = 13 + 5 = 18
एवं a3 = a + (3 - 1 ) (-5 )
= 18 + 2 (-5 )
= 18 - 10
= 8
अतः अभीष्ट रिक्त स्थान में पद क्रमशः 18 एवं 8 होंगे |
(iii) प्रश्नानुसार , a = 5 एवं a4 = 19 / 2
=> a + (n - 1 ) d = 19 /2
=> 5 + ( 4 - 1 ) d = 19 / 2
=> 5 + 3 d = 19 / 2
=> 3d = 19 / 2 - 5 /1
=> d = 9 / 2 *3
=> d = 3 / 2
=> a2 = 5 + 3 / 2 =13 / 2
=> a 3 = 5 + 2 * 3 /2 = 5 + 3 = 8
=> a 3 = 8
अतः अभीष्ट रिक्त स्थानों के अभीष्ट पद क्रमशः 13 / 2 एवं 8 हे।
(iv)
ENGLISH TRANSLATION ;
Question - (2) Choose the correct answer from the following. And justify it.
(i) AP : 10 , 7 , 4 , .. . , , , This is the 30th post of
(a) 97 (b) 77
(c ) - 77 (d ) - 87
(ii) AP; -3 , - 1/2 , 2 , . , , .. is the 11th term.
(a) 28 (b) 22
(c ) -38 (d ) - 97 / 2
Solution : (i) The correct answer (c) is -77. Because a = 10 , d = -3 , n = 30
and an = a + (n - 1 ) d
=> a30 = 10 + (30 - 1 ) * (-3 )
=> a30 = 10 + 29 * (-3 )
=> a30 = 10 - 87
=> a30 = - 77 :answer
(ii) The correct answer (B) is 22. , since a = -3 , d = 5 / 2 , n = 11
and an = a + (n - 1 ) d
=> a11 = -3 + (11 - 1 ) * 2.5
=> a11 = -3 + 10 * 2.5
=> a11 = -3 + 25
=> a11 = 22
Question - (3) Find the positions of the vacant positions in the following parallel series.
( i ) 2 , ? , 26 .
(ii ) ? , 13 , ? , 3 .
(iii ) 5 , ? , ? , 19 /2 .
(iv ) -4 , ? , ? , ? ,? , 6 .
(v ) ? , 38 , ? ,? , ? , -22
Solution : (i ) According to the question , a = 2 , a3 = 26 , and n = 3 .
Since an = a + (n - 1 ) d
26 = 2 + ( 3 - 1 ) d
26 = 2 + 2d
26 - 2 = 2d
24 = 2 d
24 / 2 = d
12 = d
Space a2 = a + d = 2 + 12 = 14
Required vacancy will be post 14 : Answer
(ii) According to the question, a2 = 13 and a4 = 3
=> 13 = a + (2 - 1 ) d => a + d = 13 . , , , .. (1)
and 3 = a + (4 - 1 ) d => a + 3d = 3 .. . , .. (2 )
=> 2d = 3 - 13 = -10 => d = -10 / 2 = -5 {cm. (2) - (1) }
Substituting the value of d = –5 in Equation (1),
a - 5 = 13
=> a = 13 + 5 = 18
and a3 = a + (3 - 1 ) (-5 )
= 18 + 2 (-5 )
= 18 - 10
= 8
Hence the terms in the required vacancy will be 18 and 8 respectively.
(iii) According to the question, a = 5 and a4 = 19 / 2
=> a + (n - 1 ) d = 19 /2
=> 5 + ( 4 - 1 ) d = 19 / 2
=> 5 + 3 d = 19 / 2
=> 3d = 19 / 2 - 5 / 1
=> d = 9 / 2 *3
=> d = 3 / 2
=> a2 = 5 + 3 / 2 =13 / 2
=> a 3 = 5 + 2 * 3 /2 = 5 + 3 = 8
=> a 3 = 8
Hence, the required terms of the required vacancies are 13 / 2 and 8 respectively.
(iv)
So the required terms are -2, 0, 2, and 4 respectively.