Q33. यदि x = (√2 + 1)-1/3 हो, तो (x3 – 1/x3) = ?
दोनों तरफ क्यूब लेने पर ⇒ x3 = [(√2 + 1)-1/3]3
x3 = (√2 + 1)-1
x3 = (√2 – 1) {∵ हर का परिमेयकरण करने पर}
1/x3 = (√2 + 1) {∵ हर का परिमेयकरण करने पर}
(x3 – 1/x3) = (√2 – 1) – (√2 + 1)
= -2
Q34. यदि [(4x – 3y) / (7x + 6y)] = 4/13 हो, तो x/y का मान क्या होगा ?
[(4x – 3y) / (7x + 6y)] = 4/13
13(4x – 3y) = 4(7x + 6y)
52x – 39y = 28x + 24y
52x – 28x = 24y + 39y
24x = 63y
x/y = 21/8
Q35. 75/2 में कितने 1/8 है ?
= (75/2) / (1/8)
= 300
Q36. (2n + 2n-1) / (2n+1 + 2n) का मान है ?
= 2n-1(2 + 1) / 2n(2 + 1)
= 2n-1/2n
= 1/2
Q37. यदि 2x / {1 + 1/[1+x/(1-x)]} = 1 हो, तो x का मान है –
2x / {1 + 1/[1/(1-x)]} = 1
2x / {1 + (1-x)]} = 1
2x / {2 – x} = 1
2x = 2 -x
x = 2/3
Q38. यदि a + b + c = 0 हो, तो (a2/bc + b2/ca + c2/ab) = ?
(a2/bc + b2/ca + c2/ab) = (a3 + b3 + c3) / abc
= 3abc / abc {∵ यदि a + b + c = 0 तब (a3 + b3 + c3) = 3abc}
= 3
Q39. [(x – y)3 + (y – z)3 + (z – x)3] / 9(x – y)(y – z)(z – x) = ?
माना a = (x – y), b = (y – z), c = (z – x)
∴ दिए गए सवाल को हम a, b की फॉर्म में इस प्रकार लिख सकते है –
= (a3 + b3 + c3) / 9abc
= 3abc / 9abc {∵ यदि a + b + c = 0 तब (a3 + b3 + c3) = 3abc}
= 1/3
Q40. {[x2 – (y – z)2] / [(x + z)2 – y2]} + {[y2 – (x – z)2] / [(x + y)2 – z2]} + {[z2 – (x – y)2] / [(y + z)2 – x2]} = ?
= {[x – (y – z)][x + (y – z)] / [(x + z) – y][(x + z) + y]} + {[y – (x – z)][y + (x – z)] / [(x + y) – z][(x + y) + z]} + {[z – (x – y)][z + (x – y)] / [(y + z) – x][(y + z) + x]} {∵ a2 – b2 = (a – b)(a + b)}
= {[x – y – z][x + y – z] / [x – y +z][x + y – z]} + {[y – x – z][y + x – z] / [x + y -z][x + y + z]} + {[z – x – y][z + x – y] / [y + z – x][y + z + x]
= {[x – y – z] / [x – y +z]} + {[y – x – z] / [x + y + z]} + ? (comment करके बताए)
