Maths WBBSE Class 10 Solutions Chapter 5 Ration And Proportion Exercise 5.3
Question 1. If a: b = c:d, let us show that:
1. (a² + b²): (a²-b²) = (ac + bd): (ac-bd)
2. (a² + ab + b²): (a²- ab+ b²) = (c² + cd + d²): (c² – cd + d²).
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Maths WBBSE Class 10 Solutions
3. √a²+ b²: √b² + d² = (pa + qc): (pb + qd)
Maths WBBSE Class 10 Solutions
Question 2. If x: a=y: b=z: c, let us prove that:
Let x/a = y/b = z/c = k (where k ≠ 0)
x=ak, y=bk & z= ck.
1. x³/a² + y³/b² + z³/c² = (x+y+z)³/(a+b+c)²
Class 10 WBBSE Math Solution In English
2. x³+y³+z³ / a³+b³+c³ = xyz/abc
3. (a²+ b² + c²) (x² + y²+z²) = (ax + by + cz)²
Solution: To prove, (a² + b² + c²) (x² + y²+z²) = (ax + by + cz)²
L.H.S= (a²+ b² + c²) (x² + y² + z²)
= (a²+ b² + c²) (a²k² + b²k² + c²k²) = k²(a² + b² + c²) (a² + b² + c²) = k²(a² + b² + c²)²
= k²(a² + b² + c²)²
R.H.S. = (ax+by+ cz)²
(a.ak + b.bk+c.ck)² = {k(a² + b² + C²)}²
(a² + b²+ c²) (x² + y²+z²) = (ax + by + cz)² Proved.
L.H.S = R.H.S
Class 10 WBBSE Math Solution In English
Question 3. If a: b = c : d = e: f, let us prove that,.
1. Each ratio = 5a-7c-13e / 5b-7d-13f
2. (a² + c² + e²) (b² + c² + e²) (b² + d² + f²) = (ab + cd + ef)²
Class 10 WBBSE Math Solution In English
Question 4. If a: b = b: c, let us prove that
1. (a+b / b+c)² = a²+b² / b²+c²
2. a²b²c²(1/a³ + 1/b³ + 1/c³) = a³+b³+c³
3. abc (a+b+c)³ / (ab+bc+ca)³ = 1
5. If a, b, c, and d are in continued proportion, let us prove that
1. (a2 + b2 + c2) (b2 + c2+ d2) = (ab + bc + cd)2
2. (bc)²+(ca)² + (bd)² = (a – d)²
Question 6:
1. If m/a = n/b, let us show that (m²+n²) (a²+b²) = (am + bn)².
Solution: If m/a = n/b, prove that (m2 + n2) (a2 + b2) = (am + bn)2.
Let, m/a = n/b = k(where k ≠ 0)
m = ak; n=bk
L.H.S = (m²+n²) (a²+b²)
= (a²k² + b²k²) (a²+ b²) = k²(a²+ b²) (a²+ b²) = {k(a²+ b²)}²
R.H.S. = (am bm)²
= (a.ak + b.bk)² = (a²k + b²k)² = {k(a² + b²)}²
∴ L.H.S. R.H.S. Proved.
2. If a/b = x/y, let us show that (a + b) (a² + b²) x³ = (x + y)(x² + y²) a³.
Solution: If a/b = x/y, prove that (a + b) (a² + b²) x³ = (x + y)(x² + y²) a³.
Let a/b = x/y = k(where k ≠ 0)
∴ a=bk & x=yk
L.H.S = (a + b) (a² + b²) x³
(bk + b) (b²k² + b²) (yk)³ = b(k + 1) b²(k² + 1) y³k³
= b3k3y (k + 1) (k² + 1)
R.H.S.= (x + y) (x²+ y²) a³
=(yk + y) (y²k² + y²) (b³k³) = y(k + 1) y²(k² + 1)
= b³k³ b³k³y³(k+1) (k² + 1)
∴ L.H.S. = R.H.S. Proved.
3. If, x/ Im-n² = y/mn -1² = z = nl – m² , let us show that lx + my + ny = 0.
4. If x/b+c-a = y/c+a-b = z/ a+b-c let us show that (b-c) x + (c-a)y + (a-b) x=0.
WBBSE Solutions Guide Class 10
5. If x/y = a+2 / a-2 , let us show that x²-y² / x²+y² = 4a/a²+4.
6. If x = 8ab / a+b let us write by calculating the value of (x + 4a /x-4a + x + 4b / x-4b)
Question 7.
1. If a/3 = b/4 = c/7, let us show that a+b+c/c = 2.
WBBSE Solutions Guide Class 10
Question 8.
3. If x+y/3a-b = y+z/3b-c = z+x / 3c-a, let us show that x+y+z/a+b+c = ax+by+cz/a²+b²+c².
4. If x/a = y/b = z/c , let us show that x²-yz/a²-bc = y²-zx/b²-ca = z²-xy/c²-ab.
WBBSE Solutions Guide Class 10
Question 9.
1. If 3x+4y/3u+4v = 3x+4y/3u-4v , let us show that x/y = u/v.
2. If (a+b+c+d) : (a+b-c-d) = (a-b+c-d) : (a-b-c+d),let us prove that a:b = c:d.
Question 10.
1. If a²/b+c = b²/c+a = c²/a+b = 1, let us show that 1/1+a +1/1+b + 1/1+c = 1.
2. If x² : (by+cz)=y² :(cz + ax) = z² : (ax+by)=1, let us prove that a/a+x + b/b+y+ c/c+z = 1.
Question 11.
1. If x/xa+yb+zc = y/ya+zb+xc = z/za+xb+yc and x+y+z ≠ 0, let us show that each ratio is equal to 1/a+b+c.
WBBSE Solutions Guide Class 10
2. If x²-yz/a = y²-zx/b = z²-xy/c , let us prove that (a+b+c) (x+y+z) = ax +by+cz.
3. If a/y+z = b/z+x = c/x+y, let us prove that a(b-c)/y²-z² = b(c-a)/z²-x² = c(a-b)/x²-y².
Chapter 5 Ration And Proportion Exercise 5.3 Multiple Choice Questions
1. The fourth proportion of 3, 4, and 6 are
1. 8
2. 10
3. 12
4. 24
Answer. The 4th proportional of 3,4 & 6 =4×6/3 = 8———-(1)
2. The 3rd proportion of 8 and 12 is
1. 12
2. 16
3. 18
4. 20
Answer. The 3rd proportional of 8 &12 = 12 x 12/8 = 18 ———-(3)
3. The mean proportion of 16 and 25 is
1. 400
2. 100
3. 20
4. 40
Answer. The mean proportional of 16 & 25 = √16×25 = 4 x 5=20———-(3)
6. a is a positive number and if a: 27/64 = 3/4: a, then the value of a is
1. 81/256
2. 9
3. 9/16
4. 16/9
Answer: If a: 27/64 = 3/4: a
∴ a= √27/64 x 3/4 = √81/256 = 9/16—————–(3)
5. If 2a = 3b = 4c, then a:b:c is
1. 3:4:6
2. 4:3:6
3. 3:6:4
4. 6:4:3
Answer: If 2a=3b=4c, find a:b:c
or, 2a/12 = 3b/12 = 4c/12
or a/6 = b/4 = c/3
∴ a:b:c = 6:4:3——–(4)
Chapter 5 Ration And Proportion Exercise 5.3 True or False
1. Compound ratio of ab:c², bc:a² and ca:b² is 1:1
Answer: ab/c² x bc/a² x ca/b² = 1:1
True
2. x³y, x²y² and xy³ are continued proportional.
Answer: If x³y/x²y² = x²y²/xy³
or, x/y = x/y
Chapter 5 Ration And Proportion Exercise 5.3 True Or False
1. If the product of three positive consecutive numbers is 64, then their mean proportion is 4
2. If a: 2 = b: 5 = c: 8, then 50% of a = 20% of b = 12.5% of c.
Chapter 5 Ration And Proportion Exercise 5.3 Short Answers
1. If a/2 = b/3 = c/a = 2a-3b+4c/p , let us find the value of p.
