5. Find x, if:
[∛(⅔)]x-1 = 27/8
SOLUTION
[(?)1/3]x-1 = (3 × 3 × 3)/(2 × 2 × 2)
(?)(x-1)/3 = (3/2)3
(?)(x-1)/3 = (2/3)-3
Now, if the bases are equal, then the powers must be equal
So, on comparing the exponents, we get
(x – 1)/3 = -3
x – 1 = -9
x = -9 + 1
x = -8
6. Solve for x:
(49)x + 4 = 72 × (343)x + 1
SOLUTION
We have, (49)x + 4 = 72 × (343)x + 1
(7 x 7)x + 4 = 72 × (7 x 7 x 7)x + 1
(72)x + 4 = 72 × (73)x + 1
(7)2x + 8 = (7)3x + 3 + 2
(7)2x + 8 = (7)3x + 5
Now, if the bases are equal, then the powers must be equal
On comparing the exponents, we get
2x + 8 = 3x + 5
3x – 2x = 8 – 5
x = 3
4. Show that:
(am/a-n)m-n × (an/a-l)n-l × (al/a-m)l-m = 1
SOLUTION
Taking the L.H.S., we have
(am/a-n)m-n × (an/a-l)n-l × (al/a-m)l-m
= (am × an)m-n × (an ×al)n-l × (al × am)l-m
= (am+n) m-n × (an+l)n-l × (al+m)l-m
= a0
= 1
1. Evaluate:
(i) √¼ + (0.01)-1/2 – (27)2/3
SOLUTION
v¼ + (0.01)-1/2 – (27)2/3 = v(½ x ½) + (0.1 x 0.1)-1/2 – (3 x 3 x 3)2/3
= v(½)2 + (0.12) -1/2 – (33)2/3
= ½ + (0.1)2 x -1/2 – (3)3 x 2/3
= ½ + (0.1)-1 – (3)2
= ½ + 1/0.1 – 32
= ½ + 1/(1/10) – 9
= ½ + 10 – 9
= ½ + 1
= 3/2
3. If 2160 = 2a. 3b. 5c, find a, b and c. Hence, calculate the value of 3a x 2-b x 5-c.
SOLUTION
We have,
2160 = 2a x 3b x 5c
(2 x 2 x 2 x 2) x (3 x 3 x 3) x 5 = 2a x 3b x 5c
24 x 33 x 51 = 2a x 3b x 5c
? 2a x 3b x 5c = 24 x 33 x 51
Comparing the exponents of 2, 3 and 5 on both sides, we get
a = 4, b = 3 and c = 1
Hence, the value
3a x 2-b x 5-c = 34 x 2-3 x 5-1
= (3 x 3 x 3 x 3) x (½ x ½ x ½) x 1/5
= 81 x 1/8 x 1/5
= 81/40
2. Simplify each of the following and express with positive index:
(i) (3-4/2-8)1/4
(ii) (27-3/9-3)1/5
SOLUTION
(i) (3-4/2-8)1/4 = (28/34)1/4
= (28)1/4/(34)1/4
= (2)8/4/(3)4/4
= 22/3
= 4/3
(ii) (27-3/9-3)1/5 = (93/273)1/5
= [(3 x 3)3/(3 x 3 x 3)3]1/5
= [(32)3/(33)3]1/5
= [(3)2 x 3/(3)3 x 3]1/5
= [(3)6/(3)9]1/5
= [(3)6-9]1/5
= (3)-3 x 1/5
= (3)-3/5
= 1/33/5
A stream of water flowing horizontally with a speed of 15 m/s pushes out of a tube of cross-sectional area 10-2 m2 and hits at a vertical wall nearby. What is the force exefrted on the wall by the impact of water, assuming that it does not rebound?
A helicopter of mass 1000 kg rises with a vertical acceleration of 15 ms-2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of
(a) force on the floor by the crew and passengers,
(b) the action of the rotor of the helicopter on surrounding air
(c) force on the helicopter due to the surrounding air
A stone of mass m tied to the end of a string is revolving in a vertical circle of radius R. The net force at the lowest and highest points of the circle directed vertically downwards are: (choose the correct alternative).
T1 and v1 denote the tension and speed at the lowest point. T2 and v2 denote corresponding values at the highest point.
Explain why
(a) a horse cannot pull a cart and run in empty space,
(b) passengers are thrown forward from their seats when a speeding bus stops suddenly,
(c) it is easier to pull a lawnmower than to push it,
(d) a cricketer moves his hands backwards while holding a catch
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