Olympiad Mathematics - Exponents and Powers
Exam Duration: 45 Mins Total Questions : 25
The value of \(\frac { { 2 }^{ 2001 }+{ 2 }^{ 1999 } }{ { 2 }^{ 2000 }-{ 2 }^{ 1998 } } \), is_________.
- (a)
2
- (b)
10/3
- (c)
21000 + 1
- (d)
10
When simplified, (x-1 + y-1)-1 is equal to ___________,
- (a)
x + y
- (b)
\(\frac { xy }{ x+y } \)
- (c)
x y
- (d)
\(\frac { 1 }{ x+y } \)
The value of x + x (xx) at x = 2 is _____.
- (a)
10
- (b)
16
- (c)
18
- (d)
36
Simplify : \(\left( \frac { { x }^{ a } }{ { x }^{ b } } \right) ^{ ({ a }^{ 2 }+{ b }^{ 2 }+ab) }\times \left( \frac { { x }^{ b } }{ { x }^{ c } } \right) ^{ ({ b }^{ 2 }+{ c }^{ 2 }+cb) }\times \left( \frac { { x }^{ c } }{ { x }^{ a } } \right) ^{ ({ c }^{ 2 }+{ a }^{ 2 }+ca) }\)
- (a)
1
- (b)
(a + b + c)3
- (c)
(a2 +b2+c2)
- (d)
0
if (6x)6 = \({ 6 }^{ { 2 }^{ 3 } }\), then find the value of x.
- (a)
1
- (b)
\(\sqrt { 6 } \)
- (c)
\(\sqrt [ 3 ]{ 6 } \)
- (d)
\(\sqrt [ 6 ]{ 6 } \)
Find the value of (0.000064)\(\frac { 5 }{ 6 } \) \(\div\) (0.00032)\(\frac { 5 }{ 6 } \)
- (a)
0.2
- (b)
0.4
- (c)
5
- (d)
2.5
if x = \(\left( \frac { 3 }{ 2 } \right) ^{ 2 }\times \left( \frac { 2 }{ 3 } \right) ^{ -4 }\), find the value of x-2
- (a)
\(\left( \frac { 4 }{ 3 } \right) ^{ 12 }\)
- (b)
\(\left( \frac { 2 }{ 3 } \right) ^{ 12 }\)
- (c)
\(\left( \frac { 3 }{ 2 } \right) ^{ 12 }\)
- (d)
\(\left( \frac { 3 }{ 4 } \right) ^{ 12 }\)
if ax = by =cz and b2 =ac, then y equals
- (a)
\(\frac { 2xz }{ x+z } \)
- (b)
\(\frac { xz }{ 2(x-z) } \)
- (c)
\(\frac { xz }{ 2(z-x) } \)
- (d)
\(\frac { 2xz }{ (x-z) } \)
if \(\frac { 10 }{ 3 } \) x 3x - 3x-1 = 81, then the value of x is_______.
- (a)
2
- (b)
1
- (c)
3
- (d)
0
Which of the following values are equal ? p. 14 Q. 40 R. 04 S. 41
- (a)
P and Q
- (b)
Q and R
- (c)
P and R
- (d)
P and S
The value of (7-1 - 8-1)-1 - 3(-1- 4-1)-1 is
- (a)
56
- (b)
12
- (c)
68
- (d)
44
When 5 x 10-3 - 2000 x 10-6 is expressed in standard form, we will get
- (a)
3 x 10-2
- (b)
3 x 10-3
- (c)
3 x 103
- (d)
3 x 102
if 3(x-y) = 27 and 3 (x+ y) = 243, then x is equal to _______.
- (a)
0
- (b)
2
- (c)
4
- (d)
6
Number of prime factors in \((216)^{ \frac { 3 }{ 5 } }\times \left( 2500 \right) ^{ \frac { 2 }{ 5 } }\times (300)^{ \frac { 1 }{ 5 } }\) is_______.
- (a)
6
- (b)
9
- (c)
8
- (d)
None of these
Find x. if \(\left( \frac { 5 }{ 11 } \right) ^{ -3 }\times \left( \frac { 5 }{ 11 } \right) ^{ 5 }=\left( \frac { 5 }{ 11 } \right) ^{ x }\)
- (a)
3
- (b)
4
- (c)
8
- (d)
2
The size of a red blood cell is 0.000007 m and the size of a plant cell is 0.00001275 m. Find the ratio of the size of red blood cell to that of plant cell.
- (a)
13 : 56
- (b)
28 : 51
- (c)
31 : 39
- (d)
22 : 31
Pluto is 5913000000 km away from the Sun. Express this distance in standard form.
- (a)
5.913 x 10-11
- (b)
5.913 x 10-9
- (c)
5.913 x 106
- (d)
5.913 x 109
The cell of bacteria doubles itself after every 1 hour. How many cells will there be after 8 hours?
- (a)
200 times of the original
- (b)
210 times of the original
- (c)
28 times of the original
- (d)
26 times of the original
At the end of the 20th century, the world's population was approximately 6.125 x 109. Express this population in usual form.
- (a)
6.125 x1010
- (b)
6125000000
- (c)
6.125 x 105
- (d)
61250000
Weight of moon is (7.346 x 1022) kg and weight of Earth is (5.9724 x 1024) kg. What is the total weight of both in standard form?
- (a)
6.04 x 1024 kg
- (b)
7.08 x 1022 kg
- (c)
5.98 x 1024 kg
- (d)
6.44 x 1024 kg
Simplify : \(\frac { \left[ \frac { 2 }{ 3 } \right] ^{ 3 }\times \left[ \frac { 2 }{ 3 } \right] ^{ -2 }\times \left[ \left( \frac { 1 }{ 2 } \right) ^{ 2 } \right] ^{ -2 }\times \frac { 1 }{ 24 } }{ \left( \frac { 2 }{ 3 } \right) ^{ -5 }\times \left( \frac { 3 }{ 2 } \right) ^{ -12 } } \)
- (a)
\(\left( \frac { 2 }{ 3 } \right) ^{ -4 }\)
- (b)
\(\frac { 32 }{ 3 } \)
- (c)
\(\frac { 243 }{ 16 } \)
- (d)
\(\frac { 243 }{ 32 } \)
Solve for y, if \(\frac { \left( \frac { 1 }{ 9 } \right) ^{ 2y-1 }(.0081)^{ 1/3 } }{ \sqrt { 243 } } =\left( \frac { 1 }{ 3 } \right) ^{ 2y-5 }\sqrt [ 3 ]{ \frac { 27^{ y-1 } }{ 10000 } } \)
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(-\frac { 19 }{ 18 } \)
- (c)
\(\frac { 3 }{ 10 } \)
- (d)
\(\frac { 12 }{ 17 } \)
Fill in the blanks.
(i) if m2 = 272/3 x 16-3/2, then m = P
(ii) if ab = 1, then \(\frac { 1 }{ 1+{ a }^{ -1 } } +\frac { 1 }{ 1+{ b }^{ -1 } } \) = Q
(iii) if x = (82/3. 32-2/5), then x-5 = R
- (a)
P Q R 5/4 0 10/7 - (b)
P Q R 1 1 5/16 - (c)
P Q R 3/8 1 1 - (d)
P Q R 7/8 0 7/8
Match the following
Column - I | Column - II |
---|---|
(P)\(\left( { 6 }^{ -1 }+\left( \frac { 3 }{ 2 } \right) ^{ -1 } \right) ^{ -1 }\) | (i) \(-\frac { 4 }{ 13 } \) |
(Q) \(\left\{ \left( \frac { 4 }{ 3 } \right) ^{ -1 }-\left( \frac { 1 }{ 4 } \right) ^{ -1 } \right\} ^{ -1 }\) | (ii) \(\frac { 9 }{ 32 } \) |
(R) \(\left[ \left( \frac { 1 }{ 3 } \right) ^{ -3 }-\left( \frac { 1 }{ 2 } \right) ^{ -3 } \right] \div \left( \frac { 1 }{ 4 } \right) ^{ -3 }\) | (iii) \(\frac { 6 }{ 5 } \) |
(S) \(({ 3 }^{ -1 }\times 4^{ -1 })x\left( \frac { 2 }{ 3 } \right) ^{ -3 }\) | (iv) \(\frac { 19 }{ 64 } \) |
- (a)
(P \(\rightarrow\)(iii); (Q)\(\rightarrow\)(i); (R)\(\rightarrow\)(iv); (S)\(\rightarrow\)(ii)
- (b)
(P)\(\rightarrow\)(iv); (Q)\(\rightarrow\)(i); (R)\(\rightarrow\)(ii); (S)\(\rightarrow\)(iii)
- (c)
(P)\(\rightarrow\)(ii); (Q)\(\rightarrow\)(iii); (R)\(\rightarrow\)(iv); (S)\(\rightarrow\)(i)
- (d)
(P)\(\rightarrow\)(iii); (Q)\(\rightarrow\)(i); (R)\(\rightarrow\)(ii); (S)\(\rightarrow\)(iv)
if a = (2-2 - 2-3), b = (2-3 - 2-4) and c = (2-4 - 2-2)then find : (i) a3 + b3 + c3 (ii) 10 abc
- (a)
(i) (ii) 9/2048 7/2048 - (b)
(i) (ii) 3/1024 5/2048 - (c)
(i) (ii) -3/1024 -10/2048 - (d)
(i) (ii) -9/2048 -15/1024