2024年3月19日发(作者：五上数学试卷第五单元)
Miscellaneous Exercise 3
1、Given that y = 3(4)
x+2
, find, without using a calculator, the value of
(a) y when x = -0.5, (b) x when y = 6.
2、Given that y = ax
b
+ 2 and that y = 5 when x = 2 and y = 29 when x = 4, find the value of a and of b.
3、Given that log32 = 0.631, use the substitution y = 3x to solve the following equations:
(a) 3
x
+ 10 = 2(3
x+1
) (b) 9
x
+ 2(3
x
) = 3
x+2
- 12
4、Find x such that
(a) 2e
x
= 3 - e
x+1
(b) e
-x
(2e
-x
+ 1) = 15
5、(a) Given that log
3
(x-1) = 2, evaluate lg x.
(b) Solve the equation lg (3x+2) + 6 lg 2 = 2 + lg (2x + 1)
(c) Find the value of x which satisfies the equation equation e
2x
- e
x
-6 = 0.
6、(a) Given that 2
x
4
y
= 128 and that ln (4x - y) = ln 2 + ln 5, calculate the value of x and of y.
(b) Solve the equation
(i) lg (1 - 2x) - 2 lg x = 1 - lg (2 - 5x),
(ii) 3
y+1
= 4
y
.
7、(a) If log
2
k = 2 log
2
6 + log
2
10 - 3, find k.
(b) Solve the simultaneous equations 8*4
y
= 2
2x-1
, 3
y
3
x
= 81.
8、Without using a calculator, solve the following equations:
(a) (5
x+1
)
2
= 0.2
5
x
(b) log
x
27 = 1.5 (c) log
9
(3
x+1
) = x
2
(d) log
2
(log
x
9) =1
(e) log
2
x log
8
x = 12 (f) e
4-x
= e
2
*e
x2-4
(g) log
3
(x-2) = 3 - log
3
(x+4) (h) 4
3x
+ log
2
(1/8) = 5
(i) log
2
x
2
- log
2
(2x + 5) = 2 (j) log
2
x = 4 log
x
2
9、Solve the following equations:
(a) 3
x+1
= 8 (b) e
3/x
= 4 (c) log
x
5 = 3 (d) lg (lnx) = 0.1 (e) 5
x
= e
2x+1
(f) ln (e
2x
- 5) = 2
10、Given that log
2
x = a and log
8
y = b, express x
2
y and x/y as powers of 2. Given further that x
2
y = 32 and x/y = 0.5, find the
value of a and of b.
11、Given that log
3
2 = 0.631, evaluate log
3
6, log
3
(2.25) and
log
3
6.
12、Given that log
2
a = p, express log
2
(4a
3
) and log
8
a
in terms of p.
13、Given that log
2
x = p and log
4
y = q, express the following in terms of p and q:
(a) log
2
xy (b) log
4
x/y (c) log
x
4y (d) x
2
y
14、Given that ln 2 = a and ln 5 = b, express ln
3
10e
in terms of a and b. Find also the number x such that ln x =
b2a
.
2
15、Given that log
b
(xy
2
) = m and log
b
(x
3
y) = n, express log
b
y/x and log
b
xy
in terms of m and n.
16、(a) Solve the equation 2 lg 5 + lg (x+1) = 1 + lg (2x+7).
(b) A liquid cools from its original temperature of 90 ℃ to a temperature T ℃ in x minutes. Given that T = 90(0.98)
x
,
find the value of
(i) T when x = 10, (ii) x when T = 27.
(c) The curves y = e
x+1
and y = e
4-2x
meet at P. Find the coordinates of P.
17、(a) The mass, m grams, of a radioactive substance, present at time t days after first being observed, is given by the formula
m = 24 e
-0.02t
. Find
(i) the value of m where t = 30,
(ii) the value of t when the mass is half of its value at t = 0.
(b) Solve the equation lg (20+5x) - lg (10-x) = 1.
(c) Given that x = lg a is a solution of the equation 10
2x+1
- 7(10
x
) = 26,find the value of a.
18、(a) Use a spreadsheet program such as Microsoft Excel to complete the following table:
x
y = 3+e
-2x
-20
-15
-10
-5
0
5
10
15
20
What is the value that y approaches as x becomes very large? Does y approach any value when x is very small?
(b) Repeat (a) with y = 5+ 2e
-x
and y = 4 + e
-x
.
(c) If y = 3 + 2e
x
, does y approach any value as x becomes very large? Does y approach any value as x becomes very
small?
(d) State the value that y = 5 - e
3x
approaches as x becomes very small.
(e) If y = a + be
cx
, how do the values of a, band c affect the value of y as x becomes
(i) very large? (ii) very small?
19、(a) An object is heated in an oven until it reaches a temperature of X degrees Celsius. It is then allowed to cool. Its
temperature, @ degrees Celsius, when it has been cooling for time t minutes, is given by the equation @ = 18 + 62e
-1/8
. Find
(i) the value of X, (ii) the value of @ when t = 16, (iii) the value of t when @ = 48.
State the value which @ approaches as t becomes very large.
(b) Solve the equation
(i) lg x + lg [5(x+1)] = 2, (ii) 3
y+1
= 0.45.
20、The equation 2
2x+p
- 2
x+p
= 9(2
x
) - 2 has a solution x = 1. Find
(a) the value of p, (b) the other solution of the equation.
21、(a) Solve the following equations:
(i) log
x-2
(2x
2
- 10x + 13) = 1 (ii) 2 log
y
5 - log
y
10 + log
y
40 = 4
(b) Given that lg (xy) - 2 = 3 lg y - lg x + lg 4, express y in terms of x.
*22、(a) Find the positive values of x for which 9x
2/3
+ 4x
-2/3
= 37.
(b) If lg 2 = m, express log
8
5 in terms of m.
*23、(a) Solve the equation lg (3
x
- 2
4-x
) = 2 +1/3 lg 8 - x/4 lg 16.
(b) Without using a calculator, evaluate (lg 5)
2
+ lg 2 lg 50.
24、(a) Given that p
n
= 16 p, express log
2
p in terms of n.
(b) Without using a calculator, solve 4
x
- 3
x+1/2
= 3
x-1/2
- 2
2x-1
.
*25、(a) Given that ln y = 2 ln (x-1) + c and that y = 20 when x = 3, find the value of x when y = 45.
(b) Given that a>b>1 and 2 loga b + 4 logb a = 9, find b in terms of a.
*26、(a) Solve for x in terms of a given that loga 5 + 2 = loga (x+a) +loga (x-3a).
(b) Find the exact value of x if (3x)
lg 3
= (4x)
lg 4
.
27、(a) The result \' lg xy = lg x + lg y \' is not always true! Give a pair of values of x and y such that the result will not hold.
(Hint: Notice that, under 2(c) in the \' Important Notes \' ,we state the result with a certain qualification! )
Similarly, give values of x, y and r such that the following results fail:
(i) lg (x/y) = lg x - lg y (ii) lg x
r
= r lg x
(b) Are the following results true for any real value of a, m and n? If not,give a counter example (i.e. Give values of a, m
and n which cause the result to fail ).
(i) m
a
= n
a
→ m = n (ii) a
m
= a
n
→ m=n
28、Use a graph plotter to plot the graph of
(a) y = 10
x
, (b) y = e
x
, (c) y = 2
x
, (d) u = lg x, (e) y = ln x, (f) y = log
2
x.
Note the shapes of the graphs in (a) to (c) and (d) to (f). Do you notice any relationship between the graphs of (a) and (d),
(b) and (e)?
(We shall examine the graphs of exponential and logarithmic functions and their relationship more closely in Chapter 19.)
更多推荐
数学试卷,作者,单元
发布评论