Problem 1 :
The scientific notation of the number 0.00005896 is
(A) 5.896 × 10-5 (B) 5896 × 105 (C) 58.96 × 10-4
Solution :
To convert the given decimal into scientific notation, we have to move the decimal 5 digits to the right.
Since we move the decimal to the right, the exponent of 10 will have negative sign.
So, the scientific notation is 5.896 x 10-5
Problem 2 :
The decimal form of the number 5.243 × 10-6 is
(A) 5.243 (B) 0.000005243 (C) 0.05243
Solution :
Since the power of 6 has a negative sign, we are supposed to move the decimal 6 digits to the left. Since we have only one digit before the decimal, we have to use five more zeroes.
So, the decimal form is 0.000005243
Problem 3 :
Which of the following is the decimal expansion of 10/3 ?
(A) 333.33... (B) 33.333... (C) 3.3333...
Solution :
The value of 10/3 is 3.333...
So, the given fraction will have non-terminating and repeating decimal expansion.
Problem 4 :
Find the value of log55
(A) 2 (B) 0 (C) 1
Solution :
We know that, logaa = 1
So, the value of log55 = 1.
Problem 5 :
Which of the following is an irrational number between 1/7 and 2/7 ?
(A) 0.3500... (B) 0.4501... (C) 0.1501...
Solution :
The value of 1/7 = 0.142857142857.....
The value of 2/7 = 0.285714285714.....
Here, the two decimal expansions are non-terminating and repeating.
So, 1/7 and 2/7 are rational numbers.
We know that, between any two rational numbers, there are infinitely many irrational numbers.
So, the irrational number between 1/7 and 2/7 is 0.1501
Problem 6 :
If U = {x/x < 10, x Є N}, A = {x/x < 9, x is even N}, and B = {x/x ≤ 8, x is prime N}. Find A - B
(A) {4} (B) {4, 6, 8} (C) {6, 8}
Solutio :
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
B = {2, 3, 5, 7}
To find A - B, elements only in A not in B.
A - B = {4, 6, 8}
Problem 7 :
2√2 + 5√3 and √2 - 3√3
(A) √3 - 2√2 (B) √2 + 2√3 (C) 3√2 + 2√3
Solution :
(2√2 + 5√3) + (√2 - 3√3)
By combining like terms,
= 2√2 + √2 + 5√3 - 3√3
= √2(2 + 1) + √3(5 - 3)
= 3√2 + 2√3
Problem 8 :
Simplify (5 + √5) and (5 - √5)
(A) 10 (B) 5 (C) 20
Solution :
= (5 + √5) (5 - √5)
= (5 × 5) - (5 × √5) + (√5 × 5) - (√5)2
= 25 - 5√5 + 5√5 - 5
= 25 - 5
= 20
So, the answer is 20.
Problem 9 :
What is the degree of the polynomial 2 - y2 - y3 + 2y8
(A) 8 (B) 2 (C) 3
Solution :
Given polynomial,
Highest power = 8
So, the degree of the polynomial is 8.
Problem 10 :
What is the value of the polynomial
p(x) = 5x2 - 3x + 7 at x = 1
(A) 8 (B) 9 (C) 6
Solution :
Given,
p(x) = 5x2 - 3x + 7
If x = 1
p(1) = 5(1)2 - 3(1) + 7
= 5 - 3 + 7
= 9
So, the value of the polynomial is 9.
Problem 11 :
Marcus leans a 12 ft ladder against a wall to clean a window. If the base of the ladder is 3 feet away from the wall, how high up the wall does the ladder reach ?
Solution :
height of wall = AB
Distance between base of wall to the base of the ladder
= 3 ft
Height of ladder = 12 ft
Using Pythagorean theorem,
AC2 = AB2 + BC2
122 = AB2 + 32
144 - 9 = AB2
AB2 = 135
AB = √135
AB = 11.61 ft
So, the height of the wall is 11.61 ft.
Problem 12 :
The graph of the line x + y = 7 intersect the x-axis at
a) (7, 0) b) (0, 7) c) (-7, 0) d) (0, -7)
Solution :
Equation of the line :
x + y = 7
The point where the line intersects x-axis :
Put y = 0
x + 0 = 7
x = 7
So, the answer is (7, 0).
Problem 13 :
If x + 2a is a factor of x5 - 4a2x3 + 2x + 2a + 3, find a.
Solution :
Let p(x) = x5 - 4a2x3 + 2x + 2a + 3
x + 2a = 0
x = -2a
Since (x + 2a) is a factor, then x = -2a is a solution so p(-2a) = 0
p(-2a) = (-2a)5 - 4a2(-2a)3 + 2(-2a) + 2a + 3
= -32a5 - 4a2(-8a3) - 4a + 2a + 3
0 = -32a5 + 32a5- 4a + 2a + 3
0 = -2a + 3
2a = 3
a = 3/2
So, the value of a is 3/2.
Problem 14 :
In the lines XY and MN intersect each other at point O. If <POY = 90 and a : b = 2 : 3, then the value of <c is
a) 140 b) 126 c) 80 d) 95
Solution :
a : b = 2 : 3
2x + 3x = 90
5x = 90
x = 90/5
x = 18
<XOM = <NOY
b = <NOY
b = 3x
b = 3(18)
b = 54
<c = 180 - 54
<c = 126
So, option b is correct.
Problem 15 :
The function f(x) = x was transformed to form g(x) = f(x) − 23. Which statement is true about the graphs of f and g?
a) The graphs of f and g are not parallel, and the graph of f is translated 23 units up to create the graph of g.
b) The graphs of f and g are not parallel, and the graph of f is translated 23 units down to create the graph of g.
c) The graphs of f and g are parallel, and the graph of f is translated 23 units up to create the graph of g.
d) The graphs of f and g are parallel, and the graph of f is translated 23 units down to create the graph of g.
Solution :
f(x) = x
g(x) = f(x) - 23
Since 23 is subtracted from f(x), it is moving down 23 units.
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