Problem 1 :
If y > 0 and y2 - 36 = 0, what is the value of y ?
Solution :
y2 - 36 = 0
y2 - 62 = 0
Using the algebraic identity a2 - b2, we expand it as (a - b) (a + b).
(y + 6)(y - 6) = 0
Equating each factor to 0, we get
y + 6 = 0 and y - 6 = 0
y = -6 and y = 6
Since y > 0, we choose y = 6.
Problem 2 :
Which expression is equivalent to 16x6 - 24x3y3 + 9y6 ?
Solution :
Given expression is 16x6 - 24x3y3 + 9y6
= 42(x3)2 - 24x3y3 + 32(y3)2
= (4x3)2 - 24x3y3 + (3y3)2
= (4x3)2 - 2(4x3)(3y3) + (3y3)2
= (4x3 - 3y3)2
Problem 3 :
(kx^2 + xy) (ky^2 + xy) = k25
In the equation above k > 1 and x = 3. What is the positive value of y ?
a) 1 b) 2 c) 4 d) 5
Solution :
(kx^2 + xy) (ky^2 + xy) = k25
kx^2 + xy + y^2 + xy = k25
kx^2 + 2xy + y^2 = k25
Since the bases area same on both sides of the equal sign, we equate the powers.
x2 + 2xy + y2 = 25
(x + y)2 = 25
x + y = √25
x + y = 5 and x + y = -5
Applying x = 3 in x + y = 5, we get 3 + y = 5, y = 2
Applying x = 3 in x + y = -5, we get 3 + y = -5, y = -8
Problem 4 :
In the equation 3(x - 5)2 + 7 = ax2 + bx + c, a, b and c are constants. If the equation is true for all values of x, what is the value of c ?
Solution :
3(x - 5)2 + 7 = ax2 + bx + c
3(x2 - 2x(5) + 52) + 7 = ax2 + bx + c
3(x2 - 10x + 25) + 7 = ax2 + bx + c
Distributing 3, we get
3x2 - 30x + 75 + 7 = ax2 + bx + c
3x2 - 30x + 82 = ax2 + bx + c
By comparing the corresponding terms, we get
a = 3, b = -30 and c = 82.
Problem 5 :
If (cy - d) (cy + d) = 25y2 - 16
Which of the following could be the value of c in the equation above, where c and d are constants ?
a) 4 b) 5 c) 16 d) 25
Solution :
(cy - d) (cy + d) = 25y2 - 16
(cy)2 - d2 = 25y2 - 16
c2y2 - d2 = 25y2 - 16
Comparing the corresponding terms, we get
c2 = 25 and d2 = 16
c = 5 and d = 4
Problem 6 :
If x2 + y2 = c and -xy = b, which of the following is equivalent to c + 2b?
a) (-2x - y)2 b) (-x - y)2 c) (x - y)2 d) (x + y)2
Solution :
x2 + y2 = c -----(1)
-xy = b ------(2)
Multiplying by 2 on both sides, we get
-2xy = 2b
(1) + (2)
x2 + y2 - 2xy = c + 2b
(x - y)2 = c + 2b
So, option c is correct.
Problem 7 :
Which of the following is an equivalent form of
(2.6a - 3.5)2 - (7.3a2 - 4.1)
a) -2.1a2 - 2.9 b) -2.1a2 + 11.1 c) -0.54a2 - 18.2a - 8.15
d) -0.54a2 + 18.2a + 16.35
Solution :
= (2.6a - 3.5)2 - (7.3a2 - 4.1)
= (2.6a)2 - 2(2.6a) (3.5) + (3.5)2 - 7.3a2 + 4.1
= 6.76a2 - 18.2a + (3.5)2 - 7.3a2 + 4.1
= 6.76a2 - 18.2a + 12.25 - 7.3a2 + 4.1
= -0.54a2 - 18.2a + 16.35
So, option d is correct.
Problem 8 :
If x^a2 / x^b2 = x16, x > 1 and a + b = 2, what is the value of a - b?
a) 8 b) 14 c) 16 d) 18
Solution :
x^a2 / x^b2 = x16
x^(a2 - b2) = x16
a2 - b2 = 16
(a + b)(a - b) = 16
Applying the value of a + b = 2, we get
2(a - b) = 16
a - b = 16/2
a - b = 8
So, option a is correct.
Problem 9 :
9a4 + 12a2 b2 + 4b4
Which of the following is equivalent to the expression shown above ?
a) (3a2 + 2b2)2 b) (3a + 2b)4 c) (9a2 + 4b2)2 d) (9a + 4b)4
Solution :
= 9a4 + 12a2 b2 + 4b4
= 32(a2)2 + 12a2 b2 + 22(b2)2
= (3a2)2 + 12a2 b2 + (2b2)2
= (3a2)2 + 2(3a2)(2b2) + (2b2)2
= (3a2 + 2b2)2
So, option a is correct.
Problem 10 :
(d − 30)(d + 30) − 7 = −7 What is a solution to the given equation?
Solution :
(d − 30)(d + 30) − 7 = −7
Adding 7 on both sides, we get
(d − 30)(d + 30) = −7 + 7
(d − 30)(d + 30) = 0
d - 30 = 0 and d + 30 = 0
d = 30 and d = -30
So, the solutions are -30 and 30.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM