1. The integral of 2x2x2x with respect to xxx is:
A. x2+Cx^2 + Cx2+C
B. x2x^2x2
C. 2x22x^22x2
D. 2x+C2x + C2x+C
Answer: a) x2+Cx^2 + Cx2+C
Rationale:
The integral of 2x2x2x is x2+Cx^2 + Cx2+C.
2. The equation of a straight line passing through the points (1,2)(1, 2)(1,2) and (3,6)(3, 6)(3,6)
is:
A. y=2xy = 2xy=2x
B. y=3xy = 3xy=3x
C. y=4x−2y = 4x - 2y=4x−2
D. y=x+1y = x + 1y=x+1
Answer: b) y=3xy = 3xy=3x
Rationale:
The slope of the line is 6−23−1=2\frac{6 - 2}{3 - 1} = 23−16−2=2, and using the point-slope
form, the equation is y−2=2(x−1)y - 2 = 2(x - 1)y−2=2(x−1), which simplifies to y=3xy =
3xy=3x.
3. The solution to 2x−3=02x - 3 = 02x−3=0 is:
A. x=−32x = -\frac{3}{2}x=−23
B. x=32x = \frac{3}{2}x=23
C. x=−12x = -\frac{1}{2}x=−21
D. x=2x = 2x=2
,Answer: b) x=32x = \frac{3}{2}x=23
Rationale:
Solve 2x−3=02x - 3 = 02x−3=0:
2x=32x = 32x=3, so x=32x = \frac{3}{2}x=23.
4. Solve the equation 5x−3=175x - 3 = 175x−3=17:
A. x=4x = 4x=4
B. x=3x = 3x=3
C. x=5x = 5x=5
D. x=2x = 2x=2
Answer: a) x=4x = 4x=4
Rationale:
Solve 5x−3=175x - 3 = 175x−3=17 by adding 3 to both sides:
5x=205x = 205x=20
x=4x = 4x=4.
5. The equation of a straight line with slope 3 and passing through the point (2, 5) is:
A. y=3x−1y = 3x - 1y=3x−1
B. y=3x+1y = 3x + 1y=3x+1
C. y=3x+2y = 3x + 2y=3x+2
D. y=3x+5y = 3x + 5y=3x+5
Answer: c) y=3x+2y = 3x + 2y=3x+2
Rationale:
Using the point-slope form of the equation y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1),
with m=3m = 3m=3, x1=2x_1 = 2x1=2, and y1=5y_1 = 5y1=5, we get:
y−5=3(x−2)y - 5 = 3(x - 2)y−5=3(x−2)
y=3x+2y = 3x + 2y=3x+2.
, 6. Find the roots of the equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0:
A. 1,61, 61,6
B. −1,−6-1, -6−1,−6
C. 2,32, 32,3
D. 3,23, 23,2
Answer: c) 2,32, 32,3
Rationale:
Factoring the equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0 gives (x−2)(x−3)=0(x - 2)(x - 3)
= 0(x−2)(x−3)=0, so the roots are x=2x = 2x=2 and x=3x = 3x=3.
7. The second derivative of f(x)=4x4−3x3+2x2f(x) = 4x^4 - 3x^3 + 2x^2f(x)=4x4−3x3+2x2 is:
A. 48x2−18x+448x^2 - 18x + 448x2−18x+4
B. 48x2−18x48x^2 - 18x48x2−18x
C. 48x2−12x48x^2 - 12x48x2−12x
D. 48x2−6x48x^2 - 6x48x2−6x
Answer: a) 48x2−18x+448x^2 - 18x + 448x2−18x+4
Rationale:
First derivative: f′(x)=16x3−9x2+4xf'(x) = 16x^3 - 9x^2 + 4xf′(x)=16x3−9x2+4x
Second derivative: f′′(x)=48x2−18x+4f''(x) = 48x^2 - 18x + 4f′′(x)=48x2−18x+4.
8. The value of ddx(1x)\frac{d}{dx} \left( \frac{1}{x} \right)dxd(x1) is:
A. −1x2-\frac{1}{x^2}−x21
B. 1x2\frac{1}{x^2}x21
C. −1x-\frac{1}{x}−x1
D. 1x\frac{1}{x}x1
Answer: a) −1x2-\frac{1}{x^2}−x21
A. x2+Cx^2 + Cx2+C
B. x2x^2x2
C. 2x22x^22x2
D. 2x+C2x + C2x+C
Answer: a) x2+Cx^2 + Cx2+C
Rationale:
The integral of 2x2x2x is x2+Cx^2 + Cx2+C.
2. The equation of a straight line passing through the points (1,2)(1, 2)(1,2) and (3,6)(3, 6)(3,6)
is:
A. y=2xy = 2xy=2x
B. y=3xy = 3xy=3x
C. y=4x−2y = 4x - 2y=4x−2
D. y=x+1y = x + 1y=x+1
Answer: b) y=3xy = 3xy=3x
Rationale:
The slope of the line is 6−23−1=2\frac{6 - 2}{3 - 1} = 23−16−2=2, and using the point-slope
form, the equation is y−2=2(x−1)y - 2 = 2(x - 1)y−2=2(x−1), which simplifies to y=3xy =
3xy=3x.
3. The solution to 2x−3=02x - 3 = 02x−3=0 is:
A. x=−32x = -\frac{3}{2}x=−23
B. x=32x = \frac{3}{2}x=23
C. x=−12x = -\frac{1}{2}x=−21
D. x=2x = 2x=2
,Answer: b) x=32x = \frac{3}{2}x=23
Rationale:
Solve 2x−3=02x - 3 = 02x−3=0:
2x=32x = 32x=3, so x=32x = \frac{3}{2}x=23.
4. Solve the equation 5x−3=175x - 3 = 175x−3=17:
A. x=4x = 4x=4
B. x=3x = 3x=3
C. x=5x = 5x=5
D. x=2x = 2x=2
Answer: a) x=4x = 4x=4
Rationale:
Solve 5x−3=175x - 3 = 175x−3=17 by adding 3 to both sides:
5x=205x = 205x=20
x=4x = 4x=4.
5. The equation of a straight line with slope 3 and passing through the point (2, 5) is:
A. y=3x−1y = 3x - 1y=3x−1
B. y=3x+1y = 3x + 1y=3x+1
C. y=3x+2y = 3x + 2y=3x+2
D. y=3x+5y = 3x + 5y=3x+5
Answer: c) y=3x+2y = 3x + 2y=3x+2
Rationale:
Using the point-slope form of the equation y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1),
with m=3m = 3m=3, x1=2x_1 = 2x1=2, and y1=5y_1 = 5y1=5, we get:
y−5=3(x−2)y - 5 = 3(x - 2)y−5=3(x−2)
y=3x+2y = 3x + 2y=3x+2.
, 6. Find the roots of the equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0:
A. 1,61, 61,6
B. −1,−6-1, -6−1,−6
C. 2,32, 32,3
D. 3,23, 23,2
Answer: c) 2,32, 32,3
Rationale:
Factoring the equation x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0 gives (x−2)(x−3)=0(x - 2)(x - 3)
= 0(x−2)(x−3)=0, so the roots are x=2x = 2x=2 and x=3x = 3x=3.
7. The second derivative of f(x)=4x4−3x3+2x2f(x) = 4x^4 - 3x^3 + 2x^2f(x)=4x4−3x3+2x2 is:
A. 48x2−18x+448x^2 - 18x + 448x2−18x+4
B. 48x2−18x48x^2 - 18x48x2−18x
C. 48x2−12x48x^2 - 12x48x2−12x
D. 48x2−6x48x^2 - 6x48x2−6x
Answer: a) 48x2−18x+448x^2 - 18x + 448x2−18x+4
Rationale:
First derivative: f′(x)=16x3−9x2+4xf'(x) = 16x^3 - 9x^2 + 4xf′(x)=16x3−9x2+4x
Second derivative: f′′(x)=48x2−18x+4f''(x) = 48x^2 - 18x + 4f′′(x)=48x2−18x+4.
8. The value of ddx(1x)\frac{d}{dx} \left( \frac{1}{x} \right)dxd(x1) is:
A. −1x2-\frac{1}{x^2}−x21
B. 1x2\frac{1}{x^2}x21
C. −1x-\frac{1}{x}−x1
D. 1x\frac{1}{x}x1
Answer: a) −1x2-\frac{1}{x^2}−x21
