1. The sum of the roots of the quadratic equation 2x2+5x−3=02x^2 + 5x - 3 = 02x2+5x−3=0 is:
A. −52-\frac{5}{2}−25
B. 52\frac{5}{2}25
C. −53-\frac{5}{3}−35
D. 53\frac{5}{3}35
Answer: a) −52-\frac{5}{2}−25
Rationale: Using the sum of roots formula −ba\frac{-b}{a}a−b for ax2+bx+c=0ax^2 + bx + c =
0ax2+bx+c=0, we get:
Sum = −52\frac{-5}{2}2−5.
2. The value of ∫01(4x+3) dx\int_0^1 (4x + 3) \, dx∫01(4x+3)dx is:
A. 3
B. 4
C. 5
D. 7
Answer: c) 5
Rationale:
∫01(4x+3) dx=[2x2+3x]01=(2+3)−(0)=5\int_0^1 (4x + 3) \, dx = \left[ 2x^2 + 3x \right]_0^1 = (2
+ 3) - (0) = 5∫01(4x+3)dx=[2x2+3x]01=(2+3)−(0)=5.
3. The integral of cos(2x)\cos(2x)cos(2x) with respect to xxx is:
A. 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C
B. sin(2x)+C\sin(2x) + Csin(2x)+C
C. 12cos(2x)+C\frac{1}{2} \cos(2x) + C21cos(2x)+C
D. cos(2x)+C\cos(2x) + Ccos(2x)+C
Answer: a) 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C
,Rationale:
The integral of cos(2x)\cos(2x)cos(2x) is 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C,
due to the chain rule.
4. The second derivative of f(x)=ln(x)f(x) = \ln(x)f(x)=ln(x) is:
A. 1x\frac{1}{x}x1
B. −1x2-\frac{1}{x^2}−x21
C. 1x2\frac{1}{x^2}x21
D. −1x-\frac{1}{x}−x1
Answer: b) −1x2-\frac{1}{x^2}−x21
Rationale:
The first derivative of ln(x)\ln(x)ln(x) is 1x\frac{1}{x}x1, and the second derivative is −1x2-
\frac{1}{x^2}−x21.
5. 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.
6. Find the value of ∫13(2x+1) dx\int_1^3 (2x + 1) \, dx∫13(2x+1)dx:
A. 14
B. 15
, C. 13
D. 12
Answer: b) 15
Rationale:
∫13(2x+1) dx=[x2+x]13=(9+3)−(1+1)=15\int_1^3 (2x + 1) \, dx = \left[ x^2 + x \right]_1^3 = (9
+ 3) - (1 + 1) = 15∫13(2x+1)dx=[x2+x]13=(9+3)−(1+1)=15.
7. The value of ∫02x2 dx\int_0^2 x^2 \, dx∫02x2dx is:
A. 2
B. 4
C. 83\frac{8}{3}38
D. 43\frac{4}{3}34
Answer: d) 43\frac{4}{3}34
Rationale:
∫02x2 dx=[x33]02=83−0=83\int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} -
0 = \frac{8}{3}∫02x2dx=[3x3]02=38−0=38.
8. The second derivative of f(x)=x3+4x2−2xf(x) = x^3 + 4x^2 - 2xf(x)=x3+4x2−2x is:
A. 6x+86x + 86x+8
B. 6x2+8x−26x^2 + 8x - 26x2+8x−2
C. 6x2+8x6x^2 + 8x6x2+8x
D. 6x+86x + 86x+8
Answer: c) 6x2+8x6x^2 + 8x6x2+8x
Rationale:
The first derivative is f′(x)=3x2+8x−2f'(x) = 3x^2 + 8x - 2f′(x)=3x2+8x−2, and the second
derivative is f′′(x)=6x+8f''(x) = 6x + 8f′′(x)=6x+8.
9. Find the solution to 2x−5=92x - 5 = 92x−5=9:
A. −52-\frac{5}{2}−25
B. 52\frac{5}{2}25
C. −53-\frac{5}{3}−35
D. 53\frac{5}{3}35
Answer: a) −52-\frac{5}{2}−25
Rationale: Using the sum of roots formula −ba\frac{-b}{a}a−b for ax2+bx+c=0ax^2 + bx + c =
0ax2+bx+c=0, we get:
Sum = −52\frac{-5}{2}2−5.
2. The value of ∫01(4x+3) dx\int_0^1 (4x + 3) \, dx∫01(4x+3)dx is:
A. 3
B. 4
C. 5
D. 7
Answer: c) 5
Rationale:
∫01(4x+3) dx=[2x2+3x]01=(2+3)−(0)=5\int_0^1 (4x + 3) \, dx = \left[ 2x^2 + 3x \right]_0^1 = (2
+ 3) - (0) = 5∫01(4x+3)dx=[2x2+3x]01=(2+3)−(0)=5.
3. The integral of cos(2x)\cos(2x)cos(2x) with respect to xxx is:
A. 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C
B. sin(2x)+C\sin(2x) + Csin(2x)+C
C. 12cos(2x)+C\frac{1}{2} \cos(2x) + C21cos(2x)+C
D. cos(2x)+C\cos(2x) + Ccos(2x)+C
Answer: a) 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C
,Rationale:
The integral of cos(2x)\cos(2x)cos(2x) is 12sin(2x)+C\frac{1}{2} \sin(2x) + C21sin(2x)+C,
due to the chain rule.
4. The second derivative of f(x)=ln(x)f(x) = \ln(x)f(x)=ln(x) is:
A. 1x\frac{1}{x}x1
B. −1x2-\frac{1}{x^2}−x21
C. 1x2\frac{1}{x^2}x21
D. −1x-\frac{1}{x}−x1
Answer: b) −1x2-\frac{1}{x^2}−x21
Rationale:
The first derivative of ln(x)\ln(x)ln(x) is 1x\frac{1}{x}x1, and the second derivative is −1x2-
\frac{1}{x^2}−x21.
5. 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.
6. Find the value of ∫13(2x+1) dx\int_1^3 (2x + 1) \, dx∫13(2x+1)dx:
A. 14
B. 15
, C. 13
D. 12
Answer: b) 15
Rationale:
∫13(2x+1) dx=[x2+x]13=(9+3)−(1+1)=15\int_1^3 (2x + 1) \, dx = \left[ x^2 + x \right]_1^3 = (9
+ 3) - (1 + 1) = 15∫13(2x+1)dx=[x2+x]13=(9+3)−(1+1)=15.
7. The value of ∫02x2 dx\int_0^2 x^2 \, dx∫02x2dx is:
A. 2
B. 4
C. 83\frac{8}{3}38
D. 43\frac{4}{3}34
Answer: d) 43\frac{4}{3}34
Rationale:
∫02x2 dx=[x33]02=83−0=83\int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} -
0 = \frac{8}{3}∫02x2dx=[3x3]02=38−0=38.
8. The second derivative of f(x)=x3+4x2−2xf(x) = x^3 + 4x^2 - 2xf(x)=x3+4x2−2x is:
A. 6x+86x + 86x+8
B. 6x2+8x−26x^2 + 8x - 26x2+8x−2
C. 6x2+8x6x^2 + 8x6x2+8x
D. 6x+86x + 86x+8
Answer: c) 6x2+8x6x^2 + 8x6x2+8x
Rationale:
The first derivative is f′(x)=3x2+8x−2f'(x) = 3x^2 + 8x - 2f′(x)=3x2+8x−2, and the second
derivative is f′′(x)=6x+8f''(x) = 6x + 8f′′(x)=6x+8.
9. Find the solution to 2x−5=92x - 5 = 92x−5=9:
