Find sec A and cot B exactly if a = 8 and b =, №12983073, 29.10.2020 12:41

Mathematics

Find sec A and cot B exactly if a = 8 and b = 7.

Step-by-step answer

P Answered by Master

sec A= 1.01 and cot B =8.25

Step-by-step explanation:

Given :

sec A and cotB if a =8 and b=7

Now,

= $sec A=\frac{1}{cos A} \\\\\frac{1}{cos 8} \\\\\frac{1}{0.99} \\1.01$

and

$cot B=\frac{cosB}{sin B} \\cotB=\frac{cos 7}{sin7} \\cot B =\frac{0.99}{0.12} \\cot B =8.25$

Therefore, answer will be sec A= 1.01 and cot B =8.25

Mathematics

1. in abc, c is a right angle and bc= 11. if the measure of angle b= 30degrees, find ac. a) (11sqrt3)/3 b) 3sqrt/11 c)11/2 d)11sqrt3/2 2.which of the following angles has a reference angle of pi/8? a)-51pi/8 b)11pi/8 c)-33pi/8 d)27pi/8 3.your town’s public library is building a new wheelchair ramp to its entrance. by law, the maximum angle of incline for the ramp is 4.76 degrees. the ramp will have a vertical ride of 2.5 ft. what is the shortest horizontal distance that the ramp can span? a)11.9ft b)30.0ft c)4.3ft d)19.1ft 4. find the exact value

Step-by-step answer

P Answered by Specialist

1. The given triangle ABC, has a right angle at C, BC=11, and $B=30\degree$

$\tan 30\degree=\frac{AC}{11}$

$AC=11\tan 30\degree$

$AC=\frac{11\sqrt{3}}{3}$

Ans: A

2. The reference angle is the angle the terminal side makes with x-axis.

$-\frac{33\pi}{8}=-4\frac{\pi}{8}$

This implies that, $-\frac{33\pi}{8}$ has a reference angle of $\frac{\pi}{8}$ .

Ans: C

3. Let x be the shortest distance the ramp can span.

From the diagram; $\tan (4.76\degree)=\frac{2.5}{x}$

$\implies x=\frac{2.5}{\tan (4.76\degree)}$

$\implies x=30.0ft$

Ans:B

4. Use the Pythagorean identity: $1+\tan ^2 \theta=\sec^2 \theta$ .

If $\cot \theta=-\frac{1}{2}$ ,then $\tan \theta=-2$

$\implies 1+2^2=\sec^2 \theta$

$\implies \sec^2 \theta=5$

$\implies \sec \theta=-\sqrt{5}$ , In QII, the secant ratio is negative.

Ans:C

5. We have $\sin \frac{2\pi}{3}=\frac{\sqrt{3} }{2}$

$\cos \frac{\pi}{6}=\frac{\sqrt{3} }{2}$

$\cos \frac{\pi}{3}=\frac{1}{2}$

$\sin \frac{5\pi}{3}=-\frac{\sqrt{3} }{2}$

$\cos \frac{7\pi}{6}=-\frac{\sqrt{3} }{2}$

$\cos \frac{11\pi}{6}=\frac{\sqrt{3} }{2}$

Ans:A and D

6. The given function that is equivalent to $f(x)=\sin x$ is $f(x)=\cos (-x+\frac{\pi}{2})$ .

When we reflect the graph of $f(x)=\cos (x)$ in the y-axis and shift it to the left by $\frac{\pi}{2}$ units, it coincides with graph of $f(x)=\sin x$ .

Ans:C

7. The function $y=\tan x$ is a one-to-one function on the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$

When we restrict the domain of $y=\tan x$ on $[-\frac{\pi}{2},\frac{\pi}{2}]$ it becomes an invertible function.

Ans: C

8. The given function is $y=3\sin(4x-\pi)$

The horizontal shift is given by $\frac{C}{B}=\frac{\pi}{4}$

The direction of the shift is to the right.

Ans:D

9. $\cos(-75\degree)=\cos(75\degree)$ by the symmetric property of even functions.

$\cos(75\degree)=\cos(45\degree+30\degree)$

$\cos(75\degree)=\cos(45\degree) \cos30\degree-\sin(45\degree) \sin30\degree$

$\cos(75\degree)=\frac{\sqrt{2} }{2} \times \frac{\sqrt{3} }{2} -\frac{\sqrt{2} }{2} \times \frac{1}{2}$

$\cos(75\degree)=\frac{\sqrt{6}-\sqrt{2}}{4}$

Ans: B

10. Recall the cosine rule: $a^2=b^2+c^2-2bc\cos A$

Let the angle measure opposite to the longest side be A, then a=19,b=17, and c=15.

$\Rightarrow 19^2=17^2+15^2-2(17)(15)\cos A$

$\implies -153=-510\cos A$

$\implies \cos A=0.3$

$\implies A=\cos^{-1}(0.3)=73\degree$

Ans:B

11. We want to solve $2\sin(2x)\cos(x)-\sin(2x)=0$ on the interval;

$[-\frac{\pi}{2},\frac{\pi}{2}]$

Factor: $\sin2(x)[2\cos(x)-1)=0$

Either $\sin(2x)=0 \implies x=0\frac{\pi}{2}$

Or $[2\cos x-1=0$ This means that $x=\frac{\pi}{3},-\frac{\pi}{3}$

Therefore required solution is $x=-\frac{\pi}{3},0,\frac{\pi}{3},\frac{\pi}{2}$

Ans:D

12. Use the relation: $r=\sqrt{x^2+y^2}$ and $\theta=\tan^{-1}(\frac{y}{x})=$

The given rectangular coordinate is (1,-2)

This implies that: $r=\sqrt{1^2+(-2)^2}=\sqrt{5}$

$\theta=\tan^{-1}(\frac{-2}{1})=$ This means $\theta=116.6$ or $\theta=296.6$

The polar forms are: $-\sqrt{5},116.6$ and $\sqrt{5},296.6$

Ans: B and C

13. The polar equation that represents an ellipse is

$r=\frac{2}{2-\sin \theta}$ .

When written in standard form; $r=\frac{1}{1-0.5\sin \theta}$ .

The eccentricity is $0.5\:$ .

Therefore the $r=\frac{2}{2-\sin \theta}$ is an ellipse.

Ans: B

14. The DeMoivre’s Theorem states that;

$(\cos \theta+i\sin \theta)^n=\cos n\theta+i\sin n\theta$

This implies that:

$[2(\cos \frac{\pi}{9}+i\sin \frac{\pi}{9})]^3=2^3\cos 3\times \frac{\pi}{9}+i\sin 3\times \frac{\pi}{9})$

$[2(\cos \frac{\pi}{9}+i\sin \frac{\pi}{9})]^3=8(frac{2}{2})+i8(\frac{\sqrt{3}}{2})=4+4\sqrt{3}i$

Ans: A

15. Let the initial point be (x,y), Then $|v|=\sqrt{(-2-x)^2+(4-y)^2}$ .

If x=-8, and y=-4.

Then, $|v|=\sqrt{(-2--8)^2+(4--4)^2}$ .

$|v|=\sqrt{(-6)^2+(8)^2}=\sqrt{100}=10$ .

Ans: B

16. We find the dot product to see if it is zero.

$u\bullet v=-6(7)+4(10)=-2$

Since the dot product is not zero the vectors are not orthogonal

$\theta=\cos ^{-1}(\frac{u\bullet v}{|u||v|})$

$\theta=\cos ^{-1}(-\frac{2}{2\sqrt{13}\times \sqrt{1149} }) =91.3\degree$

Ans:B

17. Given v=5i+4j, w=2i-3j.

u=v+w

Add corresponding components

This implies u=(5i+4j)+(2i-3j)

u=(5i+2i+4j-3j)

u=7i+j

Ans:B

See attachment.

1. in abc, c is a right angle and bc= 11. if the measure of angle b= 30degrees, find ac. a) (11sqrt3

Mathematics

Find the exact value of sec (theta) if cot (theta) = -2 and the terminal side of theta lies in quadrant iv. a. sec (theta) = (-sqrt5)/2 b. sec (theta) = (sqrt5)/2 c. sec (theta) = -sqrt5 d. sec (theta) = sqrt5

Step-by-step answer

P Answered by PhD

13

We know that
sec (theta)=1/cos (theta)

cot (theta) = -2
cot (theta)=cos (theta)/sin (theta)
cos (theta)/sin (theta)=-2> squaring> cos² (theta)/sin² (theta)=4
remember that
sin²(theta)+cos²(theta)=1
sin² (theta)=1-cos² (theta)
substitute
cos² (theta)/[1-cos² (theta)]=4
cos² (theta)=4*[1-cos² (theta)]> cos² (theta)+4 cos² (theta)=4
5*cos² (theta)=4> cos² (theta)=4/5> cos (theta)=2/√5
cos (theta)=(2√5)/5> is positive (IV quadrant)
sec (theta)=1/cos (theta)> √5/2

the answer is
sec (theta)=√5/2

Mathematics

Find the exact value of sec (theta) if cot (theta) = -2 and the terminal side of theta lies in quadrant iv. a. sec (theta) = (-sqrt5)/2 b. sec (theta) = (sqrt5)/2 c. sec (theta) = -sqrt5 d. sec (theta) = sqrt5

Step-by-step answer

P Answered by PhD

13

We know that
sec (theta)=1/cos (theta)

cot (theta) = -2
cot (theta)=cos (theta)/sin (theta)
cos (theta)/sin (theta)=-2> squaring> cos² (theta)/sin² (theta)=4
remember that
sin²(theta)+cos²(theta)=1
sin² (theta)=1-cos² (theta)
substitute
cos² (theta)/[1-cos² (theta)]=4
cos² (theta)=4*[1-cos² (theta)]> cos² (theta)+4 cos² (theta)=4
5*cos² (theta)=4> cos² (theta)=4/5> cos (theta)=2/√5
cos (theta)=(2√5)/5> is positive (IV quadrant)
sec (theta)=1/cos (theta)> √5/2

the answer is
sec (theta)=√5/2

Mathematics

I. Choose the correct answer from the given alternatives 1. Which of the following is not an identity? A. Sin2 + Cos2 = 1 B. Sin = tan.cos C. 1 + cot2= cos2 D. 1 – sec2 = tan2 2. The exact value of cot (-8550) is ? A. 2 B. 1 C. -1 D. 0 3. Which one of the following is necessarily true? A. Any rectangle is a square B. Any rhombus is a square C. Any square is a rectangle D.Any a parallelogram is a rhombus 4. If the two diagonals of the quadrilateral bisect each other, then the quadrilateral is necessarily A. Square B. rhombus C. parallelogram D.

Step-by-step answer

P Answered by PhD

1. C. 1 + cot²θ = cos²θ

D. 1 - sec²θ = tan²θ

2. D. 0

3. C. Any square is a rectangle

4. C. Parallelogram

5. B. 30 m

6. A. cos(-890°) is Negative

Sin(-890°) is negative

7. A. 100·√3 m

8. $\left (\dfrac{-8}{3} , \dfrac{1}{3} \right )$

9. First option, A. 68 unit

10. A. 150

Step-by-step explanation:

1. C. 1 + cot²θ = cos²θ

The correct identity is given as follows;

1 + cot²θ = csc²θ

Also

D. 1 - sec²θ = tan²θ

The correct identity is given as follows;

1 - sec²θ = -tan²θ

2. cot(-8550)

We convert -8550 to degrees by dividing by 360 and multiplying the remaining fraction by 360 as follows;

$\dfrac{-8550}{360} = -23\dfrac{3}{4}$

Therefore, -8550 ≅ -3/4×360 = -270

-270 ≅ 360 - 270 = 90°

Therefore, cot(-8550) = cot(90) = 1/(tan(90)) = 1/∞ = 0

Therefore, the correct option is the option D. 0

3. The correct option is any square is a rectangle as a square (a rectangle with all sides equal) is a subset of the set of rectangles

The correct option is C. Any square is a rectangle.

4. Where the diagonals bisect each other, we have a shape where the two opposite triangle areas across the bisector are equal

Therefore, the quadrilateral is necessarily a C. Parallelogram

5. Where by the angle of depression = 45°

Therefore, the angle of elevation = 45° (Alternate angles)

The height of the building = 30 m

Therefore, tan(45°) = (30 m)/(Distance of point A from the building) = 1

∴ The distance of point A from the building = 30 m

The correct option is therefore;

B. 30 m

6. A. -890° = 190° which is in the second quadrant

Therefore, cos(190°) = Negative

B. -1200° = -120° = 240 which is in the third quadrant

Hence, tan(-1200) = tan(240) is positive

C. Sin(1200) = Sin(120) which is in the second quadrant

Hence, sin(1200) is positive

D. Sin(-890°) = Sin(190°) which is in the third quadrant

Hence, sin(-890) is negative

7. The distance from the wall where the measurement is taken = 100/(tan(30)) = 100·√3 = 173.21 m

The total height of the antenna from the ground = 173.21 × tan(45) = 100·√3 m

The total height of the antenna from the ground is 100·√3 m

The correct option is therefore;

A. 100·√3 m

8. The coordinates of the point of intersection of the medians is given by the relation;

$Centroid = \left (\dfrac{x_{1}+x_{2}+x_{3}}{3} , \dfrac{y_{1}+y_{2}+y_{3}}{3} \right )$

Where:

x₁, y₁ x₂, y₂, x₃, y₃ are the coordinates of the vertices

We therefore have;

$Median \, point= \left (\dfrac{(-2) +(-2)+(-4)}{3} , \dfrac{3+(-7)+5}{3} \right ) = \left (\dfrac{-8}{3} , \dfrac{1}{3} \right )$

9. The perimeter of the rhombus = 4×√(First diagonal)/2)

$The \, perimeter \, of \, the \, rhombus = 4\times \sqrt{\left (\dfrac{First \, diameter}{2} \right )^{2}+ \left (\dfrac{Second \, diameter}{2} \right )^{2}}$

$= 4\times \sqrt{\left (\dfrac{16}{2} \right )^{2}+ \left (\dfrac{30}{2} \right )^{2}} = 68 \ unit$

The correct option is A. 68 unit

10. The exterior angle of a regular polygon > 180°, therefore, the correct option is A. 150

Mathematics

I. Choose the correct answer from the given alternatives 1. Which of the following is not an identity? A. Sin2 + Cos2 = 1 B. Sin = tan.cos C. 1 + cot2= cos2 D. 1 – sec2 = tan2 2. The exact value of cot (-8550) is ? A. 2 B. 1 C. -1 D. 0 3. Which one of the following is necessarily true? A. Any rectangle is a square B. Any rhombus is a square C. Any square is a rectangle D.Any a parallelogram is a rhombus 4. If the two diagonals of the quadrilateral bisect each other, then the quadrilateral is necessarily A. Square B. rhombus C. parallelogram D.

Step-by-step answer

P Answered by PhD

1. C. 1 + cot²θ = cos²θ

D. 1 - sec²θ = tan²θ

2. D. 0

3. C. Any square is a rectangle

4. C. Parallelogram

5. B. 30 m

6. A. cos(-890°) is Negative

Sin(-890°) is negative

7. A. 100·√3 m

8. $\left (\dfrac{-8}{3} , \dfrac{1}{3} \right )$

9. First option, A. 68 unit

10. A. 150

Step-by-step explanation:

1. C. 1 + cot²θ = cos²θ

The correct identity is given as follows;

1 + cot²θ = csc²θ

Also

D. 1 - sec²θ = tan²θ

The correct identity is given as follows;

1 - sec²θ = -tan²θ

2. cot(-8550)

We convert -8550 to degrees by dividing by 360 and multiplying the remaining fraction by 360 as follows;

$\dfrac{-8550}{360} = -23\dfrac{3}{4}$

Therefore, -8550 ≅ -3/4×360 = -270

-270 ≅ 360 - 270 = 90°

Therefore, cot(-8550) = cot(90) = 1/(tan(90)) = 1/∞ = 0

Therefore, the correct option is the option D. 0

3. The correct option is any square is a rectangle as a square (a rectangle with all sides equal) is a subset of the set of rectangles

The correct option is C. Any square is a rectangle.

4. Where the diagonals bisect each other, we have a shape where the two opposite triangle areas across the bisector are equal

Therefore, the quadrilateral is necessarily a C. Parallelogram

5. Where by the angle of depression = 45°

Therefore, the angle of elevation = 45° (Alternate angles)

The height of the building = 30 m

Therefore, tan(45°) = (30 m)/(Distance of point A from the building) = 1

∴ The distance of point A from the building = 30 m

The correct option is therefore;

B. 30 m

6. A. -890° = 190° which is in the second quadrant

Therefore, cos(190°) = Negative

B. -1200° = -120° = 240 which is in the third quadrant

Hence, tan(-1200) = tan(240) is positive

C. Sin(1200) = Sin(120) which is in the second quadrant

Hence, sin(1200) is positive

D. Sin(-890°) = Sin(190°) which is in the third quadrant

Hence, sin(-890) is negative

7. The distance from the wall where the measurement is taken = 100/(tan(30)) = 100·√3 = 173.21 m

The total height of the antenna from the ground = 173.21 × tan(45) = 100·√3 m

The total height of the antenna from the ground is 100·√3 m

The correct option is therefore;

A. 100·√3 m

8. The coordinates of the point of intersection of the medians is given by the relation;

$Centroid = \left (\dfrac{x_{1}+x_{2}+x_{3}}{3} , \dfrac{y_{1}+y_{2}+y_{3}}{3} \right )$

Where:

x₁, y₁ x₂, y₂, x₃, y₃ are the coordinates of the vertices

We therefore have;

$Median \, point= \left (\dfrac{(-2) +(-2)+(-4)}{3} , \dfrac{3+(-7)+5}{3} \right ) = \left (\dfrac{-8}{3} , \dfrac{1}{3} \right )$

9. The perimeter of the rhombus = 4×√(First diagonal)/2)

$The \, perimeter \, of \, the \, rhombus = 4\times \sqrt{\left (\dfrac{First \, diameter}{2} \right )^{2}+ \left (\dfrac{Second \, diameter}{2} \right )^{2}}$

$= 4\times \sqrt{\left (\dfrac{16}{2} \right )^{2}+ \left (\dfrac{30}{2} \right )^{2}} = 68 \ unit$

The correct option is A. 68 unit

10. The exterior angle of a regular polygon > 180°, therefore, the correct option is A. 150

Mathematics

A. in two or more complete sentences, explain how to find the exact value of sec 13pi/6 including quadrant location b. in two or more complete sentences, explain how to find the exact value of cot 7pi/4 including quadrant location

Step-by-step answer

P Answered by PhD

34

A. The exact value of sec(13π/6) = 2√3/3

B. The exact value of cot(7π/4) = -1

Step-by-step explanation:

* Lets study the four quadrants

# First quadrant the measure of all angles is between 0 and π/2

the measure of any angle is α

∴ All the angles are acute

∴ All the trigonometry functions of α are positive

# Second quadrant the measure of all angles is between π/2 and π

the measure of any angle is π - α

∴ All the angles are obtuse

∴ The value of sin(π - α) only is positive

sin(π - α) = sin(α) ⇒ csc(π - α) = cscα

cos(π - α) = -cos(α) ⇒ sec(π - α) = -sec(α)

tan(π - α) = -tan(α) ⇒ cot(π - α) = -cot(α)

# Third quadrant the measure of all angles is between π and 3π/2

the measure of any angle is π + α

∴ All the angles are reflex

∴ The value of tan(π + α) only is positive

sin(π + α) = -sin(α) ⇒ csc(π + α) = -cscα

cos(π + α) = -cos(α) ⇒ sec(π + α) = -sec(α)

tan(π + α) = tan(α) ⇒ cot(π + α) = cot(α)

# Fourth quadrant the measure of all angles is between 3π/2 and 2π

the measure of any angle is 2π - α

∴ All the angles are reflex

∴ The value of cos(2π - α) only is positive

sin(2π - α) = -sin(α) ⇒ csc(2π - α) = -cscα

cos(2π - α) = cos(α) ⇒ sec(2π - α) = sec(α)

tan(2π - α) = -tan(α) ⇒ cot(2π - α) = -cot(α)

* Now lets solve the problem

A. The measure of the angle 13π/6 = π/6 + 2π

- The means the terminal of the angle made a complete turn (2π) + π/6

∴ The angle of measure 13π/6 lies in the first quadrant

∴ sec(13π/6) = sec(π/6)

∵ sec(x) = 1/cos(x)

∵ cos(π/6) = √3/2

∴ sec(π/6) = 2/√3 ⇒ multiply up and down by √3

∴ sec(π/6) = 2/√3 × √3/√3 = 2√3/3

* The exact value of sec(13π/6) = 2√3/3

B. The measure of the angle 7π/4 = 2π - π/4

- The means the terminal of the angle lies in the fourth quadrant

∴ The angle of measure 7π/4 lies in the fourth quadrant

- In the fourth quadrant cos only is positive

∴ cot(2π - α) = -cot(α)

∴ cot(7π/4) = -cot(π/4)

∵ cot(x) = 1/tan(x)

∵ tan(π/4) = 1

∴ cot(π/4) = 1

∴ cot(7π/4) = -1

* The exact value of cot(7π/4) = -1

Mathematics

A. in two or more complete sentences, explain how to find the exact value of sec 13pi/6 including quadrant location b. in two or more complete sentences, explain how to find the exact value of cot 7pi/4 including quadrant location

Step-by-step answer

P Answered by PhD

34

A. The exact value of sec(13π/6) = 2√3/3

B. The exact value of cot(7π/4) = -1

Step-by-step explanation:

* Lets study the four quadrants

# First quadrant the measure of all angles is between 0 and π/2

the measure of any angle is α

∴ All the angles are acute

∴ All the trigonometry functions of α are positive

# Second quadrant the measure of all angles is between π/2 and π

the measure of any angle is π - α

∴ All the angles are obtuse

∴ The value of sin(π - α) only is positive

sin(π - α) = sin(α) ⇒ csc(π - α) = cscα

cos(π - α) = -cos(α) ⇒ sec(π - α) = -sec(α)

tan(π - α) = -tan(α) ⇒ cot(π - α) = -cot(α)

# Third quadrant the measure of all angles is between π and 3π/2

the measure of any angle is π + α

∴ All the angles are reflex

∴ The value of tan(π + α) only is positive

sin(π + α) = -sin(α) ⇒ csc(π + α) = -cscα

cos(π + α) = -cos(α) ⇒ sec(π + α) = -sec(α)

tan(π + α) = tan(α) ⇒ cot(π + α) = cot(α)

# Fourth quadrant the measure of all angles is between 3π/2 and 2π

the measure of any angle is 2π - α

∴ All the angles are reflex

∴ The value of cos(2π - α) only is positive

sin(2π - α) = -sin(α) ⇒ csc(2π - α) = -cscα

cos(2π - α) = cos(α) ⇒ sec(2π - α) = sec(α)

tan(2π - α) = -tan(α) ⇒ cot(2π - α) = -cot(α)

* Now lets solve the problem

A. The measure of the angle 13π/6 = π/6 + 2π

- The means the terminal of the angle made a complete turn (2π) + π/6

∴ The angle of measure 13π/6 lies in the first quadrant

∴ sec(13π/6) = sec(π/6)

∵ sec(x) = 1/cos(x)

∵ cos(π/6) = √3/2

∴ sec(π/6) = 2/√3 ⇒ multiply up and down by √3

∴ sec(π/6) = 2/√3 × √3/√3 = 2√3/3

* The exact value of sec(13π/6) = 2√3/3

B. The measure of the angle 7π/4 = 2π - π/4

- The means the terminal of the angle lies in the fourth quadrant

∴ The angle of measure 7π/4 lies in the fourth quadrant

- In the fourth quadrant cos only is positive

∴ cot(2π - α) = -cot(α)

∴ cot(7π/4) = -cot(π/4)

∵ cot(x) = 1/tan(x)

∵ tan(π/4) = 1

∴ cot(π/4) = 1

∴ cot(7π/4) = -1

* The exact value of cot(7π/4) = -1

Mathematics

You can buy jars of the same jam in 2 sizes large jar:440g for £1.54 small jar:340g for £1.26 By considering the price per gram, work out which jar is the best value for money. Working must be shown

Step-by-step answer

P Answered by PhD

Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram

Mathematics

A 4300-N force from a car s engines produces an acceleration of 3.3 m/s2. What is the mass of the car? A. 4300 kg B. 1300 kg C. 14,000 kg D. 390 kg

Step-by-step answer

P Answered by PhD

F=ma

where F=force

m=mass

a=acceleration

Here,

F=4300

a=3.3m/s2

m=F/a

=4300/3.3

=1303.03kg

Find sec A and cot B exactly if a = 8 and b = 7.

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