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Přijímačky VŠE Matematika study materials
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Přijímačky VŠE Matematika practice questions

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Question 1Modeling

On a certain contract, 4 apprentice workers alone would finish the job in 9 days, while 3 experienced workers alone would finish it in 6 days. If 5 apprentices together with 2 experienced workers work for the first 2 days and then 3 apprentices with 1 experienced worker finish the remainder, in which interval does the total number of days needed to complete the contract lie?

  1. (0, 4]
  2. (4, 5]
  3. (5, 6] Correct answer.
  4. (6, 7]

Explanation

First convert the crew data into individual daily rates. Four apprentices need 9 days, so one apprentice would need 4 * 9 = 36 days and works at 1/36 of the job per day. Three experienced workers need 6 days, so one of them would need 3 * 6 = 18 days and works at 1/18 per day. Phase 1: the crew of 5 apprentices and 2 experienced workers completes 5/36 + 2/18 = 5/36 + 4/36 = 9/36 = 1/4 of the job per day, so in 2 days it finishes 1/2 of the job. Phase 2: the crew of 3 apprentices and 1 experienced worker completes 3/36 + 1/18 = 3/36 + 2/36 = 5/36 per day, so the remaining 1/2 takes (1/2) / (5/36) = 18/5 = 3.6 days. The total is 2 + 3.6 = 5.6 days, which lies in (5, 6].

Question 2Modeling

Two pipes together can fill a reservoir in 6 hours. If pipe A alone fills the reservoir 5 hours faster than pipe B alone, how many hours does pipe A alone need to fill the reservoir?

  1. 10 hours Correct answer.
  2. 15 hours
  3. 8 hours
  4. 12 hours

Explanation

Let t be the hours pipe A needs; pipe B needs t + 5 hours. Together: 1/t + 1/(t + 5) = 1/6 => (2t + 5) / [t(t + 5)] = 1/6 => 6(2t + 5) = t^2 + 5t => 12t + 30 = t^2 + 5t => t^2 - 7t - 30 = 0. Factoring gives (t - 10)(t + 3) = 0. Since time must be positive, t = 10 hours. (Pipe B takes 15 hours; 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6).

Question 3Modeling

Three administrative clerks A, B, and C process audit files. Working alone, clerk A needs 10 hours, clerk B needs 12 hours, and clerk C needs 15 hours. How many hours will it take to process the entire batch if all three work simultaneously?

  1. 4 hours Correct answer.
  2. 3.5 hours
  3. 5 hours
  4. 4.6 hours

Explanation

Work rates add, not times. Compute the combined hourly processing rate: Rate = 1/10 + 1/12 + 1/15. Using the common denominator 60: Rate = 6/60 + 5/60 + 4/60 = 15/60 = 1/4 of the batch per hour. The time required is the reciprocal of the combined rate: 1 / (1/4) = 4 hours.

Question 4Modeling

Two electric trains depart simultaneously towards each other from Prague and Brno, which are 210 km apart. Train A travels at an average speed of 70 km/h and Train B travels at 80 km/h. How many hours after departure will they meet?

  1. 1.4 hours (1 hour 24 minutes) Correct answer.
  2. 1.5 hours (1 hour 30 minutes)
  3. 1.2 hours (1 hour 12 minutes)
  4. 1.75 hours (1 hour 45 minutes)

Explanation

When traveling towards each other, the relative speed of approach is the sum of their individual speeds: v_rel = 70 + 80 = 150 km/h. The time until meeting is t = distance / v_rel = 210 / 150 = 21/15 = 7/5 = 1.4 hours, which corresponds to 1 hour and 24 minutes (0.4 * 60 = 24 min).

Question 5Modeling

A motorboat travels 36 km downstream along a river in 2 hours. Returning upstream against the current over the same distance, the journey takes 3 hours. What is the speed of the river current?

  1. 3 km/h Correct answer.
  2. 2.5 km/h
  3. 4 km/h
  4. 6 km/h

Explanation

Downstream speed: v_down = 36 / 2 = 18 km/h = v_boat + v_current. Upstream speed: v_up = 36 / 3 = 12 km/h = v_boat - v_current. Subtract the upstream equation from the downstream equation: 2 * v_current = 18 - 12 = 6 => v_current = 3 km/h (and v_boat = 15 km/h).

Question 6Modeling

A laboratory technician has a 20% saline solution and a 50% saline solution. How many liters of the 50% solution must be mixed with 10 liters of the 20% solution to obtain a 30% saline solution?

  1. 5 liters Correct answer.
  2. 4 liters
  3. 6 liters
  4. 7.5 liters

Explanation

Let x be the volume of the 50% solution added. The mass balance of salt is: 0.20(10) + 0.50(x) = 0.30(10 + x) => 2 + 0.5x = 3 + 0.3x. Subtract 0.3x and 2 from both sides: 0.2x = 1 => x = 1 / 0.2 = 5 liters. Verification: 10 L of 20% contains 2 L salt, 5 L of 50% contains 2.5 L salt; total salt = 4.5 L in 15 L solution = 30%.

Question 7Modeling

If 8 identical 3D printers manufacture 1,200 components in 5 hours, how many hours will it take 6 of these printers to manufacture 1,800 components?

  1. 10 hours Correct answer.
  2. 8 hours
  3. 12 hours
  4. 7.5 hours

Explanation

Find the production rate per machine per hour: 1,200 / (8 * 5) = 1,200 / 40 = 30 components per printer-hour. With 6 printers, their combined output is 6 * 30 = 180 components per hour. To produce 1,800 components: time = 1,800 / 180 = 10 hours.

Question 8Modeling

A company surveyed its staff about two internal systems, a reporting system R and a planning system P. At least one of the two systems is used by 190 employees, at most one of them is used by 155 employees, the reporting system is used by 110 employees and the planning system by 130 employees. How many employees use neither system?

  1. 15 Correct answer.
  2. 35
  3. 50
  4. 60

Explanation

Start with the inclusion-exclusion principle on the union: |R U P| = |R| + |P| - |R cap P|, so 190 = 110 + 130 - |R cap P| and |R cap P| = 50 employees use both systems. Hence 110 - 50 = 60 use only reporting and 130 - 50 = 80 use only planning. The group using at most one system consists of those who use exactly one plus those who use neither: N + 60 + 80 = 155, so N = 155 - 140 = 15 employees use neither system.

Question 9Modeling

In a group of 100 business students, 50 study English, 40 study German, and 35 study French. Furthermore, 20 study both English and German, 15 study both English and French, 10 study both German and French, and 5 study all three languages. How many students study none of these three languages?

  1. 15 Correct answer.
  2. 10
  3. 20
  4. 25

Explanation

Apply the principle of inclusion and exclusion for 3 sets: |E U G U F| = |E| + |G| + |F| - (|E cap G| + |E cap F| + |G cap F|) + |E cap G cap F| = 50 + 40 + 35 - (20 + 15 + 10) + 5 = 125 - 45 + 5 = 85 students study at least one language. The number of students studying none is 100 - 85 = 15.

Question 10Modeling

What is the formal logical negation of the proposition: 'Every student in the seminar passed the mathematics entrance exam'?

  1. At least one student in the seminar did not pass the mathematics entrance exam. Correct answer.
  2. No student in the seminar passed the mathematics entrance exam.
  3. Every student in the seminar failed the mathematics entrance exam.
  4. At most one student in the seminar passed the mathematics entrance exam.

Explanation

The original statement is universally quantified: 'For all x, P(x)'. The logical negation of a universal statement is an existential statement with the negated predicate: 'There exists at least one x such that not P(x)'. Therefore, the negation is: 'At least one student in the seminar did not pass the mathematics entrance exam.'

Question 11Modeling

Given that the conditional statement 'If the company's operating profit increases by more than 10%, then all employees receive an annual bonus' is TRUE, which of the following statements must also logically be true?

  1. If at least one employee did not receive an annual bonus, then operating profit did not increase by more than 10%. Correct answer.
  2. If operating profit did not increase by more than 10%, no employee received a bonus.
  3. If all employees received an annual bonus, operating profit must have increased by more than 10%.
  4. If at least one employee received an annual bonus, operating profit increased by more than 10%.

Explanation

A conditional statement P -> Q is logically equivalent to its contrapositive ~Q -> ~P. Here, P is 'operating profit increases by > 10%' and Q is 'all employees receive a bonus'. The negation ~Q is 'not all employees receive a bonus' (i.e. 'at least one employee did not receive a bonus'). The contrapositive is: 'If at least one employee did not receive a bonus, then operating profit did not increase by more than 10%.'

Question 12Modeling

An item originally priced at 2,000 CZK was first marked up by 20%, and then its new price was discounted by 25%. What is the final price of the item?

  1. 1,800 CZK Correct answer.
  2. 1,900 CZK
  3. 1,950 CZK
  4. 1,750 CZK

Explanation

Calculate successive percentage changes: After a 20% markup, the price becomes 2,000 * 1.20 = 2,400 CZK. Then, discounting by 25% means paying 75% of the new price: 2,400 * (1 - 0.25) = 2,400 * 0.75 = 1,800 CZK. (Notice that the combined multiplier is 1.20 * 0.75 = 0.90, which is an overall 10% decrease from the original 2,000 CZK).

Question 13Modeling

A retail bookstore sells an economics textbook for 1,500 CZK, which represents a 25% profit markup on the wholesale cost price. What was the bookstore's original wholesale cost price for the textbook?

  1. 1,200 CZK Correct answer.
  2. 1,125 CZK
  3. 1,250 CZK
  4. 1,100 CZK

Explanation

Let C be the cost price. A 25% markup means Selling Price = C * (1 + 0.25) = 1.25 * C. Given that the selling price is 1,500 CZK: 1.25 * C = 1,500 => C = 1,500 / 1.25 = 1,500 * (4/5) = 1,200 CZK.

Question 14Modeling

In a seminar group of 12 students, the average score on a mathematics test was 65 points. The highest-scoring student scored 20% more points than the group average, while the lowest-scoring student scored two-thirds fewer points than the highest-scoring student. In which interval does the average score of the remaining 10 students lie?

  1. (60, 65]
  2. (65, 70] Correct answer.
  3. (70, 75]
  4. (75, 80]

Explanation

The total for all 12 students is 12 * 65 = 780 points. The highest score is 65 * 1.20 = 78 points. 'Two-thirds fewer' means the lowest student keeps one third of the highest score: 78 * (1/3) = 26 points. The two extreme students together scored 78 + 26 = 104 points, so the other 10 students scored 780 - 104 = 676 points in total. Their average is 676 / 10 = 67.6 points, which lies in the interval (65, 70].

Question 15Modeling

A student's final grade in introductory statistics is based on: two midterm tests (weighted 20% each), homework assignments (weighted 15%), and a final comprehensive exam (weighted 45%). If the student scored 80 on both midterms, 90 on homework, and 70 on the final exam, what is the student's overall weighted average score?

  1. 78 points
  2. 80 points
  3. 77 points Correct answer.
  4. 83 points

Explanation

Multiply each score by its weight and add: 0.20(80) + 0.20(80) + 0.15(90) + 0.45(70) = 16 + 16 + 13.5 + 31.5 = 77 points. The weights sum to 0.20 + 0.20 + 0.15 + 0.45 = 1, so no further division is needed. The result is below the simple average because the heaviest component, the final exam, carries the lowest score.

Question 16Modeling

A student prepared for a university entrance examination with a 25-question practice test under a reward scheme agreed with a parent: 60 CZK for each question solved correctly, and 140 CZK paid back for each question solved incorrectly. All 25 questions were answered, and after scoring the student received a net payout of 300 CZK. In which interval does the percentage of questions solved correctly lie?

  1. (80, 85)
  2. (70, 80) Correct answer.
  3. (85, 90)
  4. (90, 100)

Explanation

Let c be the number of correct answers and w the number of wrong ones, with c + w = 25, so w = 25 - c. The payout condition is 60c - 140w = 300, i.e. 60c - 140(25 - c) = 300 => 60c - 3,500 + 140c = 300 => 200c = 3,800 => c = 19 correct answers and w = 6 wrong ones. Check: 19 * 60 - 6 * 140 = 1,140 - 840 = 300 CZK. The share of correct answers is 19 / 25 = 76%, which lies in the interval (70, 80).

Question 17Modeling

An investor deposits 100,000 CZK into a 9-month fixed-term bank account earning simple interest at a rate of 4.0% p.a. If the earned interest is subject to a 15% withholding tax deducted at maturity, what is the net interest received in CZK?

  1. 2,550 CZK Correct answer.
  2. 3,000 CZK
  3. 2,700 CZK
  4. 2,400 CZK

Explanation

Gross interest for 9 months (9/12 = 0.75 year) is: Gross = Principal * rate * time = 100,000 * 0.04 * (9/12) = 4,000 * 0.75 = 3,000 CZK. The 15% withholding tax is 0.15 * 3,000 = 450 CZK. The net interest received is 3,000 - 450 = 2,550 CZK (or 3,000 * 0.85 = 2,550 CZK).

Question 18Modeling

A company purchases production machinery for 400,000 CZK. The asset depreciates at a constant rate of 10% per year based on its carrying value at the beginning of each year. What is the book value of the machinery after 3 years?

  1. 291,600 CZK Correct answer.
  2. 280,000 CZK
  3. 300,000 CZK
  4. 262,440 CZK

Explanation

Each year, the value becomes 1 - 0.10 = 0.90 of the previous year's value. After 3 years, the book value is V_3 = 400,000 * (0.90)^3 = 400,000 * 0.729 = 291,600 CZK.

Question 19Modeling

A borrower repays a debt in 12 monthly installments forming an arithmetic progression. The first installment is 5,000 CZK, and each subsequent monthly installment increases by 300 CZK. What is the total sum repaid over the 12 months?

  1. 79,800 CZK Correct answer.
  2. 78,000 CZK
  3. 81,600 CZK
  4. 76,500 CZK

Explanation

Here a_1 = 5,000, d = 300, and n = 12. The twelfth installment is a_12 = a_1 + 11d = 5,000 + 11(300) = 5,000 + 3,300 = 8,300 CZK. The total sum repaid is the arithmetic series sum: S_12 = (12/2) * (a_1 + a_12) = 6 * (5,000 + 8,300) = 6 * 13,300 = 79,800 CZK.

Question 20Modeling

A bacterial culture initially consisting of 1,000 cells doubles in population every 2 hours. What will the bacterial population be after 10 hours?

  1. 32,000 cells Correct answer.
  2. 16,000 cells
  3. 64,000 cells
  4. 10,000 cells

Explanation

In 10 hours, the number of doubling periods is n = 10 / 2 = 5. The population grows according to a geometric progression: P(10) = P_0 * 2^n = 1,000 * 2^5 = 1,000 * 32 = 32,000 cells.

Question 21Modeling

How many 4-digit banking PIN codes can be created using only odd digits {1, 3, 5, 7, 9} if no digit may be repeated in the PIN?

  1. 120 Correct answer.
  2. 625
  3. 24
  4. 240

Explanation

There are 5 odd digits available. Selecting and arranging 4 distinct digits where order matters is the number of 4-element variations from 5 elements without repetition: V(4, 5) = 5 * 4 * 3 * 2 = 120.

Question 22Modeling

In how many distinct ways can 6 delegates be seated around a circular negotiation table if two specific delegates, A and B, must sit directly next to each other? (Rotations of the table are considered identical).

  1. 48 Correct answer.
  2. 24
  3. 120
  4. 96

Explanation

Treat delegates A and B as a single composite unit. This reduces the problem to arranging 5 units around a circular table. The number of circular permutations of 5 units is (5 - 1)! = 4! = 24. Within the composite unit, delegates A and B can be seated in 2! = 2 ways (A left of B, or B left of A). Therefore, the total number of distinct seating arrangements is 24 * 2 = 48.

Question 23Modeling

A rectangular building plot measures 40 m by 30 m. A paved path of constant width 2 m is laid around the outside of the plot, following its whole boundary. What is the area of the path?

  1. 280 m^2
  2. 296 m^2 Correct answer.
  3. 312 m^2
  4. 1,496 m^2

Explanation

The path plus the plot form a larger rectangle whose sides are each extended by 2 m on both ends: (40 + 2 + 2) by (30 + 2 + 2), that is 44 m by 34 m, with area 1,496 m^2. The plot itself has area 40 * 30 = 1,200 m^2. The path is the difference: 1,496 - 1,200 = 296 m^2.

Question 24Modeling

A contractor calculated that 12 equally productive workers would lay a warehouse floor in 15 working days. After 6 days with the full crew, 3 workers were transferred to another site. How many further working days do the remaining workers need to finish the floor?

  1. 9 days
  2. 12 days Correct answer.
  3. 18 days
  4. 20 days

Explanation

Measure the job in worker-days, since the number of workers and the time needed are inversely proportional. The whole floor costs 12 * 15 = 180 worker-days. The full crew works 6 days and delivers 12 * 6 = 72 worker-days, leaving 180 - 72 = 108 worker-days of work. Only 12 - 3 = 9 workers remain, so they need 108 / 9 = 12 further working days.

Question 25Modeling

A right-angled triangle has a hypotenuse of length c = 13 cm and a perimeter of 30 cm. What is the area of the triangle?

  1. 30 cm^2 Correct answer.
  2. 60 cm^2
  3. 26 cm^2
  4. 39 cm^2

Explanation

Let the legs be a and b. The perimeter is a + b + c = 30 => a + b + 13 = 30 => a + b = 17. By the Pythagorean theorem, a^2 + b^2 = c^2 = 13^2 = 169. Expanding (a + b)^2 gives a^2 + 2ab + b^2 = 17^2 = 289. Subtracting a^2 + b^2 = 169 yields 2ab = 289 - 169 = 120 => ab = 60. The area of the right triangle is Area = (1/2)ab = (1/2)(60) = 30 cm^2. (The legs are 5 cm and 12 cm).

Question 26Modeling

A trapezoidal building plot has parallel bases of lengths 24 m and 36 m, with a perpendicular height of 15 m between them. What is the total land area of the plot?

  1. 450 m^2 Correct answer.
  2. 900 m^2
  3. 400 m^2
  4. 540 m^2

Explanation

The area of a trapezoid is S = [(a + c) / 2] * h. Here bases are a = 36 m, c = 24 m, and height h = 15 m. S = [(36 + 24) / 2] * 15 = (60 / 2) * 15 = 30 * 15 = 450 m^2.

Question 27Modeling

An outdoor running track has the shape of a circular ring (annulus) with an outer radius R = 20 m and an inner radius r = 16 m. What is the exact area of the running surface?

  1. 144*pi m^2 Correct answer.
  2. 72*pi m^2
  3. 16*pi m^2
  4. 256*pi m^2

Explanation

The area of an annulus is the difference between the outer and inner circular areas: S = pi * R^2 - pi * r^2 = pi * (R^2 - r^2) = pi * (20^2 - 16^2) = pi * (400 - 256) = 144*pi m^2.

Question 28Modeling

A cylindrical reservoir has a base radius of r = 2 m and a height of h = 3 m. Water is pumped into the initially empty reservoir at a constant rate of 400 liters per minute. Using the approximation pi approx 3.14, approximately how many minutes will it take to fill the reservoir completely? (1 m^3 = 1,000 liters).

  1. Approximately 94.2 minutes Correct answer.
  2. Approximately 47.1 minutes
  3. Approximately 376.8 minutes
  4. Approximately 75.4 minutes

Explanation

Calculate the reservoir volume: V = pi * r^2 * h = 3.14 * 2^2 * 3 = 3.14 * 4 * 3 = 37.68 m^3. Convert to litres: 37.68 * 1,000 = 37,680 litres. At 400 litres per minute the filling time is t = 37,680 / 400 = 94.2 minutes.

Question 29Modeling

A rectangular shipping container has edge lengths in the ratio 2 : 3 : 5. If the total surface area of the container is 248 m^2, what is the volume of the container?

  1. 240 m^3 Correct answer.
  2. 120 m^3
  3. 480 m^3
  4. 360 m^3

Explanation

Let the dimensions be 2k, 3k, and 5k meters. Total surface area is S = 2(ab + bc + ac) = 2[(2k)(3k) + (3k)(5k) + (2k)(5k)] = 2[6k^2 + 15k^2 + 10k^2] = 2[31k^2] = 62k^2. Set equal to 248: 62k^2 = 248 => k^2 = 4 => k = 2. The edge dimensions are 2(2) = 4 m, 3(2) = 6 m, and 5(2) = 10 m. The volume is V = a * b * c = 4 * 6 * 10 = 240 m^3.

Question 30Modeling

A solid metal cone of base radius r and height h is melted down and recast into a solid cylinder of the exact same base radius r. What is the height of the new cylinder in terms of h?

  1. h / 3 Correct answer.
  2. h / 2
  3. 3h
  4. h

Explanation

Volume of the cone is V_cone = (1/3) * pi * r^2 * h. Volume of the cylinder with base radius r and height h_cyl is V_cyl = pi * r^2 * h_cyl. Since the volume of metal is conserved: pi * r^2 * h_cyl = (1/3) * pi * r^2 * h. Dividing both sides by pi * r^2 yields h_cyl = h / 3.

Question 31Algebra Functions

For n != 0 and n != +-1, simplify the algebraic expression: [(n + 3)/(n + 1) - (n - 3)/(n - 1)] * (n^2 - 1)/(4n).

  1. -1
  2. n
  3. 1 Correct answer.
  4. 4

Explanation

Subtract the two fractions inside the brackets using the common denominator (n + 1)(n - 1) = n^2 - 1: [(n + 3)(n - 1) - (n - 3)(n + 1)] / (n^2 - 1) = [(n^2 + 2n - 3) - (n^2 - 2n - 3)] / (n^2 - 1) = 4n / (n^2 - 1). Multiplying by (n^2 - 1)/(4n) gives [4n / (n^2 - 1)] * [(n^2 - 1) / (4n)] = 1.

Question 32Algebra Functions

For real numbers x and y such that x != y and x != -y, simplify the expression: (x^3 - y^3)/(x - y) - (x^3 + y^3)/(x + y).

  1. 0
  2. 2xy Correct answer.
  3. 2(x^2 + y^2)
  4. -2xy

Explanation

Factor both numerators using sum and difference of cubes: x^3 - y^3 = (x - y)(x^2 + xy + y^2) and x^3 + y^3 = (x + y)(x^2 - xy + y^2). Dividing by the denominators yields (x^2 + xy + y^2) - (x^2 - xy + y^2) = x^2 + xy + y^2 - x^2 + xy - y^2 = 2xy.

Question 33Algebra Functions

Simplify the radical expression for a > 0: (a^(1/2) * a^(2/3)) / a^(1/6).

  1. a^(1/2)
  2. a Correct answer.
  3. a^2
  4. 1

Explanation

Apply the laws of exponents: a^(1/2) * a^(2/3) = a^(1/2 + 2/3) = a^(3/6 + 4/6) = a^(7/6). Dividing by a^(1/6) gives a^(7/6 - 1/6) = a^(6/6) = a^1 = a.

Question 34Algebra Functions

Solve the linear equation in the set of real numbers R: (4x - 1)/3 - (x + 2)/2 = (x + 4)/6.

  1. x = 0
  2. x = 1
  3. x = 2
  4. x = 3 Correct answer.

Explanation

Multiply every term by the common denominator 6: 2(4x - 1) - 3(x + 2) = x + 4. Expanding gives 8x - 2 - 3x - 6 = x + 4, which simplifies to 5x - 8 = x + 4. Transposing x to the left and -8 to the right yields 4x = 12, so x = 3. Substituting back: (12 - 1)/3 - (3 + 2)/2 = 11/3 - 5/2 = 7/6, and (3 + 4)/6 = 7/6.

Question 35Algebra Functions

What is the sum of all real roots of the quadratic equation 2x^2 - 7x + 3 = 0?

  1. 3
  2. 7/2 Correct answer.
  3. 3/2
  4. -7/2

Explanation

By Vieta's formulas, for any quadratic equation ax^2 + bx + c = 0 with real roots, the sum of the roots is x_1 + x_2 = -b/a = -(-7)/2 = 7/2. Alternatively, solving using the quadratic formula gives discriminant D = 49 - 24 = 25, yielding roots x = (7 +- 5)/4, so x_1 = 3 and x_2 = 1/2; 3 + 1/2 = 7/2.

Question 36Algebra Functions

Find all real solutions to the rational equation: (x + 2)/(x - 1) + 1/(x + 1) = 6/(x^2 - 1).

  1. x = -5 and x = 1
  2. x = -5 only Correct answer.
  3. x = 1 only
  4. No real solution

Explanation

The domain requires x != 1 and x != -1 because x^2 - 1 = (x - 1)(x + 1). Multiplying both sides by (x - 1)(x + 1) gives (x + 2)(x + 1) + (x - 1) = 6. Expanding yields x^2 + 3x + 2 + x - 1 = 6, which simplifies to x^2 + 4x - 5 = 0. Factoring gives (x + 5)(x - 1) = 0, so candidate solutions are x = -5 and x = 1. Since x = 1 makes the denominators zero, it is an extraneous root. Thus, x = -5 is the only valid solution.

Question 37Algebra Functions

Solve the linear inequality in the set of real numbers R: (3x - 5)/2 - (x - 1)/3 <= 1.

  1. (-infinity, 19/7] Correct answer.
  2. [19/7, infinity)
  3. (-infinity, 17/7]
  4. (-infinity, 3]

Explanation

Multiply the inequality by 6 (a positive number, so the inequality sign does not change): 3(3x - 5) - 2(x - 1) <= 6. Expanding gives 9x - 15 - 2x + 2 <= 6, which simplifies to 7x - 13 <= 6. Adding 13 to both sides yields 7x <= 19, so x <= 19/7. The solution set is (-infinity, 19/7].

Question 38Algebra Functions

Find the set of all real numbers x satisfying the quadratic inequality: x^2 - 2x - 15 <= 0.

  1. (-infinity, -3] U [5, infinity)
  2. [-3, 5] Correct answer.
  3. (-5, 3)
  4. [-5, 3]

Explanation

Factor the quadratic expression: x^2 - 2x - 15 = (x - 5)(x + 3). The roots are x = -3 and x = 5. The graph of y = x^2 - 2x - 15 is an upward-opening parabola, so it lies on or below the x-axis exactly between its roots, endpoints included. Therefore, the solution set is the closed interval [-3, 5].

Question 39Algebra Functions

Find the set of all real numbers x in R satisfying the rational inequality: (x + 5)/(3x - 6) < 1.

  1. (-infinity, 2] U [11/2, infinity)
  2. (2, 11/2)
  3. (-infinity, 2) U (11/2, infinity) Correct answer.
  4. (11/2, infinity)

Explanation

Subtract 1 from both sides instead of multiplying by the denominator, whose sign is unknown: (x + 5)/(3x - 6) - 1 < 0. Over the common denominator this is [(x + 5) - (3x - 6)] / (3x - 6) = (-2x + 11) / (3x - 6) < 0. The numerator vanishes at x = 11/2 and the denominator at x = 2. Sign analysis: for x < 2 the numerator is positive and the denominator negative, so the quotient is negative; for 2 < x < 11/2 both are positive, so the quotient is positive; for x > 11/2 the numerator is negative and the denominator positive, so the quotient is negative. The solution set is therefore (-infinity, 2) U (11/2, infinity).

Question 40Algebra Functions

Solve the absolute value equation in R: |2x - 5| = 3x - 2.

  1. x = 7/5 and x = -3
  2. x = 7/5 Correct answer.
  3. x = -3
  4. No real solution

Explanation

Since the absolute value is non-negative, we must have 3x - 2 >= 0, so x >= 2/3. For 2x - 5 >= 0 (x >= 5/2): 2x - 5 = 3x - 2 => x = -3, which contradicts x >= 5/2 and x >= 2/3 (extraneous). For 2x - 5 < 0 (x < 5/2): -(2x - 5) = 3x - 2 => -2x + 5 = 3x - 2 => 5x = 7 => x = 7/5 = 1.4. Since 2/3 <= 1.4 < 5/2, x = 7/5 is the unique real solution.

Question 41Algebra Functions

Find all real numbers x satisfying the absolute value inequality: |x - 2| + |x + 1| <= 5.

  1. [-2, 3] Correct answer.
  2. [-3, 2]
  3. [-1, 2]
  4. (-infinity, -2] U [3, infinity)

Explanation

Partition the real line into three intervals by zero points x = -1 and x = 2: 1) For x < -1: -(x - 2) - (x + 1) <= 5 => -2x + 1 <= 5 => -2x <= 4 => x >= -2, yielding [-2, -1). 2) For -1 <= x <= 2: -(x - 2) + (x + 1) <= 5 => 3 <= 5, which is true for all x in [-1, 2]. 3) For x > 2: (x - 2) + (x + 1) <= 5 => 2x - 1 <= 5 => 2x <= 6 => x <= 3, yielding (2, 3]. Combining the three intervals gives [-2, 3].

Question 42Algebra Functions

Solve the system of linear equations for (x, y) in R^2: 3x - 2y = 7 and 5x + 3y = -1.

  1. (1, -2) Correct answer.
  2. (-1, 2)
  3. (2, -1)
  4. (-2, 1)

Explanation

Eliminate y by multiplying the first equation by 3 and the second equation by 2: 9x - 6y = 21 and 10x + 6y = -2. Adding both equations gives 19x = 19, so x = 1. Substitute x = 1 into the first equation: 3(1) - 2y = 7 => -2y = 4 => y = -2. Thus, the solution is the ordered pair (1, -2).

Question 43Algebra Functions

Find the number of real solutions (x, y) in R^2 of the system: x - y = 2 and x^2 + y^2 = 10.

  1. 0 solutions
  2. 1 solution
  3. 2 solutions Correct answer.
  4. 4 solutions

Explanation

Express y from the first equation: y = x - 2. Substitute into the second equation: x^2 + (x - 2)^2 = 10 => x^2 + x^2 - 4x + 4 = 10 => 2x^2 - 4x - 6 = 0. Dividing by 2 gives x^2 - 2x - 3 = 0, which factors as (x - 3)(x + 1) = 0. This gives two distinct values: x_1 = 3 (with y_1 = 1) and x_2 = -1 (with y_2 = -3). Thus, there are exactly 2 real solutions: (3, 1) and (-1, -3).

Question 44Algebra Functions

Find the complete solution set of the system of equations in R^2: xy + 4x = 0 and x^2 + y^2 = 25.

  1. {(0, 5), (0, -5)}
  2. {(3, -4), (-3, -4)}
  3. {(0, 5), (0, -5), (3, -4), (-3, -4)} Correct answer.
  4. {(0, 5), (3, -4)}

Explanation

Factor the first equation: x(y + 4) = 0, so x = 0 or y = -4, and both branches must be examined. Branch 1, x = 0: the second equation gives y^2 = 25, so y = +-5, producing (0, 5) and (0, -5). Branch 2, y = -4: the second equation gives x^2 + 16 = 25, so x^2 = 9 and x = +-3, producing (3, -4) and (-3, -4). The full solution set therefore contains all four ordered pairs.

Question 45Algebra Functions

For which values of the real parameter q does the equation qx^2 + qx + 3 = 0 have two distinct real roots?

  1. q in (-infinity, 0)
  2. q in (12, infinity)
  3. q in (-infinity, 0) U (3, 12)
  4. q in (-infinity, 0) U (12, infinity) Correct answer.

Explanation

Two conditions must hold. First, the equation must actually be quadratic, so the leading coefficient must satisfy q != 0. Second, the discriminant must be strictly positive: D = q^2 - 4(q)(3) = q^2 - 12q > 0, i.e. q(q - 12) > 0. A product of two factors is positive when both are negative (q < 0) or both are positive (q > 12). Both branches already exclude q = 0, so the answer is q in (-infinity, 0) U (12, infinity).

Question 46Algebra Functions

Solve the irrational equation in R: sqrt(2x + 7) = x + 2.

  1. x = 1 Correct answer.
  2. x = -3 and x = 1
  3. x = -3
  4. No real solution

Explanation

The right-hand side must be non-negative: x + 2 >= 0 => x >= -2. Squaring both sides: 2x + 7 = (x + 2)^2 = x^2 + 4x + 4. Rearranging terms gives x^2 + 2x - 3 = 0, which factors as (x + 3)(x - 1) = 0. Candidate roots are x = -3 and x = 1. Testing x = -3: sqrt(2(-3) + 7) = sqrt(1) = 1, but x + 2 = -1 != 1 (extraneous). Testing x = 1: sqrt(2(1) + 7) = sqrt(9) = 3 and 1 + 2 = 3. Thus, x = 1 is the unique solution.

Question 47Algebra Functions

Solve the irrational equation in R: sqrt(x + 5) - sqrt(x - 3) = 2.

  1. x = 4 Correct answer.
  2. x = 3
  3. x = 5
  4. x = 7

Explanation

Domain requires x >= 3. Isolate one radical: sqrt(x + 5) = 2 + sqrt(x - 3). Square both sides: x + 5 = 4 + 4*sqrt(x - 3) + (x - 3) => x + 5 = x + 1 + 4*sqrt(x - 3). Subtract x + 1 from both sides: 4 = 4*sqrt(x - 3) => 1 = sqrt(x - 3). Squaring again gives 1 = x - 3 => x = 4. Verification: sqrt(4 + 5) - sqrt(4 - 3) = sqrt(9) - sqrt(1) = 3 - 1 = 2.

Question 48Algebra Functions

For a != b and a, b != 0, simplify the expression: (a/b - b/a) / (1/b + 1/a).

  1. a + b
  2. a - b Correct answer.
  3. (a - b)/(ab)
  4. 1/(a - b)

Explanation

Express numerator and denominator with common denominator ab: a/b - b/a = (a^2 - b^2)/(ab) = [(a - b)(a + b)]/(ab). Denominator: 1/b + 1/a = (a + b)/(ab). Dividing the two fractions gives {[(a - b)(a + b)]/(ab)} / {[(a + b)/(ab)]} = a - b.

Question 49Algebra Functions

For x != 2 and x != -2, simplify the algebraic expression: [(x^3 - 8)/(x^2 + 2x + 4)] * [(x + 2)/(x^2 - 4)].

  1. 1 Correct answer.
  2. x - 2
  3. (x + 2)/(x - 2)
  4. -1

Explanation

Factor the difference of cubes: x^3 - 8 = (x - 2)(x^2 + 2x + 4). The first factor simplifies to (x - 2). Factor the difference of squares: x^2 - 4 = (x - 2)(x + 2). The second factor simplifies to (x + 2)/[(x - 2)(x + 2)] = 1/(x - 2). Multiplying them together yields (x - 2) * [1/(x - 2)] = 1.

Question 50Algebra Functions

For which values of the real parameter k in R does the quadratic equation x^2 - (k - 2)x + (k - 5) = 0 have two real roots of opposite signs?

  1. k in (-infinity, 5) Correct answer.
  2. k in (5, infinity)
  3. k in (2, 5)
  4. k in (-infinity, 2)

Explanation

By Vieta's formulas, the product of the roots is x_1 * x_2 = c/a = k - 5. For two roots to have opposite signs (one positive, one negative), their product must be strictly negative: k - 5 < 0 => k < 5. When the product is negative, the discriminant D = b^2 - 4ac = (k - 2)^2 - 4(k - 5) = k^2 - 8k + 24 = (k - 4)^2 + 8 is strictly positive for all real k. Thus, the condition is k in (-infinity, 5).