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Korean CSAT Mathematics study materials
Korean CSAT Mathematics study resources

Korean CSAT Mathematics practice questions

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Question 1Functions, algebra and sequences

If f(x) = 2x - 5, what is f(7)?

  1. 9 Correct answer.
  2. 11
  3. 12
  4. 14

Explanation

Substitute x = 7 into the function: f(7) = 2(7) - 5 = 14 - 5 = 9. A direct substitution question checks whether you keep the order of operations simple and exact.

Question 2Functions, algebra and sequences

What are the real zeros of x^2 - 5x + 6?

  1. 1 and 6
  2. 2 and 3 Correct answer.
  3. -2 and -3
  4. -1 and -6

Explanation

Factor the quadratic: x^2 - 5x + 6 = (x - 2)(x - 3). The zeros occur when each factor is 0, so x = 2 and x = 3.

Question 3Functions, algebra and sequences

What is the value of log_2 32?

  1. 4
  2. 5 Correct answer.
  3. 16
  4. 64

Explanation

log_2 32 asks for the exponent to which 2 must be raised to get 32. Since 2^5 = 32, log_2 32 = 5.

Question 4Functions, algebra and sequences

For 0 <= theta <= pi, if sin theta = 1/2, which value can theta be?

  1. pi/6 Correct answer.
  2. pi/4
  3. pi/3
  4. pi/2

Explanation

On the unit circle, sin(pi/6) = 1/2. Although sin(5pi/6) is also 1/2, it is not listed, so pi/6 is the matching option.

Question 5Functions, algebra and sequences

An arithmetic sequence has first term 3 and common difference 4. What is the 6th term?

  1. 19
  2. 20
  3. 23 Correct answer.
  4. 27

Explanation

The nth term is a_n = a_1 + (n - 1)d. Thus a_6 = 3 + 5(4) = 23.

Question 6Functions, algebra and sequences

A geometric sequence has first term 2 and common ratio 3. What is the 4th term?

  1. 18
  2. 24
  3. 54 Correct answer.
  4. 81

Explanation

For a geometric sequence, a_n = a_1 r^(n - 1). Therefore a_4 = 2 * 3^3 = 54.

Question 7Functions, algebra and sequences

Solve for x: 3x + 4 = 19.

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

Explanation

Subtract 4 from both sides to get 3x = 15. Dividing by 3 gives x = 5.

Question 8Functions, algebra and sequences

Which inequality is equivalent to 2x - 1 > 7?

  1. x > 3
  2. x > 4 Correct answer.
  3. x < 3
  4. x < 4

Explanation

Add 1 to both sides to get 2x > 8, then divide by 2 to get x > 4. The inequality direction stays the same because we divided by a positive number.

Question 9Functions, algebra and sequences

What is the slope of the line passing through (2, 1) and (6, 9)?

  1. 1
  2. 2 Correct answer.
  3. 3
  4. 4

Explanation

Slope is change in y divided by change in x: (9 - 1) / (6 - 2) = 8 / 4 = 2.

Question 10Functions, algebra and sequences

Solve 2^(x + 1) = 16.

  1. 2
  2. 3 Correct answer.
  3. 4
  4. 5

Explanation

Since 16 = 2^4, we have x + 1 = 4. Therefore x = 3.

Question 11Functions, algebra and sequences

How many real solutions does |x - 3| = 5 have?

  1. 0
  2. 1
  3. 2 Correct answer.
  4. 5

Explanation

The equation |x - 3| = 5 means x - 3 = 5 or x - 3 = -5. These give x = 8 and x = -2, so there are two real solutions.

Question 12Functions, algebra and sequences

What is the sum 1 + 2 + 3 + 4 + 5?

  1. 10
  2. 12
  3. 15 Correct answer.
  4. 20

Explanation

Adding the terms gives 1 + 2 + 3 + 4 + 5 = 15. You can also use n(n + 1)/2 with n = 5 to get 5 * 6 / 2 = 15.

Question 13Functions, algebra and sequences

If f(x) = 2x + 1, what is f^{-1}(7)?

  1. 2
  2. 3 Correct answer.
  3. 4
  4. 5

Explanation

To find f^{-1}(7), solve 2x + 1 = 7. This gives 2x = 6 and x = 3.

Question 14Functions, algebra and sequences

On 0 <= x < 2pi, which value solves cos x = 0?

  1. 0
  2. pi/3
  3. pi/2 Correct answer.
  4. pi

Explanation

Cosine is the x-coordinate on the unit circle. It equals 0 at pi/2 and 3pi/2; among the options, pi/2 is listed.

Question 15Functions, algebra and sequences

A taxi fare is 4000 won plus 800 won per kilometer. What is the fare for 5 kilometers?

  1. 8000 won Correct answer.
  2. 8400 won
  3. 8800 won
  4. 9000 won

Explanation

The fare is 4000 + 800(5) = 4000 + 4000 = 8000 won. This is a linear model with fixed cost plus variable cost.

Question 16Functions, algebra and sequences

If f(x) = x^2 - 4x + 1, what is the minimum value of f(x)?

  1. -4
  2. -3 Correct answer.
  3. 0
  4. 1

Explanation

Complete the square: x^2 - 4x + 1 = (x - 2)^2 - 3. Since (x - 2)^2 >= 0, the minimum value is -3.

Question 17Functions, algebra and sequences

For x > 0, if log_2 x + log_2 4 = 5, what is x?

  1. 4
  2. 6
  3. 8 Correct answer.
  4. 16

Explanation

Since log_2 4 = 2, the equation becomes log_2 x + 2 = 5. Thus log_2 x = 3 and x = 2^3 = 8.

Question 18Functions, algebra and sequences

If f(x) = (x - 1)/(x + 2), what is the excluded value from the domain?

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

Explanation

A rational function is undefined where its denominator is 0. Here x + 2 = 0 gives x = -2.

Question 19Functions, algebra and sequences

The function f(x) = a(x - 1)^2 + 3 passes through (3, 11). What is a?

  1. 1
  2. 2 Correct answer.
  3. 3
  4. 4

Explanation

Substitute x = 3 and f(x) = 11: 11 = a(3 - 1)^2 + 3 = 4a + 3. Thus 4a = 8 and a = 2.

Question 20Functions, algebra and sequences

If sin theta = 3/5 and theta is acute, what is cos theta?

  1. 2/5
  2. 3/4
  3. 4/5 Correct answer.
  4. 5/4

Explanation

For an acute angle, cosine is positive. Using sin^2 theta + cos^2 theta = 1 gives cos^2 theta = 1 - 9/25 = 16/25, so cos theta = 4/5.

Question 21Functions, algebra and sequences

For 0 <= x < 2pi, how many solutions does sin x = 1/2 have?

  1. 1
  2. 2 Correct answer.
  3. 3
  4. 4

Explanation

On one full cycle, sin x = 1/2 in Quadrants I and II. The solutions are x = pi/6 and x = 5pi/6, so there are 2 solutions.

Question 22Functions, algebra and sequences

If f(x) = 2^x, what is f(x + 2) / f(x)?

  1. 2
  2. 4 Correct answer.
  3. 2^x
  4. 2x

Explanation

f(x + 2) = 2^(x + 2) = 2^x * 2^2. Dividing by f(x) = 2^x leaves 4.

Question 23Functions, algebra and sequences

For f(x) = x^2 and g(x) = x - 1, which expression equals (f o g)(x)?

  1. x^2 - 1
  2. (x - 1)^2 Correct answer.
  3. x^2 - x
  4. x - 2

Explanation

(f o g)(x) means f(g(x)). Since g(x) = x - 1 and f(t) = t^2, the result is (x - 1)^2.

Question 24Functions, algebra and sequences

The line y = mx + 2 passes through (3, 8). What is m?

  1. 1
  2. 2 Correct answer.
  3. 3
  4. 4

Explanation

Substitute the point into the equation: 8 = 3m + 2. Then 3m = 6 and m = 2.

Question 25Functions, algebra and sequences

If x^2 - 6x + k has exactly one real zero, what is k?

  1. 6
  2. 8
  3. 9 Correct answer.
  4. 12

Explanation

A quadratic has exactly one real zero when its discriminant is 0. Here b^2 - 4ac = 36 - 4k = 0, so k = 9.

Question 26Functions, algebra and sequences

An arithmetic sequence has a_3 = 11 and a_8 = 31. What is the common difference?

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

Explanation

From a_3 to a_8 there are 5 equal steps. The value increases by 31 - 11 = 20, so the common difference is 20 / 5 = 4.

Question 27Functions, algebra and sequences

A geometric sequence has a_2 = 6 and a_5 = 162, with positive common ratio. What is the common ratio?

  1. 2
  2. 3 Correct answer.
  3. 4
  4. 9

Explanation

From a_2 to a_5 there are 3 ratio steps, so 6r^3 = 162. Thus r^3 = 27 and r = 3.

Question 28Functions, algebra and sequences

What is the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, ...?

  1. 135
  2. 145
  3. 155 Correct answer.
  4. 165

Explanation

The 10th term is 2 + 9(3) = 29. The sum is 10(2 + 29)/2 = 155.

Question 29Functions, algebra and sequences

The infinite geometric series 6 + 3 + 3/2 + ... has what sum?

  1. 9
  2. 10
  3. 12 Correct answer.
  4. 15

Explanation

The first term is 6 and the common ratio is 1/2. Since |r| < 1, the sum is a/(1 - r) = 6/(1 - 1/2) = 12.

Question 30Functions, algebra and sequences

A rectangle has perimeter 30 and width 6. What is its area?

  1. 36
  2. 48
  3. 54 Correct answer.
  4. 60

Explanation

Perimeter 30 means 2(L + 6) = 30, so L + 6 = 15 and L = 9. The area is 9 * 6 = 54.

Question 31Functions, algebra and sequences

A population is modeled by P(t) = 1000(1.05)^t, where t is years. What is P(2)?

  1. 1050
  2. 1100
  3. 1102.5 Correct answer.
  4. 1200

Explanation

Substitute t = 2: P(2) = 1000(1.05)^2 = 1000(1.1025) = 1102.5. This is compound growth, not simple addition.

Question 32Functions, algebra and sequences

The cost C of producing x items is C(x) = 500 + 20x. If the revenue is R(x) = 35x, what is the minimum whole number of items that must be sold so revenue is at least cost?

  1. 25
  2. 30
  3. 34 Correct answer.
  4. 40

Explanation

Break even means revenue equals cost: 35x = 500 + 20x. Then 15x = 500, so x = 33.33..., and at least 34 whole items are needed.

Question 33Functions, algebra and sequences

A sequence is defined by a_1 = 1 and a_{n+1} = a_n + 2n. What is a_4?

  1. 7
  2. 9
  3. 11
  4. 13 Correct answer.

Explanation

Compute terms: a_2 = 1 + 2(1) = 3, a_3 = 3 + 2(2) = 7, and a_4 = 7 + 2(3) = 13. The recurrence adds 2n at the step from a_n to a_{n+1}.

Question 34Functions, algebra and sequences

For x > 0, which expression is equal to log_3(9x)?

  1. 2 + log_3 x Correct answer.
  2. 9 + log_3 x
  3. 2log_3 x
  4. log_3 x - 2

Explanation

Use log(ab) = log a + log b. Since log_3 9 = 2, log_3(9x) = 2 + log_3 x.

Question 35Functions, algebra and sequences

If 0 < a < 1, which statement best describes y = a^x as x increases?

  1. It increases without bound
  2. It decreases and stays positive Correct answer.
  3. It is constant
  4. It becomes negative

Explanation

For an exponential function with base between 0 and 1, each increase of x multiplies the value by a fraction. The function decreases but remains positive for all real x.

Question 36Functions, algebra and sequences

The graph of y = (x - 2)^2 + 1 is the graph of y = x^2 shifted how?

  1. Right 2 and up 1 Correct answer.
  2. Left 2 and up 1
  3. Right 2 and down 1
  4. Left 2 and down 1

Explanation

Replacing x by x - 2 shifts the graph right 2 units, and adding 1 shifts it up 1 unit.

Question 37Functions, algebra and sequences

If tan theta = 3/4 and theta is acute, what is sin theta?

  1. 3/5 Correct answer.
  2. 4/5
  3. 3/4
  4. 5/3

Explanation

For an acute angle with tan theta = opposite/adjacent = 3/4, use a 3-4-5 right triangle. Thus sin theta = opposite/hypotenuse = 3/5.

Question 38Functions, algebra and sequences

A student scores 70, 80, and 90 on three tests. What score is needed on a fourth test to make the average 85?

  1. 90
  2. 95
  3. 100 Correct answer.
  4. 105

Explanation

An average of 85 over four tests requires a total of 4 * 85 = 340. The first three scores total 240, so the fourth score must be 100.

Question 39Functions, algebra and sequences

If a line has slope -2 and passes through (1, 5), which equation represents it?

  1. y = -2x + 7 Correct answer.
  2. y = -2x + 5
  3. y = 2x + 3
  4. y = 2x + 7

Explanation

Use y = mx + b with m = -2. Substituting (1, 5) gives 5 = -2(1) + b, so b = 7 and y = -2x + 7.

Question 40Functions, algebra and sequences

Let f(x) = x^2 - 2x and g(x) = x + 1. What is the sum of all real solutions to f(g(x)) = 3?

  1. -2
  2. 0 Correct answer.
  3. 2
  4. 4

Explanation

Compute f(g(x)) = (x + 1)^2 - 2(x + 1) = x^2 - 1. Setting x^2 - 1 = 3 gives x^2 = 4, so x = -2 or 2; their sum is 0.

Question 41Functions, algebra and sequences

For x > 0, if (log_2 x)^2 - 5log_2 x + 6 = 0, what is the product of all possible x-values?

  1. 16
  2. 32 Correct answer.
  3. 64
  4. 128

Explanation

Let t = log_2 x. Then t^2 - 5t + 6 = 0, so t = 2 or t = 3. Thus x = 4 or 8, and the product is 32.

Question 42Functions, algebra and sequences

A quadratic function f(x) has vertex (2, -1) and passes through (0, 7). What is f(4)?

  1. 3
  2. 5
  3. 7 Correct answer.
  4. 9

Explanation

Write f(x) = a(x - 2)^2 - 1. Since f(0) = 7, 4a - 1 = 7, so a = 2. Then f(4) = 2(2)^2 - 1 = 7.

Question 43Functions, algebra and sequences

An arithmetic sequence has first term a and common difference d. If a_4 = 10 and a_9 = 25, what is a?

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

Explanation

The difference from a_4 to a_9 spans 5 steps, so 5d = 15 and d = 3. Since a_4 = a + 3d = 10, a + 9 = 10 and a = 1.

Question 44Functions, algebra and sequences

For a geometric sequence with positive first term, a_2 = 6 and a_4 = 24. If the common ratio is positive, what is a_1?

  1. 2
  2. 3 Correct answer.
  3. 4
  4. 6

Explanation

From a_2 to a_4 there are two ratio steps, so 6r^2 = 24 and r^2 = 4. With positive ratio, r = 2. Then a_1 = a_2 / r = 3.

Question 45Functions, algebra and sequences

A function is modeled by f(x) = ax + b. If f(2) = 7 and f(5) = 16, what is f(10)?

  1. 28
  2. 30
  3. 31 Correct answer.
  4. 32

Explanation

The slope is (16 - 7)/(5 - 2) = 3, so f(x) = 3x + b. Using f(2) = 7 gives b = 1, so f(10) = 31.

Question 46Functions, algebra and sequences

The sum of the first n positive integers is 78. What is n?

  1. 10
  2. 11
  3. 12 Correct answer.
  4. 13

Explanation

Use n(n + 1)/2 = 78, so n(n + 1) = 156. Since 12 * 13 = 156, n = 12.

Question 47Limits, derivatives and integration

What is lim_{x -> 2} (x^2 + 3x)?

  1. 8
  2. 10 Correct answer.
  3. 12
  4. 14

Explanation

Polynomials are continuous, so substitute x = 2 directly. The value is 2^2 + 3(2) = 4 + 6 = 10.

Question 48Limits, derivatives and integration

If f(x) = x^3, what is f'(2)?

  1. 6
  2. 8
  3. 12 Correct answer.
  4. 16

Explanation

Differentiate f(x) = x^3 to get f'(x) = 3x^2. Then f'(2) = 3(2^2) = 12.

Question 49Limits, derivatives and integration

What is the value of integral from 0 to 2 of 3x^2 dx?

  1. 4
  2. 6
  3. 8 Correct answer.
  4. 12

Explanation

An antiderivative of 3x^2 is x^3. Evaluating from 0 to 2 gives 2^3 - 0^3 = 8.

Question 50Limits, derivatives and integration

If f(x) = x^2 + 1, what is the instantaneous rate of change at x = 3?

  1. 3
  2. 5
  3. 6 Correct answer.
  4. 10

Explanation

The instantaneous rate of change is the derivative. Since f'(x) = 2x, f'(3) = 6.