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Liceum Batorego Test 2 October 13, 2017

Imię i nazwisko:

Klasa:

Grupa 1 i 2 Wynik:

Question 1 (1 pt) The graph of f (x) =

x has been first translated by a vector [−1, −1] and then reflected in the y-axis to form a graph of g(x). The equation of g(x) is given by:

A. g(x) =

−x − 1 − 1 B. g(x) =

−x + 1 − 1 C. g(x) = −√

x + 1 − 1 D. g(x) = −√

x + 1 + 1

Question 2 (1 pt)

Consider a triangle ABC with |AB| = 10, ∠ABC = 42 and angleBAC = 93. The radius of the circle circumscribing this triangle is equal to:

A. 5

2

2 B. 5 C. 5

2 D. 10

2

Question 3 (1 pt)

Consider a parallelogram ABCD with A(3, 1), B(5, 5) and D(2, 2). The co- ordinates of point C are:

A. (4, 4) B. (4, 5) C. (4, 6) D. (4, 7)

Question 4 (1 pt)

Point M with coordinates (√

2, 1) is the mid-point of the line segment AB, where A(−3√

2, −4). Point B has coordinates:

A. (−5√

2, −32) B. (−√

2, −32) C. (

2, 2) D. (5 2, 6)

Question 5 (1 pt)

A circle has been inscribed in a square ABCD with vertices A(−2, 4), B(6, 4) and C(6, 12). The centre of the circle has coordinates:

A. (2, 8) B. (4, 8) C. (2, 4) D. (4, 4)

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Liceum Batorego Test 1, page 2 of 4 October 13, 2017

Question 1 (1 pt) The graph of f (x) =

x has been first translated by a vector [−1, −1] and then reflected in the x-axis to form a graph of g(x). The equation of g(x) is given by:

A. g(x) =

−x − 1 − 1 B. g(x) =

−x + 1 − 1 C. g(x) = −√

x + 1 − 1 D. g(x) = −√

x + 1 + 1

Question 2 (1 pt)

Consider a triangle ABC with |AB| = 5, ∠ABC = 48 and ∠BAC = 87. The radius of the circle circumscribing this triangle is equal to:

A. 5

2

2 B. 5 C. 5

2 D. 10

2

Question 3 (1 pt)

Consider a parallelogram ABCD with A(3, 0), B(5, 4) and D(2, 1). The co- ordinates of point C are:

A. (4, 4) B. (4, 5) C. (4, 6) D. (4, 7)

Question 4 (1 pt)

Point M with coordinates (−√

2, −1) is the mid-point of the line segment AB, where A(−3√

2, −4). Point B has coordinates:

A. (−5√

2, −32) B. (−√

2, −32) C. (

2, 2) D. (5 2, 6)

Question 5 (1 pt)

A circle has been inscribed in a square ABCD with vertices A(−2, 4), B(6, 4) and C(6, 12). The centre of the circle has coordinates:

A. (2, 8) B. (4, 8) C. (2, 4) D. (4, 4)

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Liceum Batorego Test 1, page 3 of 4 October 13, 2017

Question 10 (4 pts) Sketch the function

f (x) =

|x2 − 1| if x < −1

1

x + 1 if − 1 ¬ x < 2

|x − 3| + 12 if x ­ 2

Consider the equation:

f (x) = | log4(m − 3)|

Find the value(s) of m for which this equation has exactly 2 solutions.

Two solutions for | log4(m − 3)| = 12 log4(m − 3) = 12 or log4(m − 3) = −12. m − 3 = 2 or m − 3 = 12.

m = 5 or m = 3.5.

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Liceum Batorego Test 1, page 4 of 4 October 13, 2017

Question 10 (4 pts) Sketch the function

f (x) =

|x3 + 1| if x < −1

1

x + 1 if − 1 ¬ x < 2

|x − 4| − 12 if x ­ 2

Consider the equation:

f (x) = log8(m − 1)

Find the value(s) of m for which this equation has exactly 2 solutions.

Two solutions for log8(m − 1) = −12 m − 1 = 212 =

2 4

m = 1 +

2 4

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