Free Printable Taste of the Exams

1.

When the curves y = log₁₀ x and y = x^–1 are drawn in the x-y plane, how many times do they intersect for values x ≥ 1?

a)

Never

b)

Once

c)

Twice

d)

More than twice.

2.

Let g(x) = max (5 - x, x + 2). The smallest possible value of g(x) is:

a)

4

b)

4.5

c)

1.5

d)

None of the above

3.

The function f(x) = |x – 2| + |2.5 – x| + |3.6 – x|, where x is a real number, attains a minimum at:

a)

x = 2.3

b)

x = 2.5

c)

x = 2.7

d)

None of the above

4.

Let p and q be the roots of the quadratic equation x² – (a – 2) x – (a + 1) = 0. What is the minimum possible value of p² + q² ?

a)

0

b)

3

c)

4

d)

5

5.

If log₃ 2, log₃ (2^x – 5), log₃ (2^x – 7/2) are in arithmetic progression, then the value of x is equal to:

a)

5

b)

4

c)

2

d)

3

6.

Consider the following two curves in the x-y plane; y = x³ + x² + 5 and y = x² + x + 5 which of the following statements is true for -2 $\le$ x $\le$ 2?

a)

The two curves intersect once.

b)

The two curves intersect twice.

c)

The two curves do not intersect.

d)

The two curves intersect thrice.

7.

If f(x) = x³ – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true

a)

–1 < p < 2

b)

0 < p < 3

c)

–2 < p < 1

d)

p > 3 or p < 0

8.

$f_1(x)=\ \ \ \ \ \ \ \ \ \ \ \ x0\le x\le1$

$=1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ge1$

$=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Otherwise$

$f_2(x)=f_1(-x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

$f_3(x)=-f_2(x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

$f_4(x)=f_3(-x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

How many of the following products are necessarily zero for every x. f₁ (x)f₂ (x), f₂(x)f₃ (x), f₂ (x)f₄ (x)? = 1

a)

0

b)

1

c)

2

d)

3

9.

$f_1(x)=\ \ \ \ \ \ \ \ \ \ \ \ x0\le x\le1$

$=1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ge1$

$=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Otherwise$

$f_2(x)=f_1(-x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

$f_3(x)=-f_2(x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

$f_4(x)=f_3(-x)\ \ \ \ \ \ \ \ \ \ \ \ for\ all\ x$

Which of the following is necessarily true?

a)

f₄ (x) = f₁ (x) for all x

b)

f₁ (x) = –f₃ (–x) for all x

c)

f₂ (–x) = f₄ (x) for all x

d)

f₁ (x) = f₃ (x) = 0 for all x

10.

Let u = (log₂ x)² – 6log₂ x + 12 where x is a real number. Then the equation x^u = 256, has.

a)

no solution for x

b)

exactly one solution for x

c)

exactly two distinct solutions for x

d)

exactly three distinct solutions for x

11.

For which value of k does the following pair of equations yield a unique solution of x such that the solution is positive? x²– y² = 0 (x – k)² + y² = 1

a)

2

b)

0

c)

$\sqrt[]{2}$

d)

$-\ \sqrt[]{2}$

12.

Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is the relation g(x+p) = g(x) necessarily true for every real x?

a)

5

b)

3

c)

2

d)

6

13.

The graph of y – x against y + x is as shown below. (All graphs in this question are drawn to scale and the same scale has been used on each axis.)Which of the following shows the graph of y against x?

a)

b)

c)

d)

e)

14.

What values of x satisfy $x^{\frac{2}{3}}+x^{\frac{1}{3}}-2\le0$ (‘x’ is a real number)?

a)

$-8\le x\le1$

b)

$-1\le x\le8$

c)

1 < x < 8

d)

1 < x $\le$ 8

e)

$-8\le x\le8$

15.

Let f(x) = max(2x + 1, 3 – 4x), where x is any real number. Then the minimum possible value of f(x) is:

a)

1/3

b)

1/2

c)

2/3

d)

4/3

e)

5/3

16.

If log_Y x = (a . log_Z y) = (b . log_X z) = ab, then which of the following pairs of values for (a, b) is not possible?

a)

(–2, 1/2)

b)

(1,1)

c)

(0.4, 2.5)

d)

(a, 1/a)

e)

(2,2)

17.

A function ƒ(x) satisfies ƒ(1) = 3600 and ƒ(1) + ƒ(2) + ... + ƒ(n) = n2 f(n), for all positive integers n > 1. What is the value of ƒ(9)?

a)

80

b)

240

c)

200

d)

100

e)

120

18.

A quadratic function ƒ(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value ƒ(x) at x = 10?

a)

–119

b)

–159

c)

–110

d)

–180

e)

–105

19.

t f(x) = ax² + bx + c, where a, b and c are certain constants and a $\ne$ 0. It is known that f(5) = –3f(2), and that 3 is a root of f(x) = 0. What is the other root of f(x) = 0?

a)

–7

b)

–4

c)

2

d)

6

e)

cannot be determined

20.

t f(x) = ax²+ bx + c, where a, b and c are certain constants and a $\ne$ 0. It is known that f(5) = –3f(2), and that 3 is a root of f(x) = 0. What is the value of a + b + c?

a)

9

b)

14

c)

13

d)

37

e)

cannot be determined

Taste of the Exams—Block V CAT 3

Total questions: 20

Worksheet time: 40mins

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