Athens Academy Islamabad · Unit Model Paper · Mathematics HSSC-II

Athens Academy Islamabad · Unit Model Paper · Mathematics HSSC-II · Chapter 1

ATHENS ACADEMY ISLAMABAD

MATHEMATICS HSSC–II

Time allowed: 2:35 Hours Total Marks Sections B and C: 80

SECTION – A: Attempt MCQs from the Unit Quiz — 20 marks

SECTION – B (Marks 48)

Q. 2 Solve the following Questions.( 12 x 4 = 48 )

(Attempt any twelve. Every question carries 4 marks. All questions are from Chapter 1 — Functions and Graphs.)

(i)

f(x)=3x+2

, find

f^{-1}(x)

and also show that

f^{-1}\big(f(x)\big)=x

Define a function

f:A\to B

and distinguish between its domain, range and codomain, with one example of each.

(ii)

Find the domain of

h(x)=\dfrac{x+4}{x^{2}-9}

Find the domain of

g(x)=\sqrt{x-5}

and state why the endpoint is included.

(iii)

Find the domain and range of

f(x)=3+\sqrt{x-2}

Find the range of

f(x)=x^{2}-4

and explain why it is not all of

\mathbb{R}

(iv)

Find

f^{-1}(x)

for

f(x)=\dfrac{x}{x-2}

Find the inverse of

f(x)=x^{3}

, and hence explain why

f^{-1}(x)\neq\dfrac{1}{f(x)}

(v)

Determine, with reason, whether

f(x)=2x^{2}+1

is one-one on

\mathbb{R}

Show that

f(x)=x^{2}

is not one-one over

\mathbb{R}

, and state how the domain may be restricted to make it one-one.

(vi)

For

f(x)=x^{2}-1

with domain

\{1,-1,2,-2\}

and codomain

\{0,3,-3\}

, state whether

f

is onto or into.

State the onto and one-one tests, and define a bijective function.

(vii)

For the parabola

y=2x^{2}+3x-2

, find the opening direction and the axis of symmetry

x=-\dfrac{b}{2a}

For

y=2x^{2}+3x-2

, find the vertex and the

y

-intercept.

(viii)

Describe the transformation from

y=|x|

y=|x-1|-1

, and give the new vertex.

State the effect on a graph of

f(x+c)

and of

f(x)+c

, explaining why the two directions differ.

(ix)

State the laws

\log_{a}(xy)

\log_{a}\!\big(\tfrac{x}{y}\big)

and

\log_{a}(x^{n})

, and hence evaluate

\log_{5}125

Prove that

\log_{a}(x^{n})=n\log_{a}x

and state the domain of

\log_{a}x

(x)

A deposit grows as

A(t)=30000(1.08)^{t}

. Find the time

t

for it to reach Rs. 150,000.

Solve

(1.08)^{t}=5

for

t

, showing how logarithms bring the exponent down.

(xi)

State the domain and range of

y=a^{x}\ (a>1)

and of

y=\log_{a}x

, and explain how the two are related.

Explain, using the line

y=x

, why the graphs of

f

and

f^{-1}

are mirror images.

(xii)

State the domain and range of

y=\sin x

and of

y=\tan x

Give the domain and range of each:

y=ax+b

y=\sqrt{x}

y=|x|

(Page 1 of 2 Mathematics)

SECTION – C (Marks 32)

Note: Solve the following Questions.( 4 x 8 = 32 )

(Attempt any four. Every question carries 8 marks. All questions are from Chapter 1 — Functions and Graphs.)

Q.3

For

f(x)=\dfrac{x}{x-2}

(a) Find $f^{-1}(x)$ , showing every step of the three-step method.

(b) Verify that $f\big(f^{-1}(x)\big)=x$ .

(a) Explain the one-one and onto tests and define a bijective function.

(b) For $f(x)=x^{2}-1$ , domain $\{1,-1,2,-2\}$ , codomain $\{0,3,-3\}$ , determine the type of $f$ , showing all images.

Q.4

For the parabola

y=2x^{2}+3x-2

(a) Find the opening direction, the axis of symmetry, the vertex, and the $y$ -intercept.

(b) Using these features, sketch the parabola, labelling the vertex and intercept.

(a) Starting from $y=|x|$ , sketch $y=|x-1|-1$ using the shift rules, labelling the vertex.

(b) State the two transformation rules used and explain why an inside-bracket change reverses direction.

Q.5

A deposit grows as

A(t)=30000(1.08)^{t}

(a) Find the time to reach Rs. 150,000, using logarithms to bring the exponent down.

(b) State the domain and range of the exponential model.

(a) State all five laws of logarithm and the domain of $\log_{a}x$ .

(b) Evaluate $\log_{5}125$ and $\log_{a}1$ , giving reasons.

Q.6

(a) Find the domain and range of $f(x)=3+\sqrt{x-2}$ , with full reasoning.

(b) Find the domain of $h(x)=\dfrac{x+4}{x^{2}-9}$ , listing every excluded value.

(a) Determine whether $f(x)=2x^{2}+1$ is one-one on $\mathbb{R}$ , with reason.

(b) Find the inverse of $f(x)=3x+2$ and show $f^{-1}\big(f(x)\big)=x$ .

—— (B) ——

(Page 2 of 2 Mathematics)

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