HSSC-I Mathematics Question Bank

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Class 11 - Mathematics / Unit 8 - Trigonometric Identities / 8.1 Fundamental Law of Trigonometry / 8.1.4

AKU-EB Model Paper 2025MEDIUM

Fill in the blanks:

Show that

\cos(\alpha+\beta)\times\cos(\alpha-\beta)=1-(\sin^2\alpha+\sin^2\beta)

.
We will apply formula
That will give us
Rewriting further gives us, which expands and simplifies to

1-(\sin^2\alpha+\sin^2\beta)

, as required.

Class 11 - Mathematics / Unit 11 - Vectors / 11.2 Vectors in Space / 11.2.4

AKU-EB Model Paper 2025MEDIUM

Fill in the blank:

The vectors

\mathbf{P}

\mathbf{Q}

and

\mathbf{R}

are given as

\mathbf{P}=2\mathbf{i}-3\mathbf{j}+4\mathbf{k}

\mathbf{Q}=\mathbf{j}-\mathbf{k}

and

\mathbf{R}=3\mathbf{i}-\mathbf{k}

.
Note: Answer the blanks using the built in math tool. i. Show that the magnitude of

\mathbf{P}-5\mathbf{Q}

\sqrt{149}

units.

\mathbf{P}-5\mathbf{Q}=

, that has magnitude 149. ii. Find the unit vector in the direction of

\mathbf{P}-5\mathbf{Q}

. It has:

\mathbf{i}

-component:

\mathbf{j}

-component:

\mathbf{k}

-component:
iii. State a vector which is parallel to the vector

\mathbf{R}

but opposite in direction.
That vector with all components is:

Class 11 - Mathematics / Unit 10 - Inverse Trigonometric Functions and Trigonometric Equations / 10.3 Trigonometric Equations / 10.3.1

AKU-EB Model Paper 2025HARD

Fill in the blanks:

Find the solution set of the trigonometric equation

\sin 2x=\cos x

, when

0\leq x\leq 2\pi

.
The solution set, in ascending order, is

\{

,,,

\}

Class 11 - Mathematics / Unit 9 - Trigonometric Functions and their Graphs / 9.1 Trigonometric Functions / 9.1.2

AKU-EB Model Paper 2025MEDIUM

Fill in the following blanks:

Find the period, domain and range of

2\sin 2x

.
Period:
Domain:
Range:

Class 11 - Mathematics / Unit 9 - Trigonometric Functions and their Graphs / 9.2 Graphs of Trigonometric Functions / 9.2.4

AKU-EB Model Paper 2025MEDIUM

Fill in the blanks:

The figure shows the graph of

y=a\sin(px)

for

0°\leq x\leq 180°

, where

a

and

p

are positive constants.
The value of

a

isand value of

p

is
The period of

y

Class 11 - Mathematics / Unit 8 - Trigonometric Identities / 8.2 Trigonometric Ratios of Allied Angles / 8.2.2

AKU-EB Model Paper 2025MEDIUM

Fill in the blanks:

Using the fundamental law of trigonometry, prove

\cos\left(\dfrac{\pi}{2}+\alpha\right)=-\sin\alpha

.
We will apply formula
That will give us
Substitutingandgives the answer.

Class 11 - Mathematics / Unit 6 - Mathematical Induction and Binomial Theorem / 6.2 Binomial Theorem / 6.2.5

AKU-EB Model Paper 2025MEDIUM

Fill in the blank:

Find the term independent of

x

in the expansion of

\left(x^2+\dfrac{4}{x}\right)^6

.
The term independent of

x

Class 11 - Mathematics / Unit 6 - Mathematical Induction and Binomial Theorem / 6.1 Mathematical Induction / 6.1.3

AKU-EB Model Paper 2025MEDIUM

Fill in the following blanks:

Show that

\dfrac{8^n-3^n}{5}

is an integer for all natural numbers

n

, using the principle of mathematical induction.
(Note:

n\in N

)
Basis step:
State the induction hypothesis:
Using the hypothesis, show that

8^{k+1}-3^{k+1}

is divisible by

5

:
State the final conclusion:

Class 11 - Mathematics / Unit 5 - Permutation and Combination / 5.2 Counting Techniques / 5.2.8

AKU-EB Model Paper 2025MEDIUM

Solve the following question:

\displaystyle\binom{n}{3}=8(n-1)

, then find the value of

n

n =

Class 11 - Mathematics / Unit 3 - Sequence and Series / 3.2 Arithmetic Sequence / 3.2.3

AKU-EB Model Paper 2025MEDIUM

Q10

Fill the following blank:

Note: Enter the answer in fractions like 2m/3
Find the sum up to

n

terms of an arithmetic sequence whose

1^{\text{st}}

term is

7

and

7^{\text{th}}

term is

37

S_n

Class 11 - Mathematics / Unit 3 - Sequence and Series / 3.2 Arithmetic Sequence / 3.2.3

AKU-EB Model Paper 2025EASY

Q11

Fill the following blank:

An arithmetic sequence satisfies the condition

4\times a_6 = a_{26}

. If the first term

a=5

, find the common difference

d

of the sequence.
The common difference

d

Class 11 - Mathematics / Unit 3 - Sequence and Series / 3.7 Geometric Series / 3.7.4

AKU-EB Model Paper 2025MEDIUM

Q12

Fill in the following blank:

Convert

0.481481481481481\ldots

into an equivalent fraction.
Note: Enter the answer as a fraction like 23/4,5/2
The equivalent fraction is:

Class 11 - Mathematics / Unit 2 - Matrices and Determinants / 2.1 Matrices / 2.1.1

AKU-EB Model Paper 2025MEDIUM

Q13

Fill in the following blanks:

Given that

Q'P' = \begin{bmatrix} m & 2 & 0 \\ 3 & 0 & 1 \\ 0 & 4 & 10 \end{bmatrix}

i. State the matrix $PQ$ .

\begin{bmatrix} (1,1) & (1,2) & (1,3) \\ (2,1) & (2,2) & (2,3) \\ (3,1) & (3,2) & (3,3) \end{bmatrix}

Entry (1,1) is
Entry (1,2) is
Entry (1,3) is
Entry (2,1) is
Entry (2,2) is
Entry (2,3) is
Entry (3,1) is
Entry (3,2) is
Entry (3,3) is
ii. Determine the value of $m$ for which the matrix $PQ$ will become singular.

m

Class 11 - Mathematics / Unit 2 - Matrices and Determinants / 2.1 Matrices / 2.1.1

AKU-EB Model Paper 2025MEDIUM

Q14

Fill in the following blanks:

For the matrices

A=\begin{bmatrix} i \\ 2i \\ 3i \end{bmatrix}

and

B=\begin{bmatrix} i & 4i \end{bmatrix}

, find

A\times B

and

(A\times B)\times(A\times B)'

.
(Note:

i=\sqrt{-1}

)
Part 1: $A \times B$

\begin{bmatrix} (1,1) & (1,2) \\ (2,1) & (2,2) \\ (3,1) & (3,2) \end{bmatrix}

Entry (1,1) is
Entry (1,2) is
Entry (2,1) is
Entry (2,2) is
Entry (3,1) is
Entry (3,2) is
Part 2: $(A\times B)\times(A\times B)'$

\begin{bmatrix} (1,1) & (1,2) & (1,3) \\ (2,1) & (2,2) & (2,3) \\ (3,1) & (3,2) & (3,3) \end{bmatrix}

Entry (1,1) is
Entry (1,2) is
Entry (1,3) is
Entry (2,1) is
Entry (2,2) is
Entry (2,3) is
Entry (3,1) is
Entry (3,2) is
Entry (3,3) is

Class 11 - Mathematics / Unit 2 - Matrices and Determinants / 2.3 Solution of Linear Equations using Matrices / 2.3.3

AKU-EB Model Paper 2025MEDIUM

Q15

Fill in the following blanks:

The matrix form of the system of simultaneous linear equations

x+3y+3z=1

y+2z=2

, and

2x+y=3

is:

\begin{bmatrix}1&3&3\\0&1&2\\2&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix}

|A|=4

, then find the value of the matrix

\begin{bmatrix}x\\y\\z\end{bmatrix}

using the matrix inversion method.
Note: Answer in fractions like 2/3,4/5.
The value of x is
The value of y is
The value of z is