Mathematics 11 past paper 2025

What are the multiplicative factors of $z^{2}+4z+5$?

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Solve the quadratic equation $3z^{2}+2z+2=0, z \in \mathbb{C}$ by completing the square method.

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If $P(x)=2x^{3}+x^{2}-13x+6$, then: a) Verify that $(x+3)$ is a factor of $P(x)$. b) Factorize $P(x)$ using synthetic division. c) Find all the roots of $P(x)$. d) Write complete f…

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The principal argument $(\theta)$ of the complex number $z=-1+i\sqrt{3}$ is:

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A triangular park has its three corners located at points $A(2,3,1)$, $B(5,7,2)$, and $C(1,4,3)$. Find the area of the park using the cross product of vectors.

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If $\vec{a}=3\hat{i}-2\hat{j}+4\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$, then: (a) Find $\vec{a} \times \vec{b}$. (b) Find $(\vec{a} \times \vec{b}) \cdot \vec{a}$ and verif…

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A system of linear equations is said to be **inconsistent** if it has:

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Verify that: $\sin \theta \cdot \sin \left(\frac{\pi}{3}-\theta\right) \cdot \sin \left(\frac{\pi}{3}+\theta\right) = \frac{1}{4} \sin 3\theta$

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If $x$ is very small such that its square and higher powers can be neglected, then show that: $\frac{(16+4x)^{\frac{3}{4}}}{(4+x) \sqrt{9-6x}} \approx \frac{2}{3}+\frac{13x}{72}$

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If $|A|$ is the determinant of a $3 \times 3$ matrix, then $|3A| = $

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Verify that: $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$

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Verify the fundamental law of trigonometry: $\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$ where $\alpha, \beta$ are real angles in standard position and $\alph…

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In an arithmetic sequence, $a_{1}=5$ and $a_{15}=75$. What is the common difference?

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Find the value of $p$ such that vectors $\hat{i}+2\hat{j}+p\hat{k}$, $3\hat{i}+p\hat{j}+4\hat{k}$ and $2\hat{i}+3\hat{j}+4\hat{k}$ are coplanar.

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Solve $x+2y-3z=4, 2x-3y+4z=5, 3x+4y-5z=6$ using the Gaussian elimination method.

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If $\frac{1}{2}, \frac{1}{x}, \frac{1}{5}$ are in harmonic sequence then what is the value of $x$?

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Find the maximum and minimum values of the function $f(\theta) = \frac{1}{3+5 \cos (2\theta+\pi)}$

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$y=3 \cos 2x; -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ a) Make table of values for given interval. b) Draw the graph of the function for given interval.

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What is the first term of an infinite geometric series whose sum is $10$, and common ratio $0.6$?

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If $P(x)=x^{4}-6x^{3}+11x^{2}-6x$, then: (a) Divide $P(x)$ by $(x-1)$ using synthetic division. (b) Solve the resulting depressed equation.

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For an arithmetic-geometric series: $1+5\left(\frac{1}{2}\right)^{1}+9\left(\frac{1}{2}\right)^{2}+13\left(\frac{1}{2}\right)^{3}+\dots$ (a) Find general term of the series. (b) Su…

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How many ways can a committee of $4$ people be chosen from a group of $6$?

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A harmonic sequence has $2^{\text{nd}}$ term $\frac{1}{6}$ and $4^{\text{th}}$ term $\frac{1}{12}$. Find general term of the sequence.

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A company has $7$ engineers and $5$ managers. In how many ways can a project team of $5$ be selected if it includes: (a) $3$ engineers and $2$ managers? (b) $4$ engineers and $1$ m…

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What is the middle term in the expansion of $(x+y)^{6}$?

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Without drawing graph find range, amplitude, period and frequency of the function $y=-4 \cos (7x-\pi)$

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Find the term containing $x^{3}$ in the expansion of $(x+2)^{6}$.

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If dot product of two vectors is zero then angle between them is:

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In an arithmetic sequence, the $5^{\text{th}}$ term is $15$, and the $12^{\text{th}}$ term is $50$. Find: a) Common difference and b) First term of the sequence.

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If $\vec{a} \cdot (\vec{b} \times \vec{c})=12$ then volume of parallelepiped with $\vec{a}, \vec{b}$ and $\vec{c}$ as co-terminal edges is:

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Verify that: $\frac{1+\sin 2\theta+\cos 2\theta}{1+\sin 2\theta-\cos 2\theta} = \cot \theta$

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A force $\vec{F}=2\hat{i}+\hat{j}+3\hat{k}$ moves an object from origin to $(2, -1, 1)$. The work done is:

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Verify that $\begin{vmatrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix} = (b-a)(c-a)(c-b)$

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If $\tan \theta = \frac{3}{4}$ with $\pi < \theta < \frac{3\pi}{2}$, find the exact values of (a) $\sin(\frac{\theta}{2})$ and (b) $\cos(\frac{\theta}{2})$ without using calculator…

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If $\sin \theta = \frac{3}{5}$, then $\sin 2\theta = $

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A bouncing ball rebounds 80% of its previous height. If the ball is dropped from a height of 50 meters, find the total distance traveled by the ball before coming to rest.

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If the graph of a function is symmetric about the origin (e.g. $\sin x$), what type of function is it?

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How many words can be formed by using the letters from the word "EDUCATION" such that all the vowels are never together?

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The range of a trigonometric function $4 \sin 4x$ is:

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Use the principle of mathematical induction to prove that $1^{3}+2^{3}+3^{3}+\dots+n^{3} = \left(\frac{n(n+1)}{2}\right)^{2}$ for all $n \in \mathbb{N}$, $n \geq 1$.

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The expression $\sin 75^{\circ} \cos 15^{\circ}$ is equivalent to:

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Find rank of the matrix $\begin{bmatrix} 1 & 1 & 0 & -2 \\ 2 & 0 & 2 & 2 \\ 4 & 1 & 3 & 1 \end{bmatrix}$

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How many 5-letter words can be formed from the word 'MATHS' without repeating any letter?

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If $z_{1}=10(\cos 100^{\circ}+i \sin 100^{\circ})$ and $z_{2}=5(\cos 40^{\circ}+i \sin 40^{\circ})$, then find the following in polar form: (a) $z_{1} \cdot z_{2}$, (b) $\frac{z_{1…

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In how many different ways can a player select $6$ numbers from a set of $49$ numbers in a lottery?

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A solar panel's sunlight reception is modeled as $P(\theta) = 100 \cos(3\theta - 90°)$, with $0° \leq \theta \leq 180°$. (a) Find angle for the maximum sunlight and the max sunligh…

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A force $\vec{F}=6\hat{i}+8\hat{j}+4\hat{k}$ acts on an object. The object moves from $A(1,2,3)$ to point $B(4,6,5)$. (a) Find the displacement vector $\vec{d}$. (b) Calculate work…

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If $P(x)=ax^{3}+bx^{2}+2x-1$, leaves remainder $5$ when divided by $(x-1)$ and leaves remainder $3$ when divided by $(x+1)$. Find values of $a$ and $b$ using remainder theorem.

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Solve $x+y+z=3; y+z=2; z=2$ using Cramer's rule.

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Use binomial theorem to approximate the value of $\sqrt{101} \times \sqrt{99}$ up to three places of decimal.

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