Mathematics 12 past paper 2024

If $f(x)=\sqrt{x+5}$ and $f(g(x))=3x$ then what is the value of $g(x)$?

MCQ1 marksannual

If $f(x)=\frac{3x+2}{2x-1}$, find $f^{-1}(x)$ and also show that $f^{-1}(f(x))=x$.

Part B4 marksannual

Let $f(x)=\begin{cases} px+2 & \text{if } 0 \leq x < 2 \\ 7-qx & \text{if } 2 \leq x < 4 \\ 2x+1 & \text{if } 4 \leq x < 6 \end{cases}$. Find $p$ and $q$ such that $f$ is continuou…

Part C8 marksannual

Without finding the inverse, what is the range of $f^{-1}(x)$, where $f(x)=3+\sqrt{x-2}$?

MCQ1 marksannual

Find $\frac{dy}{dx}$ if $2y^3-3xy^2+2x^2y+5x=6$. Also find the value of $\frac{dy}{dx}$ at $(1,1)$.

Part B4 marksannual

A box with a square base and open top is to have a volume 32 cubic dm. Find the dimensions of the box that will require the least material.

Part C8 marksannual

Which of the following functions is continuous at $x=4$?

MCQ1 marksannual

Find the derivative of $y=(2\sqrt{x}+2)(x-\sqrt{x})$.

Part B4 marksannual

Evaluate $\int \frac{2x^2-x-7}{(x+2)^2(x^2+2x+5)} dx$.

Part C8 marksannual

If $x+y=\sin(x+y)$ then $\frac{dy}{dx}$ is equal to:

MCQ1 marksannual

Find the equation of tangent of an ellipse $\frac{x^2}{128}+\frac{y^2}{18}=1$ which are parallel to the line $3x+8y+1=0$. Also find the point of contact.

Part C8 marksannual

Minimum value of the function $f(x)=x^2+2x-3$ is:

MCQ1 marksannual

If $y=\cot(q \cot^{-1} x)$ then show that $(1+x^2)y_1 - q(1+y^2) = 0$.

Part B4 marksannual

If $f(x)=e^{\ln(\sin x)}$ then what is the value of $f'(\frac{\pi}{2})$?

MCQ1 marksannual

Evaluate $\lim_{\theta \to 0} \frac{\sec \theta - 1}{\theta \sec \theta}$.

Part B4 marksannual

A factory produces two items: ceiling lights and ceiling fans by using two machines A and B. Machine A has at most 120 hours available and machine B has a maximum of 144 hours avai…

Part C8 marksannual

What evaluates $\int_{0}^{\frac{\pi}{2}} \cos x (e^{\sin x}) dx$?

MCQ1 marksannual

Examine the function $x^3-6x^2+9x+3$ for extreme values.

Part B4 marksannual

Find the centre, foci, eccentricity, vertices and equation of directrices of the conic $9x^2-y^2-12x-2y+2=0$.

Part C8 marksannual

Which of the following options represents $f'(x)=\frac{x}{x^2+1}$ and $f(0)=1$?

MCQ1 marksannual

Use the differential to approximate value of $\sin 61^\circ$.

Part B4 marksannual

Evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos^2 \theta \cot^2 \theta d\theta$.

Part C8 marksannual

If $2\int_{0}^{2}(x-k)dx=1$ then what is the value of $k$?

MCQ1 marksannual

Find the area bounded by the curve $y=x^3-9x$ and the $x$-axis.

Part B4 marksannual

If the line $\pi x + \sqrt{2}y + \sqrt{7} = 0$ is perpendicular to the line $kx + 3y + 2 = 0$, then $k$ is:

MCQ1 marksannual

Find the area of the region bounded by $10x^2-xy-21y^2=0$ and $x+y+1=0$.

Part B4 marksannual

A line that cuts the $x$-axis at $(2,0)$ and $y$-axis at $(0,-4)$, has equation:

MCQ1 marksannual

Solve the differential equation $\frac{dy}{dx}=\frac{3}{4}x^3+x-3$ if $y=0$, when $x=2$.

Part B4 marksannual

What is the angle between pair of lines represented by $x^2+2xy-y^2=0$?

MCQ1 marksannual

Find the point $P$ on the line joining $A(1,4)$ and $B(5,6)$ that is twice as far from $A$ as $B$ is from $A$ and lies on the opposite side of $A$ as $B$ does.

Part B4 marksannual

If the lines $3x-y-2=0$, $5x+ay-3=0$ and $2x+y-2=0$ are concurrent then what is the value of $a$?

MCQ1 marksannual

The length of perpendicular from origin to the line is 8 units and angle of inclination is $30^\circ$. Find the slope and $y$-intercept of the line.

Part B4 marksannual

The ordered pair $(3,2)$ is **NOT** a solution of inequality:

MCQ1 marksannual

Integrate $\int \frac{x \sin^{-1} x}{\sqrt{1-x^2}} dx$.

Part B4 marksannual

What are the foci of an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$?

MCQ1 marksannual

Find the equation of a circle passing through the point $(-2,-5)$ and touching the line $3x+4y-24=0$ at the point $(4,3)$.

Part B4 marksannual

Directrices of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are:

MCQ1 marksannual

Graph the feasible region of the following system of linear inequalities by shading and find the corner points: $3x+2y \geq 6, x+y \leq 4, x \geq 0, y \geq 0$.

Part B4 marksannual

What is the length of tangent from $(2,1)$ to the circle $3x^2+3y^2+6x-12y+1=0$?

MCQ1 marksannual

Graph the feasible region of the following system of linear inequalities by shading and find the corner points: $5x+7y \leq 35, x-2y \leq 4, x \geq 0, y \geq 0$.

Part B4 marksannual

Find the equation of parabola having Focus $(-3,4)$ and directrix $3x+2y-3=0$.

Part B4 marksannual

Prove that altitudes of a triangle are concurrent (by vector method).

Part B4 marksannual

What are the direction cosines of the vector $\sqrt{3}\hat{i}-\sqrt{3}\hat{j}+\sqrt{3}\hat{k}$?

MCQ1 marksannual

Find the value of $C$, when the line $5x+2y+C=0$ will touch the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$.

Part B4 marksannual

Find the points of intersection of the two conics $\frac{x^2}{18}+\frac{y^2}{8}=1$ and $\frac{x^2}{3}-\frac{y^2}{3}=1$.

Part B4 marksannual

If $\bar{u}=2\hat{i}+3\hat{j}+4\hat{k}$, $\bar{v}=\hat{i}+4\hat{j}+3\hat{k}$ and $\bar{w}=\hat{i}+7\hat{j}+\lambda\hat{k}$ represent the sides of a triangle, find the value of $\la…

Part B4 marksannual

Find the equation of an ellipse with foci $(\pm \sqrt{5}, 0)$ and passing through the point $(\frac{3}{2}, \sqrt{3})$.

Part B4 marksannual

Find the moment about $(1,1,1)$ of each of the concurrent forces $\hat{i}-2\hat{j}$, $3\hat{i}+2\hat{j}-\hat{k}$ and $5\hat{j}+2\hat{k}$ where $P(2,0,1)$ is the point of concurrenc…

Part B4 marksannual