Mathematics 12 past paper 2025

$\lim_{x \rightarrow \infty} \frac{(1+2x)^{2}-1}{x}=$

MCQ1 marksannual

Evaluate: $\lim_{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin^{2} \theta}$

Part B4 marksannual

The growth of bacteria is given by the function of time ($t$) (in hours) $f(t)=500 e^{0.25 t}$, then find: **a.** Initial number of bacteria at $t=0$ **b.** Number of bacteria afte…

Part C8 marksannual

$\lim_{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{\frac{x}{2}}=$

MCQ1 marksannual

If $f(x, y)=3 x^{3}+7 x^{2} y+x y^{2}+5 y^{3}$, then find: **a.** Degree of homogeneous function **b.** Verify Euler's theorem for $f(x, y)$

Part B4 marksannual

$\frac{d}{dx}\left(e^{x}+2e^{-2x}\right)=$

MCQ1 marksannual

Find the value of '$k$' if $f(x)$ is continuous at $x=1$: $f(x)= \begin{cases}\frac{3 x^{2}-3}{x-1}, & x \neq 1 \\ 2(k-1) x, & x=1\end{cases}$

Part B4 marksannual

If $\frac{dy}{dx}=\frac{3}{2}x^{\frac{1}{2}}-2x$, then $\frac{d^{2}y}{dx^{2}}=$

MCQ1 marksannual

If line $x-2y+\ell=0$ is tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$, then find: **a.** Value of $\ell$ **b.** Equation of tangent **c.** Length of latus rectum

Part B4 marksannual

Evaluate: $\int \frac{2x-1}{x^3-x^2-2x} dx$

Part C8 marksannual

If $y=\frac{1}{8} \sin 2 x$, then the value of $\frac{d^{4} y}{d x^{4}}$ is:

MCQ1 marksannual

If $y=3\theta^{3}-\theta^{\frac{-3}{2}}$ and $u=2\theta^{2}-\theta^{\frac{-1}{2}}$, then find $\frac{dy}{du}$

Part B4 marksannual

Evaluate: $\int \frac{dx}{4x^3 - x}$

Part C8 marksannual

If $\vec{r}(t)=\left(1+t^{2}\right) \hat{i}+\sin t \hat{j}+\cos 2 t \hat{k}$, then $\lim _{t \rightarrow 0} \vec{r}(t)=$

MCQ1 marksannual

Compute two iterations up to 4 decimal places of $f(x)=x^{3}-5x+1$ by using **Newton-Raphson's method** with initial start $x_{0}=0.5$

Part B4 marksannual

If $\vec{r}(t)=t^{2} \hat{i}-2 t \hat{j}$, then $\vec{v}(t)=$

MCQ1 marksannual

If $\vec{r}(t)=t^{3} \hat{i}+e^{-2 t} \hat{j}+\sin 2 t \hat{k}$ is position vector of a moving particle at time $t$, then find:\n**a.** Velocity vector $\vec{v}(t)$\n**b.** Acceler…

Part B4 marksannual

$\int \frac{\cos x}{\sin x} dx=$

MCQ1 marksannual

Find the equation of line passing through the point of intersection of lines $x+y=3$ and $x-y=1$ and parallel to line $x-3y=2$ and write it in slope intercept form.

Part B4 marksannual

If $2 \int_{0}^{2} x \, dx = 3k - 2$, then the value of '$k$' is:

MCQ1 marksannual

Evaluate: $\int(3t+1)(t-1) dt$

Part B4 marksannual

If $A(5,-5), B(-2,0)$ and $C(0,2)$ are vertices of a triangle then its centroid is:

MCQ1 marksannual

Find the equation of the circle passing through the points $(4,0)$, $(0,5)$ and $(0,0)$

Part B4 marksannual

If $4x - 2y = 3$, then the slope of the line is:

MCQ1 marksannual

Find the solution of differential equation $\frac{dy}{dx}=1+e^{x}y+y+e^{x}$

Part B4 marksannual

The equation of circle with centre $(-1,-1)$ and radius 2 is:

MCQ1 marksannual

Find the area under the graph of $f(x)=3x^{2}-2x$ over the interval $[1,3]$

Part B4 marksannual

If $x^{2}+y^{2}+4x+4y=0$ is an equation of circle then its radius is:

MCQ1 marksannual

Find the extreme values of $f(x)=\frac{1}{3}x^{3}-x^{2}-3x+6$

Part B4 marksannual

If $3x^{2}-2xy-8y^{2}=0$ is a 2nd degree homogeneous equation then find: **a.** Pair of straight lines **b.** Angle between these lines

Part B4 marksannual

If $\frac{d^{2} y}{d x^{2}}-\left(\frac{d y}{d x}\right)^{2}+y=3 x$ is a differential equation, then its order and degree respectively are:

MCQ1 marksannual

Find the area of a triangle if its vertices are $A(5,4), B(2,1)$ and $C(7,3)$

Part B4 marksannual

If $\frac{1}{x} \frac{dy}{dx} = 2$, then the solution $y(x)$ of the differential equation is:

MCQ1 marksannual

Evaluate $\int x^{2} \cos x \, dx$ using integration by parts method

Part B4 marksannual

If $f(x, y)=x^{2}+3 y$, then $\frac{\partial f}{\partial x}(1,1)=$

MCQ1 marksannual

Find the equation of tangent and normal to parabola $x^{2}=8y-8$ at $x=4$

Part B4 marksannual

If $f(x, y)=e^{x+y}$, then $\frac{\partial f}{\partial y}(1,0)$ is:

MCQ1 marksannual

Find $\frac{dy}{dx}$, if $y=(2x+3)^{\frac{3}{2}}$ by first principle

Part B4 marksannual

If $f(x)=x^{3}-2x+1$ and $x_{0}=0$, then $f^{\prime}(x_{0})$ is:

MCQ1 marksannual

Evaluate $\int \frac{e^{x} dx}{e^{2x}+1}$ by substitution method

Part B4 marksannual

The value of $\Delta x$ to approximate $\int_{0}^{2}\left(x^{2}+1\right) dx$ for $n=4$ sub-intervals by **Trapezoidal Rule** is:

MCQ1 marksannual

If $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ is an equation of hyperbola then find: **a.** Centre **b.** Vertices **c.** Foci **d.** Equation of asymptotes

Part B4 marksannual