BISE Sargodha · Class 12 · Mathematics · 2025 · First Annual · Group 1 · Code 4195

${\int }_0^1\mathrm{\mid }x\mathrm{\mid }dx=$ $\underset{‾}{}$

$y=\mathrm{T}\mathrm{a}\mathrm{n}x+c$ is the solution of the differential equation:

If. $f(x)=\sqrt{x+4}$, then $f(x-1)$is equal to $\underset{‾}{}$:

${\lim }_{x\to 0}(1+x{)}^{1\mathrm{/}x}=$ $\underset{‾}{}$:

The derivative of $x^2$ w.r.t $4x$ is$\underset{‾}{}$:

If $f(x)=e^{5x}$, then $f^{\mathrm{\prime }}(0)=$ $\underset{‾}{}$

$x-\frac{x^3}{3!}+\frac{x^5}{5!}\dots$ is the Maclaurin series of the function:

$\frac{d}{dx}\left(\frac{1}{a}{\mathrm{T}\mathrm{a}\mathrm{n}}^{-1}\frac{x}{a}\right)=$ $\underset{‾}{}$

$xdy+ydx$ is the differentials of function$\underset{‾}{}$:

$\int \frac{1}{\sqrt{a^2-x^2}}dx=$ $\underset{‾}{}$

$\mathrm{\mid }\cos \alpha \mathit{i}+\sin \alpha \mathit{j}+0\mathit{k}\mathrm{\mid }=$ $\underset{‾}{}$

$3\mathit{i}\cdot 4\mathit{j}\times 5\mathit{k}=$ $\underset{‾}{}$

If the distance of the point $(x,4)$ from the origin is $5$, then what is the value of $x$?

The inclination of the line $x-2=0$ is:

The point intersection of the lines $x-y=0$ and $2x+y=0$ is$\underset{‾}{}$

Which one of the following is a feasible solution of the inequality $5x+y<12$.

The radius of the circle $x^2+y^2=18$ is:

The circle $x^2+y^2+2gx+2fy+c=0$ passing through origin if:

The axis of the parabola $(x-1{)}^2=8(y-2)$ is$\underset{‾}{}$:

The projection of $\mathit{i}+3\mathit{j}+4\mathit{k}$ along the vector $\mathit{i}$ is$\underset{‾}{}$

Prove that ${\cosh }^{2}x-{\sinh }^{2}x=1$

For $f(x)=-2x+8$, find $f^{-1}(x)$

Evaluate ${\lim }_{x\to 2}\frac{\sqrt{x}-\sqrt{2}}{x-2}$

Determine whether the function $f(x)=x^{2\mathrm{/}3}+6$ is even or odd.

For real valued functions $f(x)=\sqrt{x+1}$, $g(x)=\frac{1}{x^2}$, $x\mathrm{\ne }0$ Find $fog(x)$ and $gof(x)$

Find $\frac{dy}{dx}$ from first principle if $y=x^m$, $m\in N$

If $y=\sqrt{x}-\frac{1}{\sqrt{x}}$, show that $2x\frac{dy}{dx}+y=2\sqrt{x}$

Differentiate ${\tan }^3\theta \mathrm{.}{\sec }^2\theta$ with respect to $\theta$

Find $f^{\mathrm{\prime }}(x)$ if $f(x)=x^3\mathrm{.}{e}^{1\mathrm{/}x}$

Find $y_2$ if $y=x^2{e}^{-x}$

Apply Maclaurin series expansion to prove that $\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots$

Find the extreme values for the function $f(x)=x^3-6x^2+9x$

Find $dy$ and $\delta y$, where $y=\sqrt{x}$, when x changes from 4 to 4.41

Evaluate $\int \cos 3x\sin 2xdx$

Find $\int a^{x^2}\mathrm{.}xdx$, $a>0$, $a\mathrm{\ne }1$

Solve the differential equation $x^2(2y+1)\frac{dy}{dx}-1=0$

Find the area between the x-axis and the curve $y=\cos \frac{1}{2}x$ from $x=-\pi$ to $\pi$

Evaluate the definite integral ${\int }_1^2\ln xdx$

Evaluate $\int x\sqrt{x^2-1}dx$

Find the points three-fifth of the way along the line segment from $A(-5,8)$ to $B(5,3)$.

Find an equation of the line through $(-8,5)$ having slope undefined.

The coordinates of three points are $A(2,3)$, $B(-1,1)$ and $C(4,-5)$. By computing area bounded by ABC, check whether the points are collinear.

Find measure of the angle between the lines represented by $2x^2+3xy-5y^2=0$

The xy-coordinate areas are translated through the point $O^{\mathrm{\prime }}(4,6)$. The coordinates of the point P are $(2,-3)$ referred to the new axes. Find the coordinates of P referred to the original ones.

Define a feasible region.

Graph the solution set of linear Inequality $5x-4y\le 20$

Find the centre and radius of a circle $5x^2+5y^2+24x+36y+10=0$

Find the Equation Tangent to circle $x^2+y^2=25$ at $(4,3)$

Find focus and vertex of parabola $y^2=-12x$

Find the equation of parabola with Focus $(-3,1)$, directrex $x=3$

Find covertices of ellipse $4x^2+9y^2=36$

Find a unit vector in the same direction of the $\vec{V}=[3,-4]$

Find the sum of vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ given four points $A(1,-1)$, $B(2,0)$, $C(-1,3)$ and $D(-2,2)$.

Find a constant a, so that $\vec{V}=\widehat{i}-3\widehat{j}+4\widehat{k}$ and $\vec{w}=a\widehat{i}+9\widehat{j}-12\widehat{k}$ are parallel.

Find a scalar $\alpha$, so that $2\widehat{i}+\alpha \widehat{j}+5\widehat{k}$ and $3\widehat{i}+\widehat{j}+\alpha \widehat{k}$ are perpendicular.

Find a vector perpendicular to each of the vectors $\vec{a}=2\widehat{i}-\widehat{j}+\widehat{k}$, $\vec{b}=4\widehat{i}+2\widehat{j}-\widehat{k}$

Find the value of $3\widehat{j}\cdot 4\widehat{k}\times \widehat{i}$

Evaluate ${\lim }_{\theta \to 0}\frac{1-\cos p\theta }{1-\cos q\theta }$

Differentiate w.r.t “x” ${\mathrm{C}\mathrm{o}\mathrm{s}}^{-1}\left(\frac{1-x^2}{1+x^2}\right)$

If $x=\sin \theta$, $y=\sin m\theta$, show that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+m^{2}y=0$

Evaluate $\int \frac{\sqrt{2}}{\sin x+\cos x}dx$

Graph the feasible region for the system of Inequalities and also find all corner points: $\{\begin{array}{l}x+y\le 5\\ -2x+y\le 2\\ x\ge 0,y\ge 0\end{array}$

Solve the differential equation $\frac{ds}{dt}+2st=0$. Also, find the particular solution if $S=4e$ when $t=0$

Find an equation of the chord of contact of the tangents drawn from $(4,5)$ to the circle $2x^2+2y^2-8x+12y+21=0$

Find equations of two parallel lines perpendicular to $2x-y+3=0$ such that the product of the x- and y-intercepts of each is 3.

Find the centre, foci, eccentricity and directrix of the ellipse $25x^2+4y^2-250x-16y+541=0$

Prove that $\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$