BISE Multan · Class 12 · Mathematics · 2025 · First Annual · Group 1

The area of a unit circle is $\underset{‾}{}$.

$x=a\cos t$ and $y=a\sin t$ are parametric equations of $\underset{‾}{}$.

If $\frac{d}{dx}f(x)=0$, then $f(x)$ is a $\underset{‾}{}$ function.

The derivative of $\sin x$ w.r.t. $\sin x$ is $\underset{‾}{}$.

If $f(x)=2^{\cos x}$, then $f^{\mathrm{\prime }}(\pi \mathrm{/}2)=\underset{‾}{}$

The derivative of $\ln \left(\frac{1}{x}\right)$ w.r.t. $x$ is $\underset{‾}{}$.

$\frac{ydx-xdy}{y^2}$ is the differential of $\underset{‾}{}$.

$\int e^{9x}dx=\underset{‾}{}$

${\int }_0^{\pi \mathrm{/}2}{e}^{\sin x}\cos xdx=\underset{‾}{}$

The order of the differential equation $\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^3+2x=0$ is $\underset{‾}{}$.

If the distance between $A(0,-2)$ and $B(0,x)$ is $1$, then $x=\underset{‾}{}$

If the slope of the line is $1$, then its inclination will be $\underset{‾}{}$.

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

Which one of the given is a feasible solution of $2x+y\le 6$?

$x^2+y^2+2hxy+2gx+2fy+c=0$ represents the equation of the circle if $\underset{‾}{}$.

If the point $(1,1)$ lies inside the circle $x^2+y^2+2x+y+c=0$, then:

For parabola, the value of eccentricity is $\underset{‾}{}$.

The magnitude of the vector $\frac{1}{\sqrt{3}}\underset{‾}{i}+\frac{1}{\sqrt{3}}\underset{‾}{j}+\frac{1}{\sqrt{3}}\underset{‾}{k}$ is $\underset{‾}{}$.

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

The projection of $\underset{‾}{i}$ along $\underset{‾}{j}$ is:

Evaluate $\underset{x\to 1}{<mrow><mi>L</mi><mi>i</mi><mi>m</mi></mrow>}\frac{x^3-3x^2+3x-1}{x^3-x}$

Determine whether the given function $f$ is even or odd. $f(x)=\frac{x^3-x}{x^2+1}$

Find $f^{-1}(-1)$, when $f(x)=3x^3+7$

Discuss the continuity of $f(x)$ at $x=c$, where $f(x)=\{\begin{array}{ll}3x-1,& \text{if }x<1\\ 4,& \text{if }x=1\\ 2x,& \text{if }x>1\end{array}$, $c=1$

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

Differentiate ${(\sqrt{x}-\frac{1}{\sqrt{x}})}^2$ w.r.t $x$

Find $\frac{dy}{dx}$, if $x^2-4xy-5y=0$

Differentiate ${\sin }^{3}x$ w.r.t ${\cos }^{2}x$

Expand $f(x)=\frac{1}{1+x}$ in the Maclaurin Series.

Find the extreme values for the function defined as $f(x)=3x^2-4x+5$

Find $y_2$, if $x=a{t}^2$, $y=b{t}^4$

Find $\frac{dy}{dx}$, if $y={\cosh }^{-1}(\sec x)$, $0\le x\le \pi \mathrm{/}2$

Find $\delta y$ and $dy$ in the given $y=x^2+2x$ when $x$ changes from $2$ to $1.8$.

Evaluate the given integral. $\int \frac{(1-\sqrt{x}{)}^2}{\sqrt{x}}dx$, $(x>0)$

Evaluate $\int \frac{e^x}{e^x+3}dx$

Evaluate $\int e^{-x}(\cos x-\sin x)dx$

Evaluate ${\int }_{-1}^3(x^3+3x^2)dx$

What is order of a differential equation give an example?

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

Transform the equation $5x-12y+39=0$ into slope intercept form.

Find $h$ such that the points $A(h,1)$, $B(2,7)$ and $C(-6,-7)$ are vertices of a right triangle with right angle at the vertex $A$.

Find $K$ so that the line joining $A(7,3)$; $B(K,-6)$ and the line joining $C(-4,5)$; $D(-6,4)$ are perpendicular.

Find an equation of the horizontal line through $(7,-9)$

Check whether the lines $4x-3y-8=0$; $3x-4y-6=0$; $x-y-2=0$ are concurrent.

Define Linear inequality.

Draw the graph $3x-2y\ge 6$

Write standard form of equation of a circle.

Check the position of point $(5,6)$ with respect to circle $2x^2+2y^2+12x-8y+1=0$

Write an equation of parabola with focus $(-3,1)$, directrix $x=3$

Find the focus and vertex of $y^2=8x$

Find eccentricity of hyperbola $x^2-y^2=9$

If $\vec{u}=2\widehat{i}-7\widehat{j}$, $\vec{v}=\widehat{i}-6\widehat{j}$, $\vec{w}=-\widehat{i}+\widehat{j}$ find the value of $2\vec{u}-3\vec{v}+4\vec{w}$

Find a vector whose magnitude is $4$ and parallel to the vector $2\widehat{i}-3\widehat{j}+6\widehat{k}$

Find $\underset{‾}{a}\times \underset{‾}{b}$ if $\underset{‾}{a}=2\widehat{i}-\widehat{j}-\widehat{k}$, $\underset{‾}{b}=4\widehat{i}+2\widehat{j}-\widehat{k}$

Calculate work done if force is $\underset{‾}{F}=\widehat{i}+3\widehat{j}+5\widehat{k}$ and displacement $\underset{‾}{d}=2\widehat{i}+4\widehat{j}-\widehat{k}$

Find the value of $[\widehat{i}\widehat{i}\widehat{k}]$

Find the projection of vector $\vec{v}=a\widehat{i}+b\widehat{j}+c\widehat{k}$ along vector $\widehat{i}$

Define continuous function and discuss the continuity of function $f(x)=\{\begin{array}{ll}\frac{x^2-9}{x-3},& \text{if }x\mathrm{\ne }3\\ 6,& \text{if }x=3\end{array}$ at $x=3$

Find $\frac{dy}{dx}$ by making suitable substitution if $y=\sqrt{\frac{1-x}{1+x}}$

Show that $y=\frac{<mi\; mathvariant="normal">\ell </mi><mi>n</mi>x}{x}$ has maximum value at $x=e$

Evaluate $\int \frac{1}{\sqrt{a^2+x^2}}dx$, $a>0$

Solve the differential equation ${\sec }^{2}x\tan ydx+{\sec }^{2}y\tan xdy=0$

Minimize $z=3x+y$, subject to the constraints $3x+5y\ge 15$, $x+6y\ge 9$, $x\ge 0$, $y\ge 0$

Find the equations of tangent to the circle $x^2+y^2=2$ parallel to the line $x-2y+1=0$

If two vertices of an equilateral triangle are $A(-3,0)$ and $B(3,0)$. Find the third vertex. How many such triangles are possible?

Find centre, foci, vertices, eccentricity and equation of directrices $\frac{y^2}{4}-x^2=1$

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