BISE Lahore · Class 12 · Mathematics · 2025 · Group 2 · Subjective

Show that the parametric equations $x=a\sec \theta$, $y=b\tan \theta$ represent the equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Without finding the inverse, state the domain and range of $f^{-1}$, where $f(x)=2+\sqrt{x-1}$

Define continuity of a function at a number C.

Evaluate $\underset{\theta \to 0}{<mrow><mi>L</mi><mi>i</mi><mi>m</mi></mrow>}\frac{1-\cos \theta }{\sin \theta }$

Find $fog(x)$ and $gof(x)$, when $f(x)=2x+1$ and $g(x)=\frac{3}{x-1}$, $x\mathrm{\ne }1$

Find $\frac{dy}{dx}$, if $x=a{t}^2$ and $y=2at$

Prove that $\frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2}$, $\mathrm{\forall }x\in R$

If $x=y\sin y$, find $\frac{dy}{dx}$

Find $y_2$, if $y=\ln \left(\frac{2x+3}{3x+2}\right)$

Apply the Maclaurin Series expansion to prove that $\cos x=1-\frac{x^2}{\underset{‾}{\mid 2}}+\frac{x^4}{\underset{‾}{\mid 4}}-\frac{x^6}{\underset{‾}{\mid 6}}+---$

Determine the intervals in which f is increasing or decreasing, $f(x)=\cos x$; $x\in (-\frac{\pi }{2},\frac{\pi }{2})$

Differentiate $x^2-\frac{1}{x^2}$ w.r.t $x^4$

Find $\delta y$ of the function $y=x^2-1$ when x changes from 3 to 3.02.

Evaluate $\int \frac{\sin \theta }{1+{\cos }^2\theta }d\theta$

Find $\int x\sin xdx$

Evaluate the integral $\int \frac{1}{x\ln x}dx$

Find the definite integral ${\int }_1^2\frac{x}{x^2+2}dx$

Evaluate $\int \frac{e^{m{\tan }^{-1}x}}{1+x^2}dx$

Solve the differential equation $(x-1)dx+ydy=0$

Define inclination of a line.

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

Find an equation of line through $A(-6,5)$ having slope 7.

Convert $15y-8x+3=0$ into normal form of line

Define homogeneous function.

Write the procedure for graphing a linear inequality in two variables.

Draw a graph of linear inequality $x+2y<6$

Show that the equation $5x^2+5y^2+24x+36y+10=0$ represents a circle. Also find its centre and radius.

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 $(1,2)$ and vertex $(3,2)$.

Find foci and eccentricity of ellipse $9x^2+y^2=18$

Find the centre and directrix of hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$

Find a unit vector in the direction of the vector $\underset{‾}{v}=2\underset{‾}{i}+6\underset{‾}{j}$

Find the vector from the point A to the origin where $\overrightarrow{AB}=4\underset{‾}{i}-2\underset{‾}{j}$ and B is the point $(-2,5)$.

Find a and b so that the vectors $3\underset{‾}{j}-\underset{‾}{i}+4\underset{‾}{k}$ and $a\underset{‾}{i}+b\underset{‾}{j}-2\underset{‾}{k}$ are parallel.

Find the projection of $\underset{‾}{a}$ along $\underset{‾}{b}$ when $\underset{‾}{a}=3\underset{‾}{i}+\underset{‾}{j}-\underset{‾}{k}$, $\underset{‾}{b}=2\underset{‾}{i}-\underset{‾}{j}+\underset{‾}{k}$

Find a vector perpendicular to vectors $\underset{‾}{a}=2\underset{‾}{i}+\underset{‾}{j}+\underset{‾}{k}$, $\underset{‾}{b}=4\underset{‾}{i}+2\underset{‾}{j}-\underset{‾}{k}$

Differentiate between scalar quantity and vector quantity.

If $f(x)=\{\begin{array}{ll}3x& \text{if }x\le -2\\ x^2-1& \text{if }-2<x<2\\ 3& \text{if }x\ge 2\end{array}$ Discuss continuity at $x=2$ and $x=-2$

Differentiate $\frac{x^2+1}{x^2-1}$ w.r.t. $\frac{x-1}{x+1}$

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

Evaluate the integral $\int \frac{5x+8}{(x+3)(2x-1)}dx$

Evaluate ${\int }_0^{\pi \mathrm{/}4}{\cos }^{4}tdt$

Maximize $f(x,y)=2x+3y$ subject to the constraints $2x+y\le 8$, $x+2y\le 14$; $x\ge 0$, $y\ge 0$

Find a joint equations of the lines through the origin and perpendicular to the lines $x^2-2xy\tan \alpha -y^2=0$

Find an equation of circle which passes through the points $A(5,10)$, $B(6,9)$ and $C(-2,3)$

If $\underset{‾}{a}=4\underset{‾}{i}+3\underset{‾}{j}+\underset{‾}{k}$ and $\underset{‾}{b}=2\underset{‾}{i}-\underset{‾}{j}+2\underset{‾}{k}$, find a unit vector perpendicular to both $\underset{‾}{a}$ and $\underset{‾}{b}$. Also find the sine of angle between them.

Show that an equation of the parabola with focus at $(a\cos \alpha ,a\sin \alpha )$ and directrix $x\cos \alpha +y\sin \alpha +a=0$ is $(x\sin \alpha -y\cos \alpha {)}^2=4a(x\cos \alpha +y\sin \alpha )$.