BISE Multan · Class 11 · Mathematics · 2025 · First Annual · Group 2

The transitive property of $\mathbb{R}$ is $\underset{‾}{}$.

The multiplicative inverse of $-3i$ is $\underset{‾}{}$.

If $A\subseteq B$, then $A\cup B$ is $\underset{‾}{}$

Let $A=[a_{ij}{]}_{3\times 4}$, then number of elements of $A$ is $\underset{‾}{}$.

Let $A=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$, then $A$ is called $\underset{‾}{}$.

The sum of the roots $a{x}^2+bx+c=0$ is $\underset{‾}{}$.

If $\omega$ is cube root of unity, then ${\omega }^9$ is $\underset{‾}{}$.

The fraction $\frac{(x-1{)}^2}{(x+1{)}^3}$ is called $\underset{‾}{}$.

The G.M between $x$ and $y$ is $\underset{‾}{}$.

If $a_n=2n+1$, then $a_5=\underset{‾}{}$

The value of $\lfloor 5$ is $\underset{‾}{}$.

The middle term of $(a+b{)}^{16}$ is $\underset{‾}{}$.

$\cos (-{60}^{\circ })$ is $\underset{‾}{}$.

$\mathrm{cosec}(\pi \mathrm{/}2+\theta )=\underset{‾}{}$

$1+\cos 2\theta =\underset{‾}{}$

Period of $\tan 2x$ is $\underset{‾}{}$.

$\sqrt{\frac{s(s-a)}{bc}}=\underset{‾}{}$

With usual notation $\frac{abc}{4\mathrm{\Delta }}=\underset{‾}{}$

${\tan }^{-1}(-1)=\underset{‾}{}$

The solution of $\sin x-\frac{1}{2}=0$ is $\underset{‾}{}$.

Find the closure property w.r.t addition of $\{1,-1\}$

Factorize $9a^2+16b^2$

Write two proper subsets of $\{a,b,c\}$

Construct truth table of $(p\wedge \sim p)\to q$

If $A=\left[\begin{array}{cc}1& -1\\ a& b\end{array}\right]$ and $A^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ find the value of $a$ and $b$.

Define identity matrix.

If $A$ is a square matrix of order $3$ then show that $\mathrm{\mid }kA\mathrm{\mid }=k^3\mathrm{\mid }A\mathrm{\mid }$

If $\left[\begin{array}{ccc}1& 2& 0\\ 3& 2& -1\\ -1& 3& 2\end{array}\right]$ show that $A+A^{\mathrm{\prime }}$ is symmetric.

Convert into quadratic equation $3^{2x-1}-12.3^x+81=0$

The sum of a positive number and its reciprocal is $\frac{26}{5}$. Find the number.

Find roots of the equation by using quadratic formula $5x^2-13x+6=0$

Define Reciprocal equation.

Resolve $\frac{x^2+1}{(x-1)(x+1)}$ into partial fraction.

If $\frac{x^2+15}{(x-1)(x^2+2x+5)}=\frac{A}{x-1}+\frac{B+Cx}{x^2+2x+5}$ and $A=2$ find values of $B$ and $C$

Resolve $\frac{7x+25}{(x+3)(x+4)}$ into partial fraction.

Find G.M. between $-2i$ and $8i$.

If $a,b,c$ and $d$ are in G.P prove that $a-b,b-c,c-d$ are in G.P.

If $\frac{1}{a},\frac{1}{b}$ and $\frac{1}{c}$ are in A.P show that $b=\frac{2ac}{a+c}$

Find the sum of $15$ terms of geometric sequence $1,\frac{1}{3},\frac{1}{9},\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}$

Prove that ${}^n{C}_r={}^n{C}_{n-r}$

State additive and multiplicative laws of probability.

How many signals can be given by $6$ flags of different colours when any number of flags can be used at a time?

Check the validity of $\frac{1}{3}+\frac{1}{3^2}+\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}+\frac{1}{3^n}=\frac{1}{2}(1-\frac{1}{3^n})$ for $n=3,4$

Find $6^{th}$ term in the expansion of ${(x^2-\frac{3}{2x})}^{10}$

Prove that: $\cos \theta +\tan \theta \sin \theta =\sec \theta$

Verify: $\sin 2\theta =2\sin \theta \cos \theta$, when $\theta ={45}^{\circ }$

Convert $\frac{11\pi }{27}$ into measure of sexagesimal system.

If $\alpha ,\beta$ and $\gamma$ are the angles of a triangle $ABC$, then prove that $\tan (\alpha +\beta )+\tan \gamma =0$

Prove that: $\tan (\frac{\pi }{4}-\theta )+\tan (\frac{3\pi }{4}+\theta )=0$

Show that $\sin 3\alpha =3\sin \alpha -4{\sin }^3\alpha$

Find the period of $\mathrm{cosec}10x$

Prove that $r=(s-a)\tan \frac{\alpha }{2}$

Prove that $R=\frac{abc}{4\mathrm{\Delta }}$

Show that $r_2=s\tan \frac{\beta }{2}$

In $\mathrm{\Delta }ABC$, $a=7$, $b=3$, $c=5$. Find $\alpha$

Show that ${\cos }^{-1}\frac{12}{13}={\sin }^{-1}\frac{5}{13}$

Define Trigonometric Equations. Give one example.

Show that $\mid \begin{array}{ccc}b+c& a& a^2\\ c+a& b& b^2\\ a+b& c& c^2\end{array}\mid =(a+b+c)(a-b)(b-c)(c-a)$

Show that the roots of $(mx+c{)}^2=4ax$ will be equal, if $c=\frac{a}{m}$; $m\mathrm{\ne }0$

Resolve $\frac{9}{(x+2{)}^2(x-1)}$ into partial fractions.

The sum of an infinite geometric series is $9$ and the sum of the squares of its terms is $\frac{81}{5}$. Find the series.

Find the value of $n$ and $r$ when ${}^n{C}_r=35$ and ${}^n{P}_r=210$

Use binomial theorem to prove that $1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12}+\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}=\sqrt{2}$

Find the values of all trigonometric functions for $\frac{-71}{6}\pi$

Reduce ${\cos }^4\theta$ to an expression involving only function of multiples of $\theta$, raised to the first power.

Prove that: $r_1+r_2+r_3-r=4R$

Prove ${\sin }^{-1}\frac{77}{85}-{\sin }^{-1}\frac{3}{5}={\cos }^{-1}\frac{15}{17}$