BISE Multan · Class 11 · Mathematics · 2025 · Second Annual · Group 1

Prove the rule of addition $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$

Prove that the sum of any two conjugate complex numbers is a real number.

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

Write down the power of set of $\{+,-,\times ,÷\}$

Without expansion show that $\mid \begin{array}{ccc}6& 7& 8\\ 3& 4& 5\\ 2& 3& 4\end{array}\mid =0$

Find $x$ and $y$ if $\left[\begin{array}{cc}x+3& 1\\ -3& 3y-4\end{array}\right]=\left[\begin{array}{cc}2& 1\\ -3& 2\end{array}\right]$

Solve the matrix equation for $A$, $\left[\begin{array}{cc}4& 3\\ 2& 2\end{array}\right]A-\left[\begin{array}{cc}2& 3\\ -1& -2\end{array}\right]=\left[\begin{array}{cc}-1& -4\\ 3& 6\end{array}\right]$

Find the inverse of matrix $\left[\begin{array}{cc}3& -1\\ 2& 1\end{array}\right]$

Solve $x(x+7)=(2x-1)(x+4)$ by factorization.

Define exponential equations.

Find three cube roots of unity.

Solve the equation $x^3+x^2+x+1=0$

Define rational fraction.

Resolve into partial fraction for $\frac{x^4}{1-x^4}$ without finding unknown constants.

Give the main points for resolving a rational fraction into partial fraction.

Find the sequence of $a_n-a_{n-1}=n+1$ and $a_4=14$

Define series with an example.

What do you mean for a sequence and real sequence?

The $n^{th}$ term of the A.P. is $3n-1$, find the A.P.

Give the fundamental principle of counting

Define permutation with the help of an example.

What do you mean for sample space?

Using Binomial theorem, find the value of $\sqrt[4]{17}$ to three decimal places.

Write down the general form of the binomial series.

Convert in radian of ${75}^{\circ }{6}^{\mathrm{\prime }}{30}^{\mathrm{\prime }\mathrm{\prime }}$

Find $\mathrm{\ell }$, if $\theta ={65}^{\circ }{20}^{\mathrm{\prime }}$ and $r=18mm$

Prove that $\tan \theta +\cot \theta =\mathrm{cosec}\theta \sec \theta$

Prove that $\sin (\frac{\pi }{2}+\alpha )=\cos \alpha$

Show that $\cos (\theta -{180}^{\circ })=-\cos \theta$

Show that $2\sin \alpha \cos \beta =\sin (\alpha +\beta )+\sin (\alpha -\beta )$

Find the period of $\cot \frac{x}{2}$

Let in a triangle $ABC$, $b=3$, $c=5$ and $\alpha ={120}^{\circ }$ find $a$

State any two law of cosine.

Define inscribed circle.

Find the area of triangle $ABC$ in which $a=200$, $b=120$, $\gamma ={150}^{\circ }$

Find the value of ${\cos }^{-1}(-\frac{1}{2})$

Find the solution of equation which lies in $[0,2\pi ]$ of $\sin x+\frac{\sqrt{3}}{2}=0$

Evaluate $\mid \begin{array}{cccc}3& 4& 2& 7\\ 2& 5& 0& 3\\ 1& 2& -3& 5\\ 4& 1& -2& 6\end{array}\mid$

Show that the roots of $x^2+(mx+c{)}^2=a^2$ will be equal if $c^2=a^2(1+m^2)$

Resolve $\frac{x^4}{1-x^4}$ into partial fractions.

Sum the series upto $n$ terms. $1\times 4+2\times 7+3\times 10+\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}$

In how many ways can $8$ books including $2$ on English be arranged on a shelf in such a way that the English books are never together.

Use mathematical induction to prove $\frac{1}{3}+\frac{1}{3^2}+\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}+\frac{1}{3^n}=\frac{1}{2}[1-\frac{1}{3^n}]$

$\mathrm{\forall }\cap \mathrm{\Sigma }{Z}^{+}$

Prove that, $\frac{\tan \theta +\sec \theta -1}{\tan \theta -\sec \theta +1}=\tan \theta +\sec \theta$

Prove that, $\sin \frac{\pi }{9}\sin \frac{2\pi }{9}\sin \frac{\pi }{3}\sin \frac{4\pi }{9}=\frac{3}{16}$

Define In-Circle and prove that $r=\frac{\mathrm{\Delta }}{s}$

Prove that ${\tan }^{-1}\frac{120}{119}=2{\cos }^{-1}\frac{12}{13}$