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- Unit 01: Complex Numbers (Solutions)
- Unit 02: Matrices and Determinants (Solutions)
- Unit 03: Vectors (Solutions)
- Unit 04: Sequence and Series (Solutions)
- Unit 05: Miscellaneous Series (Solutions)
- Unit 06: Permutation, Combination and Probability (Solutions)
- Unit 07: Mathmatical Induction and Binomial Theorem (Solutions)
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- Question 1, Exercise 1.1
- Question 2 & 3, Exercise 1.1
- Question 4, Exercise 1.1
- Question 5, Exercise 1.1
- Question 6, Exercise 1.1
- Question 7, Exercise 1.1
- Question 8, Exercise 1.1
- Question 9 & 10, Exercise 1.1
- Question 11, Exercise 1.1
- Question 1, Exercise 1.2
- Question 2, Exercise 1.2
- Question 3 & 4, Exercise 1.2
- Question 5, Exercise 1.2
- Question 6, Exercise 1.2
- Question 7, Exercise 1.2
- Question 8, Exercise 1.2
- Question 9, Exercise 1.2
- Question 1, Exercise 1.3
- Question 2, Exercise 1.3
- Question 3 & 4, Exercise 1.3
- Question 5, Exercise 1.3
- Question 6, Exercise 1.3
- Question 1, Review Exercise 1
- Question 2 & 3, Review Exercise 1
- Question 4 & 5, Review Exercise 1
- Question 6, 7 & 8, Review Exercise 1
- Question 1, Exercise 2.1
- Question 2, Exercise 2.1
- Question 3, Exercise 2.1
- Question 4, Exercise 2.1
- Question 5 & 6, Exercise 2.1
- Question 7, Exercise 2.1
- Question 8, Exercise 2.1
- Question 9, Exercise 2.1
- Question 10, Exercise 2.1
- Question 11, Exercise 2.1
- Question 12, Exercise 2.1
- Question 13, Exercise 2.1
- Question 1, Exercise 2.2
- Question 2, Exercise 2.2
- Question 3, Exercise 2.2
- Question 4, Exercise 2.2
- Question 5, Exercise 2.2
- Question 6, Exercise 2.2
- Question 7, Exercise 2.2
- Question 8,9 & 10, Exercise 2.2
- Question 11, Exercise 2.2
- Question 12, Exercise 2.2
- Question 13, Exercise 2.2
- Question 14 & 15, Exercise 2.2
- Question 16 & 17, Exercise 2.2
- Question 18, Exercise 2.2
- Question 19, Exercise 2.2
- Question 1, Exercise 2.3
- Question 2, Exercise 2.3
- Question 3, Exercise 2.3
- Question 4, Exercise 2.3
- Question 1, Exercise 3.2
- Question 2, Exercise 3.2
- Question 3 & 4, Exercise 3.2
- Question 5 & 6, Exercise 3.2
- Question 7, Exercise 3.2
- Question 7, Exercise 3.2
- Question 9 & 10, Exercise 3.2
- Question 11, Exercise 3.2
- Question 12, 13 & 14, Exercise 3.2
- Question 1, Exercise 3.3
- Question 2 and 3 Exercise 3.3
- Question 4 and 5 Exercise 3.3
- Question 6 Exercise 3.3
- Question 7 & 8 Exercise 3.3
- Question 9 & 10, Exercise 3.3
- Question 11, Exercise 3.3
- Question 12 & 13, Exercise 3.3
- Question 1 Exercise 3.4
- Question 2 Exercise 3.4
- Question 3 Exercise 3.4
- Question 4 Exercise 3.4
- Question 5 Exercise 3.4
- Question 6 Exercise 3.4
- Question 7 & 8 Exercise 3.4
- Question 9 Exercise 3.4
- Question 1 & 2 Exercise 3.5
- Question 3 & 4 Exercise 3.5
- Question 5(i) & 5(ii) Exercise 3.5
- Question 5(iii) & 5(iv) Exercise 3.5
- Question 6 Exercise 3.5
- Question 7 Exercise 3.5
- Question 8 Exercise 3.5
- Question 9 Exercise 3.5
- Question 1 Review Exercise 3
- Question 2 & 3 Review Exercise 3
- Question 4 & 5 Review Exercise 3
- Question 6 & 7 Review Exercise 3
- Question 8 & 9 Review Exercise 3
- Question 10 Review Exercise 3
- Question 1 and 2 Exercise 4.1
- Question 3 and 4 Exercise 4.1
- Question 5 Exercise 4.1
- Question 6 Exercise 4.1
- Question 1 and 2 Exercise 4.2
- Question 3 and 4 Exercise 4.2
- Question 5 and 6 Exercise 4.2
- Question 7 Exercise 4.2
- Question 8 Exercise 4.2
- Question 9 Exercise 4.2
- Question 10 Exercise 4.2
- Question 11 Exercise 4.2
- Question 12 & 13 Exercise 4.2
- Question 14 Exercise 4.2
- Question 15 Exercise 4.2
- Question 16 Exercise 4.2
- Question 17 Exercise 4.2
- Question 1 Exercise 4.3
- Question 2 Exercise 4.3
- Question 3 & 4 Exercise 4.3
- Question 5 & 6 Exercise 4.3
- Question 7 & 8 Exercise 4.3
- Question 9 & 10 Exercise 4.3
- Question 11 & 12 Exercise 4.3
- Question 13 & 14 Exercise 4.3
- Question 1 Exercise 4.4
- Question 2 & 3 Exercise 4.4
- Question 4 & 5 Exercise 4.4
- Question 6 & 7 Exercise 4.4
- Question 8 Exercise 4.4
- Question 9 Exercise 4.4
- Question 10 Exercise 4.4
- Question 11 Exercise 4.4
- Question 12 Exercise 4.4
- Question 1 Exercise 4.5
- Question 2 Exercise 4.5
- Question 3 Exercise 4.5
- Question 4 Exercise 4.5
- Question 5 & 6 Exercise 4.5
- Question 7 & 8 Exercise 4.5
- Question 9 & 10 Exercise 4.5
- Question 11 & 12 Exercise 4.5
- Question 13 & 14 Exercise 4.5
- Question 15 & 16 Exercise 4.5
- Question 1 Exercise 5.1
- Question 2 & 3 Exercise 5.1
- Question 4 & 5 Exercise 5.1
- Question 6 Exercise 5.1
- Question 7 & 8 Exercise 5.1
- Question 9 Exercise 5.1
- Question 1 Exercise 5.2
- Question 2 & 3 Exercise 5.2
- Question 4 & 5 Exercise 5.2
- Question 1 Exercise 5.3
- Question 2 Exercise 5.3
- Question 3 Exercise 5.3
- Question 4 Exercise 5.3
- Question 5 Exercise 5.3
- Question 6 Exercise 5.3
- Question 1 Exercise 5.3
- Question 2 & 3 Exercise 5.4
- Question 4 Exercise 5.4
- Question 1 Review Exercise 5
- Question 2 & 3 Review Exercise
- Question 4 Review Exercise
- Question 5 & 6 Review Exercise
- Question 7 Review Exercise
- Question 8 Review Exercise
- Question 9 Review Exercise
- Question 10 Review Exercise
- Question 1 and 2 Exercise 6.1
- Question 3 & 4 Exercise 6.1
- Question 5 Exercise 6.1
- Question 4 Exercise 6.1
- Question 5 Exercise 6.1
- Question 1 and 2 Exercise 6.2
- Question 3 and 4 Exercise 6.2
- Question 5 and 6 Exercise 6.2
- Question 7 and 8 Exercise 6.2
- Question 9 Exercise 6.2
- Question 10 Exercise 6.2
- Question 11 Exercise 6.2
- Question 12 Exercise 6.2
- Question 13 Exercise 6.2
- Question 14 and 15 Exercise 6.2
- Question 1 Exercise 6.3
- Question 2 Exercise 6.3
- Question 3 Exercise 6.3
- Question 4 Exercise 6.3
- Question 5 and 6 Exercise 6.3
- Question 7 and 8 Exercise 6.3
- Question 9 Exercise 6.3
- Question 9 Exercise 6.3
- Question 1 Exercise 6.4
- Question 2 Exercise 6.4
- Question 3 Exercise 6.4
- Question 4 Exercise 6.4
- Question 5 Exercise 6.4
- Question 6 Exercise 6.4
- Question 7 Exercise 6.4
- Question 1 and 2 Exercise 6.5
- Question 3 and 4 Exercise 6.5
- Question 5 and 6 Exercise 6.5
- Question 7 Exercise 6.5
- Question 8 Exercise 6.5
- Question 9 Exercise 6.5
- Question 10 Exercise 6.5
- Question 1 Review Exercise 6
- Question 2 Review Exercise 6
- Question 3 & 4 Review Exercise 6
- Question 5 & 6 Review Exercise 6
- Question 7 & 8 Review Exercise 6
- Question 9 & 10 Review Exercise 6
- Question 11 Review Exercise 6
- Question 1 Exercise 7.1
- Question 2 Exercise 7.1
- Question 3 Exercise 7.1
- Question 4 Exercise 7.1
- Question 5 Exercise 7.1
- Question 6 Exercise 7.1
- Question 7 Exercise 7.1
- Question 8 Exercise 7.1
- Question 9 Exercise 7.1
- Question 10 Exercise 7.1
- Question 11 Exercise 7.1
- Question 12 Exercise 7.1
- Question 13 Exercise 7.1
- Question 14 Exercise 7.1
- Question 15 Exercise 7.1
- Question 1 Exercise 7.2
- Question 2 Exercise 7.2
- Question 3 Exercise 7.2
- Question 4 Exercise 7.2
- Question 5 Exercise 7.2
- Question 6 Exercise 7.2
- Question 7 Exercise 7.2
- Question 8 Exercise 7.2
- Question 9 Exercise 7.2
- Question 10 Exercise 7.2
- Question 11 Exercise 7.2
- Question 1 Exercise 7.3
- Question 2 Exercise 7.3
- Question 3 Exercise 7.3
- Question 4 Exercise 7.3
- Question 5 and 6 Exercise 7.3
- Question 7 and 8 Exercise 7.3
- Question 9 Exercise 7.3
- Question 10 Exercise 7.3
- Question 11 Exercise 7.3
- Question 12 Exercise 7.3
- Question 13 Exercise 7.3
- Question 14 Exercise 7.3
- Question 1 Review Exercise 7
- Question 2 Review Exercise 7
- Question 3 & 4 Review Exercise 7
- Question 5 & 6 Review Exercise 7
- Question 7 & 8 Review Exercise 7
- Question 9 and 10 Review Exercise 7
- Question 11 Review Exercise 7
- Question 1, Exercise 10.1
- Question 2, Exercise 10.1
- Question 3, Exercise 10.1
- Question, Exercise 10.1
- Question 5, Exercise 10.1
- Question 6, Exercise 10.1
- Question 7, Exercise 10.1
- Question 8, Exercise 10.1
- Question 9 and 10, Exercise 10.1
- Question11 and 12, Exercise 10.1
- Question 13, Exercise 10.1
- Question 1, Exercise 10.2
- Question 2, Exercise 10.2
- Question 3, Exercise 10.2
- Question 4 and 5, Exercise 10.2
- Question 6, Exercise 10.2
- Question 7, Exercise 10.2
- Question 8 and 9, Exercise 10.2
- Question 1, Exercise 10.3
- Question 2, Exercise 10.3
- Question 3, Exercise 10.3
- Question 5, Exercise 10.3
- Question 5, Exercise 10.3
- Question 1, Review Exercise 10
- Question 2 and 3, Review Exercise 10
- Question 4 & 5, Review Exercise 10
- Question 6 & 7, Review Exercise 10
- Question 8 & 9, Review Exercise 10
Fulltext results:
- Unit 04: Sequence and Series (Solutions)
- ===== Unit 04: Sequence and Series (Solutions) ===== This is a forth unit of the book Mathematics 11 p... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... th or general term of an arithmetic sequence. * Solve problems involving arithmetic sequence. * Kno... rs is equal to n times their arithmetic mean. * Solve real life problems involving arithmetic series.
- Unit 06: Permutation, Combination and Probability (Solutions)
- nit 06: Permutation, Combination and Probability (Solutions) ===== This is a sixth unit of the book Ma... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... time and know the notation. * Apply $^nP_r$ to solve relevant problems of finding the number of an a... of $n$ different objects taken $r$ at a time. * Solve problems involving combination. * Define the
- Unit 07: Mathmatical Induction and Binomial Theorem (Solutions)
- t 07: Mathmatical Induction and Binomial Theorem (Solutions) ===== This is a seventh unit of the book ... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... ions. <panel type="default" title="Exercise 7.1 (Solutions)"> * [[math-11-kpk:sol:unit07:ex7-1-p1|Question 1]] * [[math-11-kpk:sol:unit07:ex7-1-p2|Ques
- Unit 03: Vectors (Solutions)
- ===== Unit 03: Vectors (Solutions) ===== This is a third unit of the book Mathematics 11 published by ... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... tors. <panel type="default" title="Exercise 3.2 (Solutions)"> * [[math-11-kpk:sol:unit03:ex3-2-p1|Question 1]] * [[math-11-kpk:sol:unit03:ex3-2-p2|Ques
- Unit 02: Matrices and Determinants (Solutions)
- ===== Unit 02: Matrices and Determinants (Solutions) ===== This is a second unit of the book Mathemati... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... a+ib$. * Define $|z| = \sqrt{a^2+b^2}$ as the absolute value or modulus of a complex number $z=a+ib$ ... real and imaginary parts of complex numbers. * Solve simultaneous linear equations with complex coef
- Unit 01: Complex Numbers (Solutions)
- ===== Unit 01: Complex Numbers (Solutions) ===== This is a first unit of the book Mathematics 11 publi... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... ib$. * Define $|z| = \sqrt{a^2+b^2}$ as the absolute value or modulus of a complex number $z=a+ib$ ... eal and imaginary parts of complex numbers. * Solve simultaneous linear equations with complex coef
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions)
- etric Identities of Sum and Difference of Angles (Solutions) ===== This is a tenth unit of the book Ma... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... nes) <panel type="default" title="Exercise 10.1 (Solutions)"> * [[math-11-kpk:sol:unit10:ex10-1-p1|Question 1]] * [[math-11-kpk:sol:unit10:ex10-1-p2|Q
- Unit 05: Miscellaneous Series (Solutions)
- ===== Unit 05: Miscellaneous Series (Solutions) ===== This is a fifth unit of the book Mathematics 11 ... awar, Pakistan. On this page we have provided the solutions of the questions. After reading this unit ... ots $ <panel type="default" title="Exercise 5.1 (Solutions)"> * [[math-11-kpk:sol:unit05:ex5-1-p1|Question 1]] * [[math-11-kpk:sol:unit05:ex5-1-p2|Ques
- Question 7, Exercise 10.2 @math-11-kpk:sol:unit10
- ====== Question 7, Exercise 10.2 ====== Solutions of Question 7 of Exercise 10.2 of Unit 10: Trigonomet... \sin }^{4}}\theta =\dfrac{1}{\sec 2\theta }$. ====Solution==== \begin{align}L.H.S&={{\cos }^{4}}\theta ... }\dfrac{\theta }{2}=\dfrac{2}{\sin \theta }$. ====Solution==== \begin{align}L.H.S&=\tan \dfrac{\theta }... eta }{1-\cos 2\theta }={{\cot }^{2}}\theta $. ====Solution==== \begin{align}L.H.S&=\dfrac{1+\cos 2\thet
- Question 5, Exercise 1.3 @math-11-kpk:sol:unit01
- ====== Question 5, Exercise 1.3 ====== Solutions of Question 5 of Exercise 1.3 of Unit 01: Complex Numb... awar, Pakistan. =====Question 5(i)===== Find the solutions of the equation ${{z}^{2}}+z+3=0$. ====Solution==== Given: $${{z}^{2}}+z+3=0.$$ According to th... &=\dfrac{-1\pm \sqrt{11}}{2}i\end{align} Thus the solutions of the given equation are $-\dfrac{1}{2}\pm
- Question 6, Exercise 1.3 @math-11-kpk:sol:unit01
- ====== Question 6, Exercise 1.3 ====== Solutions of Question 6 of Exercise 1.3 of Unit 01: Complex Numb... awar, Pakistan. =====Question 6(i)===== Find the solutions of the equation ${{z}^{4}}+{{z}^{2}}+1=0$. ====Solution==== $$z^4+z^2+1=0$$ $$z^4+2z^2+1-z^2=0$$ $... qrt{3}}{2}i$$ =====Question 6(ii)===== Find the solutions of the equation ${{z}^{3}}=-8$. ====Solutio
- Question 1 Exercise 5.3 @math-11-kpk:sol:unit05
- ====== Question 1 Exercise 5.3 ====== Solutions of Question 1 of Exercise 5.4 of Unit 05: Mascellaneou... {1}{2.3}+\dfrac{1}{3.4}+\ldots$ to $n$ terms. ====Solution==== The general term of the series is: $$T_n=\dfrac{1}{n(n+1)}$$ Resolving $T_n$ into partial fractions $$\dfrac{1}{n(n... {1}{3.5}+\dfrac{1}{5.7}+\ldots$ to $n$ terms. ====Solution==== Here $n$ term of the series is: $u_n=\df
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- ====== Question 7 Exercise 6.4 ====== Solutions of Question 7 of Exercise 6.4 of Unit 06: Permutation, ... obability of getting doublet of even numbers. ====Solution==== The sample space rolling a pair of dice ... probability of getting a sum less than $6$. ====Solution==== The sample space rolling a pair of dice ... e probability of getting a sum more than $7.$ ====Solution==== The sample space rolling a pair of dice
- Question 1 and 2 Exercise 6.2 @math-11-kpk:sol:unit06
- ====== Question 1 and 2 Exercise 6.2 ====== Solutions of Question 1 and 2 of Exercise 6.2 of Unit 06: P... n. =====Question 1(i)===== Evaluate $^6 P_6$ ====Solution==== \begin{align}^6 P_6&=\dfrac{6 !}{(6-6) !... =====Question 1(ii)===== Evaluate $^{20} P_2$ ====Solution==== \begin{align}^{20} P_2&=\dfrac{20 !}{(20... ====Question 1(iii)===== Evaluate $^{16} P_3$ ====Solution==== \begin{align}^{16} P_3&=\dfrac{16 !}{(16
- Question 1 and 2 Exercise 4.1 @math-11-kpk:sol:unit04
- ====== Question 1 and 2 Exercise 4.1 ====== Solutions of Question 1 and 2 of Exercise 4.1 of Unit 04: ... nd infinite sequences\\ $2,4,6,8, \ldots ,50$ ====Solution==== It is finite sequence whose last term i... and infinite sequences. $1,0,1,0,1, \ldots$. ====Solution==== It is infinite sequence, the last term m... nfinite sequences: $...,-4,0,4,8, \ldots, 60$ ====Solution==== This is infinite sequence. =====Questio