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- Definitions: FSc Part1 KPK
- Important Questions
- Multiple Choice Questions (MCQs)
- Solutions: Math 11 KPK
- 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:
- Question 18, Exercise 2.2 @math-11-kpk:sol:unit02
- ingular matrix.\\ $$A=\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]$$ $$|A|=a_{11}a_{22}-a_{12}a_{21}$$ $$AdjA=\left[ \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \\ \end{matrix} \right]$$ $$A^{-1}=\dfrac{1}{|A|}AdjA$$ $$A^{-1}=\dfrac{1}{a_{11}}a_{22}-a_{12}a_{21}\left[ \begin{matrix} a_{2
- Unit 04: Sequence and Series (Solutions) @math-11-kpk:sol
- === This is a forth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 4.1 (Solutions)"> * [[math-11-kpk:sol:unit04:ex4-1-p1|Question 1 & 2]] * [[math-11-kpk:sol:unit04:ex4-1-p2|Question 3 &4 ]] * [[math-11-kpk:sol:unit04:ex4-1-p3|Question 5]] * [[math-1
- Unit 07: Mathmatical Induction and Binomial Theorem (Solutions) @math-11-kpk:sol
- = This is a seventh unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 7.1 (Solutions)"> * [[math-11-kpk:sol:unit07:ex7-1-p1|Question 1]] * [[math-11-kpk:sol:unit07:ex7-1-p2|Question 2 ]] * [[math-11-kpk:sol:unit07:ex7-1-p3|Question 3]] * [[math-1
- Unit 06: Permutation, Combination and Probability (Solutions) @math-11-kpk:sol
- === This is a sixth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 6.1 (Solutions)"> * [[math-11-kpk:sol:unit06:ex6-1-p1|Question 1 & 2]] * [[math-11-kpk:sol:unit06:ex6-1-p2|Question 3 & 4 ]] * [[math-11-kpk:sol:unit06:ex6-1-p3|Question 5]] </panel> <p
- Unit 03: Vectors (Solutions) @math-11-kpk:sol
- === This is a third unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 3.2 (Solutions)"> * [[math-11-kpk:sol:unit03:ex3-2-p1|Question 1]] * [[math-11-kpk:sol:unit03:ex3-2-p2|Question 2]] * [[math-11-kpk:sol:unit03:ex3-2-p3|Question 3 & 4]] * [[mat
- Unit 02: Matrices and Determinants (Solutions) @math-11-kpk:sol
- == This is a second unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 2.1 (Solutions)"> * [[math-11-kpk:sol:unit02:ex2-1-p1|Question 1]] * [[math-11-kpk:sol:unit02:ex2-1-p2|Question 2]] * [[math-11-kpk:sol:unit02:ex2-1-p3|Question 3]] * [[math-11
- Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) @math-11-kpk:sol
- === This is a tenth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... lt" title="Exercise 10.1 (Solutions)"> * [[math-11-kpk:sol:unit10:ex10-1-p1|Question 1]] * [[math-11-kpk:sol:unit10:ex10-1-p2|Question 2]] * [[math-11-kpk:sol:unit10:ex10-1-p3|Question 3]] * [[math
- Unit 01: Complex Numbers (Solutions) @math-11-kpk:sol
- === This is a first unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 1.1 (Solutions)"> * [[math-11-kpk:sol:unit01:ex1-1-p1|Question 1]] * [[math-11-kpk:sol:unit01:ex1-1-p2|Question 2-3]] * [[math-11-kpk:sol:unit01:ex1-1-p3|Question 4]] * [[math-1
- Unit 05: Miscellaneous Series (Solutions) @math-11-kpk:sol
- === This is a fifth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, P... ult" title="Exercise 5.1 (Solutions)"> * [[math-11-kpk:sol:unit05:ex5-1-p1|Question 1]] * [[math-11-kpk:sol:unit05:ex5-1-p2|Question 2 & 3 ]] * [[math-11-kpk:sol:unit05:ex5-1-p3|Question 4 & 5]] * [[ma
- Question 13, Exercise 2.1 @math-11-kpk:sol:unit02
- ====Solution==== $$A=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23... trix} \right]$$ $$A^t=\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32... t=( A+A^t )$$ $$A+A^t=\left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{2... \end{matrix} \right]+\left[ \begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32
- Question 3, Exercise 2.2 @math-11-kpk:sol:unit02
- $. ====Solution==== Let $$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23... 33} \\ \end{bmatrix}$$ Then \begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12... 3}\left( a_{21}a_{32}-a_{22}a_{31} \right)\\ &=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}
- Question 1, Exercise 1.3 @math-11-kpk:sol:unit01
- plex coefficient. \begin{align}&z-4w=3i\\ &2z+3w=11-5i\end{align} ====Solution==== Given that \begin{align}z-4w&=3i …(i)\\ 2z+3w&=11-5i …(ii)\end{align} Multiply $2$ by (i), we get\\... s_{-}3w&=\mathop-\limits_{+}5i&\mathop+\limits_{-}11 \\ \hline 0&-11w&=11i &-11\\ \end{array} \] \begin{align}-11w&=11i-11\\ \implies w&=\dfrac{11-11i}{
- Question 11, Exercise 2.1 @math-11-kpk:sol:unit02
- ====== Question 11, Exercise 2.1 ====== Solutions of Question 11 of Exercise 2.1 of Unit 02: Matrices and Determinants. ... KPTB or KPTBB) Peshawar, Pakistan. =====Question 11===== Let $A=\begin{bmatrix}0 & 1 & -2 \\-1 & 0 &... & 0 \end{bmatrix}$ and $B=\begin{bmatrix}0 & -6 & 11 \\6 & 0 & -7 \\-11 & 7 & 0 \end{bmatrix}$. Veri
- Question 3, Exercise 2.1 @math-11-kpk:sol:unit02
- \right]\\ \implies A( B+C )&=\left[ \begin{matrix}11 & 15 & 8 \\3 & \,\,20 & \,6 \\\end{matrix}\righ... left[ \begin{matrix}\quad 2 & 13 & 3 \\-2 & 15 & 11 \\\end{matrix} \right].\end{align} and \begin{a... AC &=\left[ \begin{matrix}2 & 13 & 3 \\-2 & 15 & 11 \\ \end{matrix} \right]+\left[ \begin{matrix}9 ... begin{matrix}\quad2+9 & 13+2 & 3+5\\-2+5 & 15+5 & 11-5\\ \end{matrix} \right] \\ \implies AB+AC&=\lef
- Question 4, Exercise 2.1 @math-11-kpk:sol:unit02
- \end{matrix} \right]\\ &=\left[ \begin{matrix} 11 & 8 & 8 \\ 8 & 11 & 8 \\ 8 & 8 & 11 \\ \end{matrix} \right]\end{align} Now we take $$2A=\left[ \begin{matrix} 2... frac{1}{3}A^2-2A-9I \\ =&\left[ \begin{matrix} 11 & 8 & 8 \\ 8 & 11 & 8 \\ 8 & 8 & 11 \\ \