MATHS MOST IMPORTANT QUESTION XI BIEK 2023

 

UNIT NO # 01 (COMPLEX NUMBER)

UNIT NO # 02 (MATRICES AND DETERMINANTS)

EXERCISE 2.1

1. Which of the following are symmetric or skew-symmetric matrices. (i) \(\begin{bmatrix}0 & -5 & 0\\-1 & 4 & 3 \\0 & 8 & 5\end{bmatrix}\)  (ii) \(\begin{bmatrix}0 & 1 & 3\\-1 & 0 & 5 \\-3 & -5 & 0\end{bmatrix}\)
2. Find the index of the matrix \(\begin{bmatrix}1& 1 & 3\\5 & 2 & 6 \\-2 & -1 & -3\end{bmatrix}\)

EXERCISE 2.2 

1.    If \(A=\begin{bmatrix}-2 & 1 & 0\\-1 & 4 & 3 \\0 & 8 & 5\end{bmatrix}\) then find: \(A^{2}-5A+4I\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
2.  Prove the identity: \(\left\{\begin{bmatrix}1& w& w^2 \\w & w^2 &1\\ w^2 & 1 & w\end{bmatrix} + \begin{bmatrix} w& w^2 & 1 \\w^2 & 1 & w\\ w & w^2 & 1 \end{bmatrix}\right\}\begin{bmatrix}1 \\ w \\ w^2 \end{bmatrix}=\begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}\).
3. If \(A= \begin{bmatrix}3& 2& 1 \\4 & -2 & 5\\ -1 & 0 & -7\end{bmatrix}\) and \(B= \begin{bmatrix}9& 4& 6 \\7 & -8 & 5\\ -1 & 0 & 3\end{bmatrix}\) then verify: (i) Commutative property under addition  (ii) \((AB)^{t}=B^{t}A^{t}\)

UNIT NO # 03 (VECTORS)

UNIT NO # 04 (SEQUENCE AND SERIES)

EXERCISE 4.2

1. \(a_{n-3}=2n-5\), find the nth term of the sequence.
2. Find the 21st term of the A.P if its 6th term is 11 and the 15th term is 47.
3. Which term of the sequence -2,4,10,..., is 148?
4. Find the nth term of the sequence \((\frac{6}{7})^{2},(\frac{11}{7})^{2},(\frac{16}{7})^{2},...\) (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
5. An object falling from rest, falls 12 meters during the first second, 24 meters during the next second month; 36 meters during the third second, and so on. How much will it fall during the 8th second?
6. A boy saves Rs. 200 at the end of the first week and increases his savings to Rs. 25 weekly. After how many weeks, his weekly saving will be Rs. 2000. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
7. A man deposits Rs 13,000 in a bank in the first month; Rs. 14,500 in the second month Rs. 16,000 in the third month and so on, Find how much he has to deposit in the bank at the end of a year. 

EXERCISE 4.3

1. Insert three A.M's between 3 and 11
2. if 5 and 8 are two A.M's between c and d, find c and d.
3. Find n so that \(\frac{a^{n+5}+b^{n+5}}{a^{n+4}+b^{n+4}}\) may be the A.M. between a and b. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 

EXERCISE 4.4

1. \(S_{n}=n(2n-1)\) , then find the series.
2. Find the sum of the first 100 natural numbers which are neither exactly divisible by 3 nor by 7. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
3. The sum of n terms of two arithmetic series is in the ratio of \(3n+2: n+1\). Find the ratio of their 8th term. (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
4. The sum of the three numbers in A.P. is 27 and their product is 405. Find the numbers.
5. Find five numbers in A.P.  whose sum is 25 and the sum of whose squares is 135.
6. The sum of Rs. 42,000 is distributed among five persons so that each person after the first receives Rs. 80 less than the preceding person, How much does each person receive? (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
7. A well-digging company charges Rs. 1,250 for the first meter, Rs. 1500 for the second meter, Rs, 1750 for the third meter, and so on. What is the depth of a well that costs Rs. 50,000.
8. A man borrows Rs. 25,000 and agrees to repay with a total profit of Rs. 10,000 in 10 installments being less than the preceding by Rs. 200. What should be his first installment?

EXERCISE 4.5

1. In a G.P, \(a_{1}=\frac{5}{9}, a_{6}=\frac{15625}{9}\). Find its nth term.
2. Find the nth term of the geometric sequence if \(\frac{a_{5}}{a_{3}}=\frac{4}{9} and a_{2}=\frac{4}{9}\). (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
3. How many terms are in G.P? \(\frac{1}{3},\frac{1}{12}, \frac{1}{48},..., \frac{1}{196608}\).
4. Find three consecutive numbers in G.P. whose sum is 39 and whose product is 729.
5. The number of bacteria in culture increased in G.P from 515,000 to 15,45,000 in 7 days. Find the daily rate to increase, assuming the rate of increase to be constant.
6. Find the profit on Rs.1000 for 5 years at 4% per annum compound profit.

EXERCISE 4.6

1. insert two G.M. between 3 and 81.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
2. The A.M. between two numbers is 29 and the geometric mean is 21, find the numbers.
3. Find n so that \(\frac{a^{n-2}+b^{n-2}}{a^{n-3}+b^{n-3}}\) may be the A.M. between a and b.
4. The A.M. of two positive integer numbers exceeds their (positive) G.M. by 2 and their sum is 20. Find the numbers.  (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
5. The A.M. between two numbers is 5 and their (positive) G.M. is 4. Find the numbers.

EXERCISE 4.7

1. Find the sum: (i)  7 + 77 + 777+... to n terms.  (ii) 0.2 + 0.22 + 0.222 + ... to n term
2. If \(a_{n}=(\frac{1}{4})^{n}\) than find its sum up to n terms.
3. Find the Vulgar fraction equivalent to the following recurring decimals. (i) \(1.\dot{7}\dot{4}\)  (ii) \(1. \dot{2}\dot{5}\dot{9}\).(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
4. If \(a =\frac{b}{2}+\frac{b^{2}}{4}+\frac{b^{3}}{8}+... \) and \(0<b<2\) than prove that \(b=\frac{2a}{1+a}\).  (Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
5. The sum of an infinite geometric series is half the sum of the squares of site terms. If the sum of its first two terms is \(4\frac{1}{2}\), find the series.

EXERCISE 4.8

1. If the 7th and 10th terms of an H.P. are \(\frac{1}{3}\) and \(\frac{5}{21}\) respectively, find its 14th and 20th terms.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
2. If the sum of first and sixth terms of H.P. is \(\frac{31}{116}\). Find the harmonic sequence if the first term is  \(\frac{1}{4}\).
3. The first term of H.P is \(\frac{-1}{3}\) and fifth term is \(\frac{1}{5}\). Find its 9th term.
4. If the pth term of an H.P. is q, the qth term is p: prove that the (p+q)th term is \(\frac{pq}{(p+q)}\). 

EXERCISE 4.9

1. Insert four H.Ms between: 4 and 20.(Prepared By PROF Dr. ZUBAIR & Dr. NISAR AHMED) 
2. Find n so that \(\frac{x^{n-5}+y^{n-5}}{x^{n-6}+y^{n-6}}\) may be the A.M. between x and y.
3. If 5 is the harmonic mean between 2 and b, find b?
4. Find the arithmetic and geometric means between 1 and 9. Also verify \(AH= G^{2}\)
5. The A.M of two numbers is 8 and H.M is 6. Find the numbers.
6. The H.M of two number is \(\frac{24}{5}\) and G.M is 6. Find the numbers.

UNIT NO # 05 (MISCELLANEOUS SERIES)

UNIT NO # 06 (PERMUTATION, COMBINATION AND PROBABILITY)

UNIT NO # 07 (MATHEMATICAL INDUCTION AND BINOMIAL THEOREM)

UNIT NO # 08 (FUNCTIONS AND GRAPHS)

UNIT NO # 09 (LINEAR PROGRAMMING LP)

UNIT NO # 10 (TRIGONOMETRIC IDENTITIES OF SUM AND DIFFERENCE OF ANGLES)

UNIT NO # 11 (APPLICATION OF TRIGONOMETRY)

UNIT NO # 12 (GRAPH OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS AND SOLUTION OF TRIGONOMETRIC EQUATIONS)