APPLIED MATHEMATICS - XII (Multiple choice Questions)(MCQs)
MODULAR ARITHMETIC
1) The least positive integers x satisfying 28 ≡ x(mod 6) is
a) 2 b) 4 c) 3 d) 1
2) If x is the least non negative integer satisfying 218 ≡ x(mod 7), then x²+ 1 is equals to
a) 1 b) 2 c) 5 d) 50
3) The unit's digit of 7¹⁰⁰ is
a) 1 b) 7 c) 2 d) 4
4) The smallest non negative integer congruent is 2796 (mod 7) is
a) 1 b) 2 c) 5 d) 3
5) The least non-negative remainder when 3²⁴² is divided by 13 is
a) 3 b) 6 c) 9 d) 1
6) The remainder when 2¹⁰⁰ is divided by 5 is
a) 1 b) 2 c) 3 d) 4
7) The remainder when 5⁶¹ is divided by 7 is
a) 1 b) 2 c) 4 d) 5
8) The unit digits in 11²⁴ is
a) 7 b) 8 c) 9 d) 1
9) The unit digit in 6⁶⁰⁰ is
a) 2 b) 4 c) 6 d) 8
10) The units digit in 3³⁰⁰ is
a) 1 b) 2 c) 3 d) 6
11) The value of x in the set {0,1,2,......,10} satisfying 3 - x ≡ 5(mod 11) is
a) 10 b) 9 c) 8 d) 2
12) The value of x in the set {0,1,2,.....,6} such that 53814 ≡ x is
a) 2 b) 4 c) 5 d) 3
1) b 2) b 3) a 4) d 5) c 6) a 7) d 8) d 9) c 10) a 11) b 12) c
ALLIGATION
1) 20 litres of a mixture contains milk and water in the ratio of 3:1. The amount of milk and water in the ratio 3:1. The amount of milk, in litres, to be added to the mixture so as to have milk and water in the ratio 4:1
a) 7 b) 4 c) 5 d) 6
2) A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk respectively, is
a) 5:4 b) 4:5 c) 1:5 d) 1:4
3) In what ratio must rice at Rs29.30 per kg mixed with rice at Rs30.80 per kg so that the mixture be worth Rs30 per kg ?
a) 7:8 b) 8:7 c) 3:8 d) 8:3
4) In what ratio water be mixed with milk to gain 50/3% on selling the mixture at cost price?
a) 1:6 b) 6:1 c) 2:3 d) 4:3
5)! How much water must be added to 60 litres of milk at 3/2 litres for Rs20 so as to have mixture worth Rs 32/3 a litre?
a) 10 litre b) 15 litres c) 5 litres d) 20 litres
6) Two vessels A and B contains milk and water mixed in the ratio 5:2 and 7:6 respectively. The ratio in which these two mixture be mixed to get a new mixture containing 900/13% milk, is
a) 2:7 b) 3:5 c) 5:2 d) 5:7
1) c 2) d 3) b 4) a 5) b 6) a
BOATS & STREAM AND PIPES & CISTERNS
1) A man can row upstream at 10 kmph and downstream at 18 kmph. Man's rate in still water in kmph is
a) 14 b) 4 c) 12 d) 10
2) A boat covers 8km in 1 hour along the stream and 2km in 1 hour against stream. The speed of the stream in kmph is
a) 3 b) 5 c) 2 d) 4
3) A man rows downstream 32 km and 14km upstream . If he takes 6 hours to cover each distance, then the speed of the stream in kmph is
a) 1/2 b) 1 c) 3/2 d) 2
4) A man rows at a speed of 8 kmph in still water to a certain distance upstream and back to the starting point in a river which flows at 4 kmph. The average speed of the journey in kmph, is
a) 12 b) 6 c) 4 d) 8
5) A boat goes downstream at u kmph and upstream at v kmph. The speed of the stream, in kmph, is
a) (u- v)/2 b) (u- v) c) (u+ v)/2 d) (u+ v)
6) A boat goes downstream at u kmph and upstream at v kmph. The speed of the boat in still water, in kmph, is
a) (u- v)/2 b) (u- v) c) (u+ v)/2 d) (u+ v)
7) A man rows 40 km upstream in 8 hours hours and a distance of 36 downstream in 6 hours. The speed of the stream is
a) 1/2 kmph b) 11/2 kmph c) 1 kmph d) 11 kmph
8) A boat rows 1 km in 5 minutes, along the stream and 6km in 1 hour against the stream . The speed of the stream is
a) 6 kmph b) 10 kmph c) 12 kmph d) 3 kmph
9) The speed of a boat in still water is 10 kmph. It is can travel 26km downstream and 14 km upstream in the same time, the speed of the stream, in kmph , is
a) 2 b) 3 c) 4 d) 5/2
10) A boat goes 12 km upstream in 48 minutes. If the speed of the stream is 2 kmph , the speed of boat in still water is
a) 13 kmph b) 17 kmph c) 6.5 kmph d) 8.5 kmph
11) A man can row a boat in still water at 15 kmph and speed of water current is 5 kmph . The distance covered by the boat downstream in 24 minutes is
a) 4 km b) 8 km c) 6 km d) 16 km
12) A man rows d km uptream and back again in t hours. If he can row in still water at u kmph and the rate of the stream is v kmph , then t =
a) 2ud/(u²- v²) b) (u²- v²)/d c) 2ud/(u²+ v²) d) uv/d
13) If a man goes 18km downstream in 4 hours and returns against the stream in 12 hours, then the speed of the stream in kmph is
a) 1 b) 3/2 c) 7/4 d) 3
14) If a boat goes 7 km upstream in 42 minutes and the speed of the stream 3 kmph, then the speed of the boat in still water is
a) 4.2 kmph b) 9 kmph c) 13 kmph d) 21 kmph
15) Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively. If both the pipes are opened together, the time taken by them to half -full the cistern is
a) 12 minutes b) 6 minutes c) 9 minutes d) 18 minutes
16) An outlet pipe can empty a cistern in 3 hours . The time taken by it to empty 3/2rd of the cistern is
a) 2 hour b) 3 hours c) 4 hour d) 6 hours
17) A pipe fills 3/7th part of a tank in 1 hour. The rest of the tank can be filled in
a) 7/3 hours b) 7/4 hours c) 4/3 hours d) 3/4 hours
18) A pipe can fill a tank in m hours and another pipe can empty it in n hours (n> m). If both the pipes are opened together, the tank will be filled in
a) (m - n) hours b) mn/(m+ n) c) mn/(m - n) d) mn/(n- m)
19) a cistern has two taps which filled it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the three pipes are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern?
a) 10 minutes b) 8 minutes c) 20 minutes d) 5 minutes
20) A pipe can fill a tank in 3 hours. Because of a leak in the tank the pipe takes 3 hours to 30 minutes to fill the tank . How much time will the leak take to empty the full tank?
a) 7 hours b) 13/2 hours c) 1/2 hours d) 21 hours
21) There is a leak in the bottom of the cistern . Before the leak, it could be filled in 9/2 hours. it now takes half hour longer. if the cistern is full, how long will the leakage take to empty the the full cistern ?
a) 30 hours b) 5 hours c) 45 hours d) 15 hours
22) A pipe A can fill a tank in 25 minutes and pipe B can empty the full tank in 50 minutes. The time taken by two pipes to fill the tank is
a) 20 minutes b) 25minutes c) 30 minutes d) 10 minutes
23) (3/4)th of a cistern is filled by a pipe in 12 minutes . How much time does it take to fill 1/2 of the cistern ?
a) 16 minutes b) 15 minutes c) 8 minutes d) 10 minutes
24) A pipe can empty 5/6 if a cistern in 20 minutes. What part of the cistern will be empties in 9 minutes .
a) 3/5 b) 4/5 c) 3/8 d) 5/8
25) A pipe can fill an empty cistern in 5 hours. A leak develops in the cistern due to which full cistern is emptied in 30 hours. With the leak, the cistern can be filled in
a) 8 hour b) 6 hours c) 10 hours d) 12 hours
26) Two pipes A and B can fill a tank in 30 hours and 20 hours respectively. Pipe B is kept open for half the time and both pipes are kept open for the remaining time. The number of hours taken to fill the tank is
a) 25 hours b) 40 hours c) 35 hours d) 15 hours
27) Two pipes A and B can fill a cistern in 10 minutes and 15 minutes respectively. Both the pipes are opened together, but after 3 minutes pipe B in turned off. How much time will the cistern take to be full?
a) 12 minutes b) 8 minutes c) 6 minutes d) 11 minutes
28) Two pipes A and B can fill a cistern in 10 minutes and 15 minutes respectively. Pipe C can empty the full cistern in 5 minutes. Pipes A and B are kept open for 4 minutes and then outlet pipe C is also opened. The cistern is emptied by the outlet pipe C in
a) 24 hours b) 20 hours c) 18 hours d) 30 hours
29) Pipe A can fill a tank in 6 times faster than a pipe B. If 5 can fill a tank in 21 minutes, then time taken by both the pipes to fill the cistern is
a) 3 minutes b) 9 minutes c) 9/2 minutes d) 7 minutes
30) Pipe A and B together can fill a cistern in 6 minutes. If A takes 5 minutes less than B to fill the cistern, then the time in which B alone can fill the cistern is
a) 15 minutes b) 10 minutes c) 30 minutes d) 25 minutes
31) A cistern normally takes 10 hours to be filled by a tap but because of one open outlet pipe, it takes 5 hours more
a) 20 hours b) 24 hours c) 30 d) hours d) 12
32) Two pipes can fill a tank separately in 12 and 20 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom. 30 minutes extra are taken for the cistern to be filled up. If the cistern is full, in what time would the leak empty it?
a) 120 hours b) 100 hours c) 115 hours d) 112 hours
33) Two pipes A and B can fill a cistern in 36 and 48 minutes respectively. Both pipes are opened together, after how many minutes should B be turned off, so that the cistern be filled in 24 minutes?
a) 6 minutes b) 16 minutes c) 10 minutes d) 12 minutes
34) Two pipes A and B can fill a tank in 20 and 16 hours respectively. Pipe B alone is kept open for 1/4 of time and both pipes are kept open for the remaining time. In how many hours, the tank will be full ?
a) 55/3 hours b) 20 hours c) 10 hours d) 25/2 hours
35) A tank has a leak which would empty it in 10 hours . A tap is turned on which delivers 4 litre a minute into the tank and now it emptied in 12 hours. The capacity of the tank is
a) 648 litres b) 1440 liters c) 1200 litres d) 1800 litres
36) 3 pipes A, B and C can fill a tank in 10, 20 and 30 hours respectively. if A is open for all the time and B and C are open for 1 hour each alternatively, then the tank will be full in
a) 6 hours 13/2 hour c) 7 hours d) 15/2 hours
37) A cistern is filled in 20 minutes by three pipes A, B and C. The pipe C is twice as fast as B and pipe B is thrice as fast as pipe A. How much time will pipe A alone take to fill the tank?
a) 200 minutes b) 205 minutes c) 352 minutes d) 180 minutes
38) Three pipes A, B and C can fill separately a cistern in 12, 16 and 20 minutes. Pipe A is opened first and pipe B and C are opened after every two minutes interval. The time after which the cistern will be full, is (after the opening of pipe A)
a) 36 minutes b) 32-1/47 c) 34-1/47 d) 36-1/47
39) Two pipes A and B can fill a cistern in 30 and 20 minutes respectively. They started filled a cistern together but pipe A is closed few minutes later and pipe C fills the remaining part of the cistern in 5 minutes. After how many minutes was pipe A closed ?
a) 9 minutes b) 10 minutes c) 12 minutes d) 14minutes
1) B 2) A 3) c 4) c 5) a 6) d 7) c 8) b 9) c 10) c 11) b 12)d 13) B 14) C 15) B 16)A 17) C 18) C 19) a 20) d 21) c 22) b 23) C 24) c 25) b 26) d 27) b 28) b 29) a 30) a 31) C 32) a 33) b 34) c 35) b 36) c 37) a 38) d 39) a
PARTNERSHIP
1) A, B, C invested Rs45000, Rs70000 and Rs90000 respectively to start a business. At the end of 2 years , they earned a profit of Rs164000. B's share in the profit is
a) Rs3600 b) Rs56000 c) Rs64000 d) Rs72000
2) A and B started a business investing 85000 and Rs15000 respectively. In what ratio the profit earned after 2 years be divided between A and B respectively?
a) 3: 6 b) 15:23 c) 17:3 d) 3:5
3) In a partnership, A 2000 invests of the capital for 1/6 of the time, B invests of 1/3 of the capital for 1/3 th of the time and C invests the rest of the capital for the whole time. Out of a profit of Rs4600, B's share is
a) Rs650 b) Rs800 c) Rs 960 d) Rs1000
4) A and B enter into partnership with Rs50000 and Rs60000 respectively. C joins them after x months, contributing Rs70000 and B leaves x months before the end of the year. if they share the profit and the ratio 20:18 :21, then x is
a) 3 b) 6 c) 8 d) 9
5) A, B, and C are three partners. They altogether invested Rs14000 in business. At the end of the year, A got Rs337.50, B Rs 1125 C Rs637.50 as profit. The difference between the investment of B and A was
a) 2200 b) 3200 c) 4200 d)5250
6) A, B, and C subscribe Rs50000 for a business. A subscribes Rs4000 more than B and Rs5000 more than C. Out of total profit of Rs35000, A receives
a) 8400 b) 11000 c) 13600 d) 14700
7) A,B and C are partners in the business. Their shares are in the proportion of 1/3 :1/4 : 1/5. A withdraws half of his capital after 15 months and after another 15 months , a profit of Rs4340 is divided . The shars of C is
a) 1240 b) 1245 c) 1360 d) 1550
8) Anu is a working partner and Vimala is a sleeping partner in a business. Anu put Rs 5000 and Bimla puts Rs6000. Anu receives 12.5% of the profit for managing the business and the rest is divided in proportion to their capitals. What does each get out of a profit of Rs880?
a) 400,480 b) 450, 430 c) 460, 420 d) 470, 410
9) x and y are partners in a business. x 1/3 of the capital for 9 months and y received 2/5 of the profit. For how long was y's money used in the business?
a) 2 months b) 3 months c) 4 months d) 5 months
10) The ratio of investment of two partners A and B 11:12 and the ratio of their profits is 2:3. If A invested the money for 8 months, then for how much time B invested his money?
a) 11 months b) 10 months c) 9 months d) 5 months
11) A, B and C enters into a partnership. Their capital contribution is in the ratio 21: 18 : 14. At the end of the business form they share profits in the ratio 15: 8 : 9. The ratio of time for which A, B and C invest their capitals is
a) 81: 56:90 b) 90:56:81 c) 56:90:81 d) 81:90:56
1) b 2) c 3) b 4) a 5) d 6) d 7) a 8) c 9) b 10) a 11) b
RACES AND GAMES
1) In a race of 400 meter, A can give B a start 20 m and C a start 39 m. How much start can B gives to C in the same 20 race?
a) 20m b) 15m c) 18m d) 25m
2) In a 2 km race, A can give B a start of 200 metres and C a start 560 metres . In the same race, how much start can B gives to C?
a) 500m b) 350m c) 300m d) 400m
3) In a 100m race , A can give B a start of 10m and C a start of 28m. How much start can B gives to C in the same race ?
a) 18m b) 20m c) 27m d) 9 m
4) In a 500 m race, the race of speeds of two contestants A and B is 3:4. If A gets a start of 140 m, then he wins by
a) 60 b) 40m c) 20m d) 10 m
5) If in a race , A beats B by 10m and C by 30m, then in a 180m race , B will beat C by
a) 5.4m b) 4.5m c) 5m d) 6m
6) In a hundred metre race A and B are two participants. If A runs at 5 kmph and A gives B a start of 8m and still beats him by 8seconds, then the speed of B is
a) 5.15 kmph b) 4.14 kmph c) 4.25 kmph d) 4.4 kmph
7) If in a 600m race , A can beat B by 50m and in a 500 m race, B can beat C by 60m. Then, in a 400 m race, A will beat C by
a) 70m b) 77m c) 232/7 m d) 155/2 m
8) In a kilometre race, A beats B by 50 metres or 10 seconds. What time does A take to complete the race?
a) 200 seconds b) 190 seconds c) 210 seconds d) 150 seconds
9) A runs a 1000m race in 4.5 minutes will B runs the same race in 5 minutes. How many metres start can A give to B in a 1000 m race ?
a) 150m b) 125m c) 130 m d) 100 m
10) In a 1000m race , A beats B by 100m and in a 800m race B beats C by 100m. By how many metres will A beat C in a 600m race .
a) 57.5m b) 127.5m c) 150.7m d) 98.6m
11) A is 7/3 times as fast as B. If A gives B a start of 80 meters, how long should the race course be so that the both of them reach at the same time ?
a) 170 metres b) 140 metres c) 160 metres d) 150 m
12) In a 1000m race , A can beat B by 100m. In a race of 400m, B can beat C by 40m. By how many metres will A beat C in a race of 500 m?
a) 85m b) 95m c) 105m d) 115m
13) In a 500m race, the ratio of the speeds of the two contestant A and B is 3:4. If A has a start of 140m, then A wins by
a) 60m b) 40 m c) 20m d) 10m
14) In a 400 m race, A gives B a start 5 seconds and beats him by 15 m. In other race of 400 m, A beats B by 50/7 second 0s. Their respective speeds are
a) 6m/sec, 7m/sec b) 5m/sec, 7m/sec c) 8m/sec, 7m/sec d) 9m/sec, 7m/sec
15) In a race of 200 pm, B can give a start of 10m to A and C can give a start of 20m to B. The start that C can give to A in the same race is
a) 27m b) 29m c) 30m d) 25m
16) In a game of 100 points, A can give B 10 points and C 18 points. Then, B can give C
a) 35 : 12 b) 45:41 c) 55:25 d) 35:41
17) In a game, A can give B 25 points, A can give C 40 points and B can give C 20 points. How many points make the game ?
a) 120 b) 100 c) 150 d) 80
1a 2d 3b 4c 5d 6b 7c 8b 9d 10b 11b 12b 13c 14c 15b 16b 17b
NUMERICAL INEQUALITIES AND LINEAR INEQUATIONS
1) If x<7, then
a) -x<-7 b) -x≤ -7 c) -x > - 7 d) -x ≥ -7
2) If 3x +17< -13, then
a) x ∈(10, ∞) b) x ∈[10, ∞) c) x ∈(-∞,10) d) x ∈[-10,10)
3) Given that x, y and b are real numbers and x < y, b> 0, then
a) x/b < y/b b) x/b ≤ y/b c) x/b > y/b d) x/b ≥ y/b
4) If x is a real number and |x|< 5, then
a) x≥5 b) -5< x < 5 c) x≤ -5 d) -5≤ x ≤ 5
5) If x and a are real numbers such that a > 0 and |x| > a, then
a) x∈(-a, ∞) b) x ∈[-∞,a) c) x∈(-a, a) d) x ∈(-∞, -a)U(a, ∞)
6) If | x -1|> 5, then
a) x ∈(-4, 6) b) x∈[-4,6] c) x ∈(-∞, -4) U(6,∞) d) x ∈(-∞, - 4) U[6, ∞)
7) If | x +2|≤ 9, then
a) x ∈(-7, 11) b) x∈[-11,7] c) x ∈(-∞, -7) U(11,∞) d) x ∈(-∞, - 7) U[11, ∞)
8) The inequality representing the following graph is
a) |x| < 3 b) |x| ≤ 3 c) |x| > 3 d) |x| ≥ 3
9) The linear inequality representing the solution set given in figure is
a) |x| < 5 b) |x| > 5 c) |x| ≥ 5 d) |x| ≤ 5
<----------------•--------•---------------->
-∞ ∞
10) The solution set of the inequation | x + 2|≤ 5 is
a) (-7,5) b) [-7,3] c) [-5,5] d) (-7,3)
11) If |x -2|/(x -2) ≥ 0, then
a) x ∈[2,∞ ) b) x∈(2, ∞] c) x ∈(-∞, 2) d) x ∈(-∞, 2)
12) If |x -2|/(x -2) ≥ 0, then
a) x ∈[2,∞ ) b) x∈(2, ∞] c) x ∈(-∞, 2) d) x ∈(-∞, 2)
13) Solution of a linear inequality in variable x is represented on the number line as shown in the figure. The solution can also be described as
<-----------------------° ---------------------->
-∞ 9/2 ∞
a) x ∈(-∞,5 ) b) x∈(-∞,5] c) x ∈[5,∞) d) x ∈(5,∞)
14) The shaded part of the number line in figure can also be represented as
a) x ∈(9/2,∞ ) b) x∈[9/2, ∞) c) x ∈[-∞, 9/2) d) x ∈(-∞, 9/2)
<-----------------------° ---------------------->
-∞ 5/2 ∞
15) The shaded part of the number line in figure can also be described as
<------------°---------° ---------------------->
-∞ 1 2 ∞
a) (-∞,1) U(2, ∞) b) (-∞, 1] U[2,∞) c) (1,2) d) [1,2]
1c 2a 3a 4b 5d 6c 7b 8b 9c 10b 11b 12d 13d 14b 15a
FILL IN THE BLANKS
1) If x ≥ -3, then x +5______2.
2) If - x ≤ -4, then 2x______8.
3) If 1/(x-2 < 0, then x______2.
4) If |x - 1|≤ 2, then -1______x< 3.
5) If |3x - 7|> 2, then x______5/3 or, x_____3.
6) If -4x ≥2, then x______-3.
7) If -3x/4 ≤ -3, then x______4.
8) If x > y and z< 0, then - xz______ - yz.
9) The solution set of the inequation |x +1|< 3 is______.
10) The solution set of the inequation |x +2|> 5 is______.
11) If (x -3)/|x - 3| ≥ 0, then x belongs to the interval ______ .
12) The solution set of the inequation (| x| +1)/(|x| - 1) < 0 is _____ .
1 ≥ 2) ≥ 3) < 4) ≤ , ≤ 5) < , > 6) ≤ 7) ≥ 8) > 9) (-4,2) 10) (-∞,7) U(3, ∞) 11) (3,∞) 12) (-1,1)
VERY SHORT ANSWER QUESTIONS
1) Write the solution set of the inequation x²/(x -2) > 0.
2) Write the solution set of the inequation x + 1/x ≥ 2.
3) Write the set of the values of x satisfying the inquation (x²- 2x +1)(x -4)≥ 0.
4) Write the solution set of the equation |2 - x| = x -2.
5) Write the set of values of x satisfying |x -1| ≤ 3 and |x - 1|≤ 1.
6) Write the solution set of the inquation |1/x - 2| < 4.
7) Write the number of integral solutions of (x +2)/(x²+1) > 1/2.
8) Write the set of values of x satisfying the inequation 5x +2 < 3x + 8 and (x + 2)/(x -1) < 4.
9) Write the solution set of |x + 1/x| > 2.
10) Write the solution set of the inquation |x - 1|≥|x -3|.
1) [2, ∞) 2) (0,∞) 3) ( -∞,4) 4) (2, ∞) 5) [2,4] 6) (-∞, -1/2) U (1/6, ∞) 7) 3 8) (-∞,1) U (2,3) 8) R - {-1,0,1} 10) [2, ∞)
MATRICES
1) 1 0 0
If A= 0 1 0 then A² is equal to
a b -1
a) a null matrix b) a unit matrix c) - A d) A
2) If A and B are symmetric matrices of the same order then ABᵀ - BAᵀ is a
a) skew-symmetric matrix
b) null matrix
c) symmetric matrix d) none
3) If A and B are two Matrix such that AB= A and BA= B, then B² is equals to
a) B b) A c) 1 d) 0
4) If AB= A and BA= B, where A and B are square Matrix, then
a) B²= BA and A²= A
b) B²≠ B and A²= A
c) A²≠ A and B²= B
d) A²≠ A and B²≠ B
5) If A and B are two matrices such AB = B and BA= A, then A²+ B² is equal to
a) 2AB b) 2BA c) A+ B d) AB
6) The matrix 1 0 0
A = 0 2 0
0 0 4 is
a) identity Matrix b) symmetric matrix c) skew-symmetric matrix diagonal matrix
7) If the matrix AB zero , then
a) it is not necessary that either A= O or, B= O
b) A= O or B= O
c) A= 0 and B= O
d) all the above statements are wrong
8) The matrix 0 -5 8
A= 5 0 12
-8 -12 0
a) diagonal Matrix b) symmetric matrix c) skew-symmetric matrix d) scalar matrix
9) null matrix b) singular matrix c) unit matrix d) non-singular matrix
10) If n 0 0 a₁ a₂ a₃
A= 0 n 0 & B= b₁ b₂ b₃
0 0 n c₁ c₂ c₃ then AB is equal to
a) B b) nB c) Bⁿ d) A+ B
11) If A = 2 -1 3 2 3
-4 5 1 & B= 4 -2
1 5 then
a) only AB is defined b) only BA is defined c) AB and BA both are defined d) AB and BA both are not defined
12) 1 2 x 1 -2 y
If A= 0 1 0 & B= 0 1 0
0 0 1 0 0 1 and AB = I₃ , then x+ y equal to
a) 0 b) -1 c) 2 d) none
13) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)½= A²+ B², then values of a and b are
a) a= 4, b= 1 b) a= 1, b= 4 c) a= 0, b= 4 d) a= 2, b= 4
14) If A= x y
z -x is such that A²= I, then
a) 1+ x²+ yz = 0 b) 1- x²+ yz = 0 c) 1- x²- yz = 0 d) 1+ x²- yz = 0
15) If S= [sᵢⱼ] is a scalar matrix such that sᵢⱼ = k and A is a square matrix of the same order, then AS = SA = ?
a) Aᵏ b) k + A c) kA d) kS
16) if A is a square Matrix such that A²= A, then (I+ A)³ - 7A is equals to
a) A B) I - A c) I d) 3A
17) If a matrix A is both symmetric and skew-symmetric, then
a) A is a diagonal Matrix b) A is a zero matrix c) A is a scalar matrix d) A is a square matrix
18) The matrix 0 5 -7
-5 0 11
7 -11 0 is
a) a skew-symmetric b) a symmetric matrix c) a diagonal Matrix d) an upper triangular matrix
19) If A is a square Matrix, then AA is a
a) skew-symmetric matrix b) symmetric matrix c) diagonal Matrix d) none
20) If A and B are symmetric matrices, then ABA is
a) symmetric matrix b) skew-symmetric matrix c) diagonal Matrix d) scalar matrix
21) If A= 5 x
y 0 and A= Aᵀ, then
a) x= 0, y= 5 b) x+ y= 5 c) x=0 d) none
22) If A is 3 x 4 matrix and B is a matrix such that AᵀB and BAᵀ are both defined . Then , B is of the type
a) 3 x 4 b) 3 x 3 c) 4 x 4 d) 4 x 3
23) If A= [aᵢⱼ] is a square Matrix of even order such that aᵢⱼ = i²- j², then
a) A is a skew-symmetric Matrix and |A| = 0.
b) A is symmetric Matrix and |A| is a square
c) A is symmetric Matrix and |A| = 0. d) none
24) If A and B are square matrices of the same order, then (A+ B)A- B) is equal to
a) A²- B² b) A²- BA - AB - B² c) A²- B² + BA - AB d) A² - BA + B² + AB
25) 2 0 -3
If A= 4 3 1
-5 7 2 is expressed as the sum of a symmetric matrix and skew-symmetric matrix, then the symmetric matrix is
a) 2 2 -4 b) 2 4 -5
2 3 4 0 3 7
-4 4 2 -3 1 2
c) 4 4 -8 d) 1 0 0
4 6 8 0 1 0
-8 8 4. 0 0 1
26) Out of the following matrices , choose that Matrix which is a Scalar Matrix.
a) 0 0 b) 0 0 0 c) 0 0 d) 0
0 0 0 0 0 0 0 0
0 0 0
27) The number of all possible matrices ordered 3 x 3 with each entry 0 OR 1 is
a) 27 b) 18 c) 81 d) 512
28) Which of the given values of x and y make the following pairs of the matrices equal?
a) 3x+7 5 and 0 y-2
y+1 2 -3x 8 4
a) x= -1/3 y= 7 b) x= -2/3 y= 7 c) x= -1/3 y= -2/5 d) not possible to find
29) If A= 0 2 and kA = 0 3a
3 -4 2b 24 then the values of k, a, b are respectively
a) -6,-12,-18 b) -6,4,9 c) -6,-4,-9 d) -6,12,18
30) If matrix A= [aᵢⱼ]₂ ₓ ₂ , where aᵢⱼ = { 1, if i≠ j
0, if I+ j , then A² is equal to
a) I b) A c) O d) - I
31) The trace of the matrix A = 1 -5 7
0 7 9
11 8 9 is
a) 17 b) 25 c) 3 d) 12
32) If A= [aᵢⱼ] is a Scalar Matrix of order n x n such that aᵢⱼ = k for all i, then trace of A is equal to
a) nk b) n + k c) n/k d) none
33) The matrix A= 0 0 4
0 4 0
4 0 0 is a
a) square Matrix b) diagonal Matrix c) unit Matrix d) none
34) The number of possible matrices of order 3 x 3 with each entry 2 or 0 is
a) 9 b) 27 c) 81 d) none
35) If 2x+ y 4x = 7 7y - 13
5x -7 4x y x+ 6 , then the value of x + y is
a) 3, 1 b) 2, 3 c) 2, 4 d) 3, 3
36) If A is a square Matrix such that A²= I, then (A - I)³+ (A + I)³ - 7A is equals to
a) A B) I - A c) I + A d) 3A
37) If A and B are two matrices of order 3 x m and 3 x n respectively and m= n, then the order of 5A - 2B is
a) m x 3 b) 3 x 3 c) m x n d) 3 x n
38) If A is a matrix of order m x n and B is a matrix such that ABᵀ and BᵀA are both defined , then the order of Matrix B is
a) m x n b) n x n c) n x m d) m x n
39) If A and B are matrices of the same order, then ABᵀ- BᵀA is a
a) skew-symmetric matrix b) null matrix c) unit matrix d) symmetric matrix.
Answers
1b 2d 3a 4a 5c 6d 7a 8c 9a 10 b 11 c 12 a 13 b 14c 15 c 16 c 17b 18a 19d 20a 21c 22a 23d 24c 25a 26a 27d 28d 29c 30a 31a 32a 33d 34d 35b 36a 37d 38d 39a
FILL IN THE BLANKS
1) IF A and B are two matrices of order a x 3 and 3 x b respectively such that AB exists and is of order 2 x 4. Then , (a, b)= ______
2) If P and Q are two Matrices of orders 3 x n and n x p respectively then the order of matrix PQ is______
3) If A= -1 2 3x
2y 4 -1
6 5 0 is a symmetric matrix, then the value of 2x + y is _____
4) If a, b are positive integers such that a< b and [a b] [a
b = 25, then (a, b)= ___
5) If A= 1/3 2 & B= 3 6
0 2x -3 0 -1 and AB = I, then x = _____
6) If A= x 1
-1 -x satisfies the equation A²= O, then x= ______
7) If m x n matrix and B is a matrix such that both AB and BA are defined , then the order of B is ______
8) 0 2 0 1 2 3
If A =. 0 0 3 & B= 3 4 5
-2 2 0 5 -4 0
Then (AB)₃₃ = _____
9) 3 0 0
If A= 0 3 0
0 0 3
then A⁴= ____
10) If A= diag(2, -1, 3), B= diagonals (-1, 3, 2), then A²B is ____
11) if A= 1
-1 & B= 2 1 -1
2
Then AB = ___::'
12) If A= x 1
1 0 and A² is the identity Matrix, then x =______
13) If A= eˣ eʸ & B= 1 1
eʸ eˣ 1 1 and eA = B
then x= _____, y=_____
14) If A= 0 2 and kA = 0 3a
3 -4 2b 24 , then (k,a,b)= _____
15) The negative of a matrix obtained by multiplying it by______
16) If A is 3 x 4 matrix and B is a matrix such that AᵀB and BAᵀ are both defined . Then the order of B is _______
17) 1
If A= 2
3 then AAᵀ= ____
18) If A is a non-singular matrix, then (Aᵀ)⁻¹ = ____
19) If A, B and C are m x n, n x p and p x q matrices respectively such that (BC)A is defined, then m= ____
20) 2 0 0
If A = 0 2 0
0 0 2 such that A⁵= λA, then λ= _____
21) If the Matrices A= a b and B= 1 1
c d 0 1commute with each other, then C=___
22) If A= 4 x+ 2
2x -3 x+1 is a symmetric matrix, than x= ____
23) If A and B are two skew-symmetric matrices of same order, then AB is symmetric if _____
24) If A and B are matrices of the same order, then (3A - 2B)ᵀ is equals to ____
25) Additional Matrices is defined if border of the matrices is____
26) if A and B are symmetric Matrices of the same order, then AB is symmetric iff___
27) If A is symmetric Matrix, then BᵀAB is____
28) if A is a skew-symmetric matrix, then A² is a ____ matrix.
29) If A is a symmetric matrix, then A³ is a _____Matrix.
30) If A is a skew-symmetric matrix, then kA is a ____(k is any scalar).
31) if A and B are symmetric matrices of the same order, then
a) AB - BA is a ____ .
b) BA - 2AB is a _______.
32) In applying one or more operations while finding A⁻¹ by elementary row operations , we obtain all zeros in one or more row, then A⁻¹ _____ .
33) The product of any Matrix by the scalar____ is the null matrix .
34) A Matrix which is not a square Matrix is called _____ matrix.
35) The sum of the two symmetric matrices is always _____Matrix.
36) A and B are square matrices of the same order, then_____ .
a) (AB)ᵀ= _____
b) (kA)ᵀ= _____
c) {k(A- B)}ᵀ= _____
Where k is any scalar.
37) ______ matrix is both symmetric and skew-symmetric matrix.
38) matrix multiplication is _____over Matrix addition.
Answer
1) (2,4) 2) 3 x p 3) 5 4) (3,4) 5) 1 6) ±1 7) n x m 8) 4 9) 13/3 10) diag(-4, 3, 18)
11) 2 1 -1
-2 -1 1
4 2 -2
12) 0 13) -1,-1 14) (-6,-4,-9) 15) -1 16) 3x4
17) 1 2 3
2 4 6
3 6 9
18) (A⁻¹)ᵀ 19) q 20) 16 21) 0 22) 5 23) AB = BA 24) 3Aᵀ - 2Bᵀ 25) same 26) AB = BA 27) Symmetric 28) Symmetric 29) Symmetric 30) skew-symmetric 31) skew-symmetric, neither Symmetric nor skew-symmetric 32) does not exist 33) zero 34) rectangular 35) skew-symmetric 36) BᵀAᵀ , kAᵀ, k(Aᵀ - Bᵀ) 37) null matrix 38) distributive
Very Short Questions
1) if A is an m x n matrix B is n x p matrix does AB exists ? If yes, write its order.
2) If A=2 1 4 & B= 3 -1
4 1 5 2 2
1 3 , Write the order of AB and BA.
3) If A= 4 3 and B= -4
1 2 3 , write AB.
4) If A= 1
2 write AAᵀ.
3
5) Give an example of two non-zero 2 x 2 matrices A and B such that AB= O.
6) if A= 2 3
5 7, find A+ Aᵀ.
7) If A= 0 a -3
2 0 -1
b 1 0 is skew-symmetric, find the values of x and y.
8) Let A and B be matrices of order 3x2 and 2x4 respectively. Write the order of matrix AB.
9) Let A = 3 5
7 9 is written as A= P + Q, where as A = P + Q, where P is symmetric and Q is skew-symmetric matrix, then write the matrix P.
10) If A= 1 0 B= x 0
y 5 1 -2 with the relation A+ 2B = I, where I is unit matrix. Find x and y.
11) If A= 1 -1
-1 1 , satisfies the matrix equation A²= kA, write the value of k.
12) If A= 1 1
1 1 satisfies A⁴= k A, then write the value of k
13) If A= -1 0 0
0 -1 0
0 0 -1 find A².
14) If A= -1 0 0
0 -1 0
0 0 -1 find A³
15) If A= -3 0
0 -3 find A⁴.
16) If A= x 2 B= 3
4 with the relation AB= 2, find x.
17) If A= [aᵢⱼ] is a 2x2 matrix such that aᵢⱼ= i + 2j, write A.
18) Write matrix A satisfy A+ B = C where
B= 2 3 C= 3 -6
-1 4 -3 8
19) If A= [aᵢⱼ] is a square matrix such that aᵢⱼ = i²- j², then write whether A is symmetric or skew-symmetric.
20) For any square matrix write whether AAᵀ is symmetric or skew-symmetric.
21) If A= [aᵢⱼ] is a skew-symmetric matrix, then write the value of ∑aᵢⱼ .
22) If A= [aᵢⱼ] is a skew-symmetric matrix, then write the value of ∑ ∑aᵢⱼ
ⁱ ʲ
23) If A and B are symmetric matrices, then write the condition for which AB is also symmetric.
24) If B is a skew-symmetric matrix, write whether the matrix ABAᵀ is symmetric or skew-symmetric.
25) If B is symmetric matrix , write whether the matrix ABAᵀ is symmetric or skew-symmetric.
26) If A is a skew-symmetric and n ∈ N that(Aⁿ)ᵀ = kAⁿ, write the value of k.
27) If A is a symmetric matrix and n ∈ N, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.
28) If A is a skew-symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.
29) if A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of the two.
30) If A and B are symmetric matrix of the same order, write whether AB - BA is symmetric or skew-symmetric or neither of the two.
31) Write a square matrix which is both symmetric as skew-symmetric.
32) Find the values of x and y if
A= 1 3 B= y 0 C= 5 6
0 x 1 2 1 8 with the relation 2A+ B = C.
33) If x+3 4 = 5 4
y-4 x+ y 3 9 find x and y.
34) Find the value of x from the following:
2x - y 5 = 6 5
3 y 3 -2
35) x - y 2 = 2 2
x 5 3 5 then find x.
36) 3x - y -y = 1 2
2y- x 3 -5 3 find x
37) If matrix A= (1 2 3), write AAᵀ.
38) 2x + y 3y= 6 0
0 4 6 4 find x.
39) If A= 1 2
3 4 find A+ Aᵀ.
40) a+ b 2 = 6 5
5 b 2 2 find a.
41) If A is matrix of order 3x4 and B is a matrix of order 4x3. Find order of the matrix of AB.
42) if A= 2 1 3 B= -1 0 -1 C= 1
-1 1 0 0
0 1 1 -1 with the relation ABC = K, then write the order of matrix K.
43) If A= 1 2 B= 3 1 C= 7 11
3 4 2. 5 k 23 with the relation AB= C then write the value of k.
44) if I is the identity Matrix and I is a square Matrix such that A²= A, then what is the value of (I+ A)²- 3A ?
45) If 1 2
0 3 is written as B+ C, where B is a symmetric matrix and C is a skew-symmetric matrix, then find B.
46) if A is 2x3 matrix and B is a matrix such that AᵀB and BAᵀ both are defined , then what is the order of B ?
47) What is the total number of 2x 2 matrices with each entry 0 and 1 ?
48) if x x - y = 3 1
2x+y 7 8 7 then find the value of y.
49) if a matrix has 5 elements, write all possible orders it can have.
50) For a 2x2 metrix A= [aᵢⱼ] whose elements are given by aᵢⱼ= i/j, write the value of a₁₂.
51) If A=2 B= -1 C= 10
3 1 5 with the relation xA+ yB = C, find the value of x.
52) if A= 9 -1 4 B= 1 2 -1
-2 1 3 0 4 9 with the relation A= X + B, then find the matrix X.
53) If a - b 2a+ c = -1 5
2a- b 3c +d 0 13 find the value of b.
54) For what value of x, is the matrix
0 1 -2
-1 0 3
x -3 0 a skew-symmetric matrix?
55) If matrix A = 2 -2
-2 2 and A²= pA, then write the value of p.
56) If A is a square matrix such that A²= A, then write the value of 7A - (I + A)³, where I is the identity Matrix .
57) If A=3 4 B= 1 y C= 7 0
5 x 0 1 10 5 with the relation 2A+ B = C, then find x - y.
58) If A= x 1 B= 1 0
-2 0 = O, find x.
59) If a+ 4 3b = 2a+ 2 b+2
8 -6 8 a- 8b write the value of a- 2b.
60) Write a 2x 2 matrix which is both symmetric and skew-symmetric.
61) If xy 4 = 8 w
z+6 x+ y 0 6, write the value of (x + y + z).
62) Construct a 2x2 Matrix A= [aᵢⱼ] whose elements aᵢⱼ are given by
aᵢⱼ = |-3i + j| , if i≠ j
(1+ j)², if I= j
63) If A= x + y B= 2 1 C= 1
x - y 4 3 -2 with the relation A= BC, then write the value of (x,y)
64) Matrix 0 2b -2
3 1 3
3a 3 -1 is given to be symmetric, find the values of a and b.
65) Write the number of all possible matrices of order 2x2 with each entry 1, 2 or 3.
1) yes, mx p 2) 2x2 and 3x3
3) -7
2
4) 1 2 3
2 4 6
3 6 9
5) A= 2 0 B= 0 0
3 0 2 -1
6) 4 8
8 11 7) -2,3 8) 3x4
9) 3 6
6 9 10) 0, -2 11) 2 12) 8 13) - A or I₃ 14) A
15) 81 0
0 81 16) -2
17) 3 5
4 6
18) 1 -9
-2 4 19) skew-symmetric 20) symmetric 21) 0 22) 0 23) AB= BA 24) skew-symmetric 25) symmetric 26) (-1)ⁿ 27) symmetric 28) symmetric 29) skew-symmetric 30) skew-symmetric 31) null matrix 32) 3,3 33) 2,7 34) 2 35) 1 36) 1 37) 14 38) 3,0
39) 2 5
5 8 40) 4 41) 3x3 42) 1x1 43) 17 44) I
45) 1 1
1 3 46) 2x3 47) 16 48) 2 49) 1x5, 5x1 50) 1/2 51) 3
52) 8 -3 5
-2. -3 -6 53) 2 54) 2 55) 4 56) -1 57) 2, -8 58) 2 59) 0
60) 0 0
0 0 61) 0
62) 4 1/2
5/2 16 63) (-1,1) 64) -2/3,3/2 65) 3⁴= 81
DETERMINANTS
Fill in the blanks
1) If A= diagonal (1, 2, 3), then |A|= ____
2) If the matrix A= 1 3 x + 2
2 4 8
3 5 10 is singular, then x=____
3) The set of real values of a for which the matrix
A= a 2
2 4 is not-singular is _____
4) If A= ln x -1
- ln x 2 and if det(A)= 2, then x=_____
5) If I is the Identity metrix of order 10, then determinant of I is= ____
6) Let A= [aᵢⱼ] be a 3x3 Matrix such that |A| =5. If Caᵢⱼ= Cofactor of aᵢⱼ in A. Then a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ = _____
7) In the above question, a₁₁C₂₁+ a₁₂C₂₂ + a₁₃C₂₃= ____
8) The value of the determinant
∆= 1 2 3
4 5 6
3x 6x. 9x is____
9) If A is a matrix of order 3x3, then the number of minors in A is ____
10) If x=-9 is a roots of
x 3 7
2 x 2= 0
7 6 x then other two roots are ____
1) 6 2) 4 3) -{1} 4) e² 5) 1 6) 5 7) 0 8) 0 9) 9 10) 2,7
SHORT QUESTIONS
1) If A is a singular matrix, then write the value of |A|.
2) For what value of x, the matrix
at 5- x x+1
2 4 is singular ?
3) Write the value of the determinant
2 3 4
2x 3x 4 4x
5 6 8
4) State whether the matrix
2 3
6 4 is singular or nonsingular
5) If A= 1 2 B= 1 0
3 -1 -1 0 find |AB|.
6) if A= [aᵢⱼ] is a 3x3 diagonal Matrix such a₁₁ = 1, a₂₂= 2 and a₃₃ = 3, then find |A|.
7) If A= [aᵢⱼ] is 3x3 scalar matrix such that a₁₁=2, then write the value of |A|.
8) if I₃ denotes identity Matrix of order 3x3, write the value of its determinant.
9) If the Matrix 5x 2
-10 1 is a singular, find the value of x.
10) Write the factors of a₁₂ in the matrix
2 -3 5
6 0 4
1 5 -7
11) if 2x + 5 3
5x + 2 9 = 0, find x.
12) Find the value of x from the following
x 4
2 2x = 0
13) Write the value of the determinant
2 3 4
5 6 8
6x 9x 12x
14) What is the value of the determinant
0 2 0
2 3 4
4 5 6
15) For what value of x is the matrix
6- x. 4
3-x 1 singular
16) if A= 5 3 8
2 0 1
1 2 3 Write the cofactor of the element a₃₂.
17) if x+ 1 x - 1 = 4 -1
x -3 x + 2 1 3 then write the value of x.
18) If 2x x + 3 = 1 5
2(x + 1) x + 1 3 3 then write the value of x.
19) If 3x 7 = 8 7
- 2 4 6 4 find the value of x.
20) If 2x 5 = 6 -2
8 x 7 3 write the value of x.
21) Write the value of the determinant
p p+1
p -1 p
22) If x ∈ N and x+3 -2
-3x 2x = 8, then the value of x.
1) 0 2) 3 3) 0 4) nonsingular 5) 0 6) 6 7) 8 8) 1 9) -4 10) 46 11) -13 12) ±2 13) 0 14) 8 15) 2 16) 11 17 2 18) 1 19) -2 20) ±6 21) 1 22) ±2
INVERSE AND APPLICATION OF MATRICES
1) If A is invertible matrix, then which of the following is not true.
a) (A²)⁻¹= (A⁻¹)²
b) |A⁻¹| = |A|⁻¹
c) (Aᵀ)⁻¹= (A⁻¹)ᵀ
d) |A|≠ 0.
2) if A is an invertebral matrix of order 3, then which of the following is not true
a) |adjacent A|= |A|²
b) (A⁻¹)⁻¹= A
c) If BA= CA, then B≠ C. Where B and C are square matrices of order 3
d) (AB)⁻¹= B⁻¹A⁻¹, where B= [aᵢⱼ] ₃ₓ₃ and|B|≠ 0
3) If A= 3 4 B= -2 -2
2 4 0 -1 , then (A+ B)⁻¹
a) is a skew-symmetric metrix
b) A⁻¹+ B⁻¹ c) does not exist d) none
4) If S= a b
c d then adj A is
a) -d -b b) d. -b c) d b d) d c
-c a -c a c a b a
5) If A is a singular matrix, then adj A is
a) non-singular b) singular c) symmetric d) not defined
6) If A, B are two nx n non-singular matrices , then
a) AB is non-singular b) AB is singular c) (AB)⁻¹=A⁻¹B⁻¹ d) (AB)⁻¹does not exist
7) If A= a 0 0
0 a 0
0 0 a then the value of |adj A| is
a) a²⁷ b) a⁹ c) a⁶ d) a²
8) If A= 1 2 -1
-1 1 2
2 -1 1 then det(adj(adj A)) is
a) 14⁴ b) 14³ c) 14² d) 14
9) If B is a non-singular Matrix and A is a square Matrix, then (B⁻¹AB) is equal to
a) Det (A⁻¹) b) Det (B)⁻¹ c) Det(A) d) Det(B)
10) For any 2x2 matrix, if A(adjA)= 10 0
0 10 then |A| is equal to
a) 20 b) 100 c) 10 d) 0
11) If A⁵= O such that Aⁿ ≠ 1 for 1≤ n ≤4 , then (I - A)⁻¹equals
a) A⁴ b) A³ c) I + A d) none
12) if A satisfies the equation x³- 5x²+ 4x + K = 0, then A⁻¹ exists if
a) K≠ 1 b) K ≠ 2 c) K≠ -1 d) K≠ 0
13) if for the matrix A, A³= I, then A⁻¹=
a) A² b) A³ c) A d) none
14) if A and B are square matrices such that B= - A⁻¹BA, then (A+ B)²=
a) O b) A²+ B² c) A²+ 2AB + B² d) A+ B
15) If A= 2 0 0
0 2 0
0 0 2 then A⁵=
a) 5A b) 10A c) 16A d) 32A
16) For non-singular square Matrix A, B and C of the same order (AB⁻¹C)⁻¹=
a) A⁻¹BC⁻¹ b) C⁻¹B⁻¹A⁻¹ c) CBA⁻¹ d) C⁻¹BA⁻¹
17) The matrix 5 10 3
2 -4 6
-1 -2 b is a singular matrix , if the value of b is
a) -3 b) 3 c) 0 d) non-existent
18) if d is the determinant of a square metrix A if order, then the determinant of its adjoint is
a) dⁿ b) dⁿ⁻¹ c) dⁿ⁺¹ d) d
19) If A is a matrix of order 3 and|A|=8, then |adj A|=
a) 1 b) 2 c) 2³ d) 2⁶
20) If A²- A + I = O, then the inverse of A is
a) A⁻² b) A+ I c) I - A d) A'- I
21) If A and B are invertible matrix, which of the following statement is not correct.
a) adjA= |A|A⁻¹
b) det(A⁻¹)= (detA⁻¹)
c) (A+ B)⁻¹ = A⁻¹ + B⁻¹
d) (AB)⁻¹ = B⁻¹A⁻¹
22) if A is a square Matrix such that A²= I, then A⁻¹ is equals to
a) A+ I b) A c) 0 d) 2A
23) Lat A= 1 2 B=1 0
3 -5 0 2 and X be a matrix such that A= BX, then X is equals to
a) 1 2 b) -1 2 c) 2 4
3/2 -5/2 3/2 5/2 3 -5 d) none
24) If A= 2 3
5 -2 be such that A⁻¹ = kA, then k equals to
a) 19 b) 1/19 c) -19 d) -1/19
25) If A= 1/3 1/3 2/3
2/3 1/3 -2/3
x/3 2/3 y/3 satisfies AᵀA = I, then x+ y=
a) 3 b) 0 c) -3 d) 1
26) If A= 1 0 1
0 0 1
a b 2 then aU + bA + 2A² equals to
a) A B) -A c) abA d) none
27) If A= 1 - tanx B= 1 tanx C= a -b
tanx 1. -tanx. 1. b a as relation AB⁻¹= C, then
a) a= 1, b=1 b) a cos2x , b= sin2x c) a= sin2x , b= cos2x d) none
28) If a matrix A is such that 3A³+ 2A²+ 5A + I = 0, then A⁻¹ is equal to
a) -(3A²+ 2A +5)
b) 3A²+ 2A +5
c) 3A²- 2A -5 d) none
29) if A is unvertible matrix, then det(A⁻¹) is equal to
a) det(A) b) 1/det(A) c) 1 d) none
30) If A= 2 k -3
0 2 5
1 1 3 then A⁻¹ exists if
a) k= 2 b) k≠ 2 c) k≠ -2 d) none
1a 2d 3d 4b 5b 6a 7c 8a 9c 10c 11d 12d 13a 14b 15c 16d 17d 18b 19d 20c 21c 22b 23a 24b 25c 26d 27b 28d 29b 30d
FILL IN THE BLANKS
1) If A is a Unit Matrix of order n, then A(adj A)=____
2) if A a non-singular square Matrix such that A³= I, then A⁻¹=____
3) if A and B are square matrices of the same order AB = 3I, then A⁻¹=___
4) if the matrix A= 1 a 2
1 2 5
2 1 1 is not invertible, than a=____
5) if A is a singular Matrix, then A (adj A)=____
6) let A be a square matric of order 3 such that |A|= 11 and B be the matrix of cofactors of elements of A, then |B|²=_____
7) If A is a square matrix of order 2 such that A (adj A)=
10 0
0 10 then |A|=_____
8) If A is an invertible matrix of order 3 and |A|= 3, then |adj A|=____
9) If A is an invertible Matrix of order 3 and |A|= 5, then adj(adj A)=____
10) If A is an invertible matrix of order 3 and|A|= 4, then |adj (adj A)|=_____
11) If A= diagonal (1, 2, 3), then |adj (adj A)|=_____
12) if A is a square Matrix of order 3 such that |A|= 5/2, then |A⁻¹|=____
13) If A is a square Matrix such that A(adj A)= 10I, then |A|=____
14) Let A be a square matrix of order 3 and B= |A| A⁻¹. If |A|= -5, then |B|=___
15) If k is a scalar and I is a unit matrix of order 3, then adj(kI)=____
16) If A= cosx sinx and A(adjA)= k 0
-sinx cosx 0 k then k= ____
17) If A is a non-singular Matrix of order 3, then adj (adj A) is equal to _____
18) If A= [aᵢⱼ]₂ₓ₂ , where aᵢⱼ = i+j, if i≠ j
i²-2j , if I=j
Then A⁻¹=____
19) If A= 0 3
2 0 and A⁻¹ = k(adj A), then k=_____
20) If A is a 3x3 no-singular matrix such that AAᵀ = AᵀA and B= A⁻¹Aᵀ, then BBᵀ =_____
21) If A and B are two square metrices of the same order such that B= -A⁻¹BA, then (A+ B)²=_____
22) If A is a singular matrix of order 3x3, then (A³)⁻¹=_____
23) If A be a square matrix such that |adj A|= |A|², then the order of the A is _____
24) If A= x 5 2
2 y 3
1 1 z , xyz= 80, 3x+ 2y + 10z = 20 and adj A = kI, then k=____
1) A 2) A² 3) B/3 4) 1 5) null matrix 6) 11⁴ 7) 10 8) 9 9) 5A 10) 4⁴ 11) 6⁴ 12) 2/5 13) 10 14) 25 15) k²I 16) I 17) |A| A
18) 0 1/3
1/3 1/3
19) -1/6 20) I 21) A²+ B² 22) (A⁻¹)³ 23) 3x3 24) 81
SHORT QUESTIONS
1) Write the adjoint of the matrix
A= -3 4
7 -2
2) if A is a square matrix such that A(adj A)= 5I, where I denotes the identity matrix of the same order. Then , find the value of |A|.
3) if A is a square matrix of order 3 such that |A|= 5. Write the value of |adj A|.
4) If A is a square matrix of order 3 such that |adj A|= 64, find |A|.
5) If A is a non-singular square Matrix such that |A|= 10, find |A⁻¹|.
6) If A, B, C are three non-null square matrix of the same order, write the condition on A such that AB = AC=> B = C.
7) If A is a non-singular square matrix such that
A⁻¹= 5 3
-2 -1 , then find (Aᵀ)⁻¹.
8) If adj A= 2 3 and adj B= 1 -2
4 -1 -3 1, find adj AB.
9) If A is a symmetric matrix , write whether Aᵀ is symmetric or skew-symmetric.
10) If A is a square metrix of order 3 such that |A|= 2, then write the value of adj(adj A).
11) If A is a square Matrix of order 3 such that |A|= 3, then find the value of |adj (adj A)|.
12) If A is a square Matrix of order 3 such that adj(2A)= k adj(A), then write the value of k.
13) If A is a square matrix, then write the matrix adj(Aᵀ) - (adj A)ᵀ.
14) Let A be a 3x3 square metrix such that A(adj A)= 2I, where I is the identity matrix . write the value of |adj A|.
15) if A is a non-singular symmetric matrix , write whether A⁻¹ is symmetric or skew-symmetric .
16) If A= cosx sinx and A(adj A)= k 0
-sinx cosx 0 k then find the value of k.
17) If A is an invertible Matrix such that |A⁻¹|= 2, find the value of |A|.
18) if A is a square matrix such that A(adj A)=
5 0 0
0 5 0
0 0 5 then write the value of |adj A|.
19) If A= 2 3
5 -2 be such that A⁻¹ = k A, then find the value of k.
20) let A be a square matrix such that A²- A + I = O, then write A⁻¹ in terms of A.
21) If Cᵢⱼ is the cofactor of the element aᵢⱼ of the matrix
A= 2 -3 5
6 0 4
1 5 -7 then write the value of a₃₂C₃₂.
22) Find the inverse of the matrix
3 -2
-7 5 fp
23) Find the inverse of the matrix
Cosx sinx
-sinx cosx
24) If A= 1 -3
2 0 write adj A.
25) If A= a b B= 1 0
c d 0 1, find adj(AB(.
26) If A= 3 1
2 -3 then find|adj A|.
27) If A= 2 3
5 -2 write A⁻¹ in terms of A.
28) Write A⁻¹ for A= 2 5
1 3
29) use elementary colum operation C₂-->C₂+ 2C₁ in the following matrix equation:
30)
31) If A is a square matrix with |A|= 4 then find the value of |A. (adj A)|.
1) -2 -4
-7 -3
2) 5 3) 25 4) ±8 5) 1/10 6) A must be invertible or |A|≠ 0
7) 5 -2
3 -1
8) -6 5
-2 -10
9) symmetric 10) 2A 11) 81 12) 4 13 null matrix 14) 4 15) symmetric 16) 1 17) 1/2 18) 25 19) 1/19 20) A⁻¹ = (I - A) 21) 110
22) 5 2
7 3
23) cosx - sinx
Sinx cosx
24) 0 3
-2 1
25) d. -b
-c a
26) -11 27) A⁻¹ = A/19
28) A⁻¹ = 3 -5
-1 2
INCREASING AND DECREASING FUNCTION
1) The function f(x)= xˣ decreases on the interval
a) (0, e) b) (0, 1) c) (0, 1/e) d) (1/e, e)
2) The function f(x)= 2 logₑ(x -2) - x²+ 4x +1 increases on the interval
a) (1,2) b) (2,3) c) (1,3) d) (2,4)
3) Let f(x)= x³- 6x²+ 15x +3. then
a) f(x)> 0 for all x ∈R
b) f(x)> f(x+1) for all x ∈R
c) f(x) is invertible
d) f(x)< 0 for all x ∈R
4) The function f(x) = x² e⁻ˣ is monoatomic increasing when
a) x ∈R -[0,2] b) 0< x< 2 c) 2 < x < ∞ d) x < 0
5) Function f(x)= x³- 27x + 5 is monotonically increasing when
a) x<-3 b) |x|>3 c) x≤-3 d) |x|≥3
6) Function f(x)= 2x³- 9x²+ 12x + 29 is monotonically decreasing when
a) x< 2 b) x > 2 c) x >3 d) 1 < x < 2
7) if the function f(x)= kx³- 9x²+ 9x +3 is monotonically increasing in every interval, then
a) k < 3 b) k ≤3 c) k >3 d) ≥3
8) The function f(x)= x/(1+ |x|) is
a) strictly increasing b) strictly decreasing c) neither increasing nor decreasing d) none
9) Function f(x)= aˣ is increasing on R, if
a) a> 0 b) a <0 c) 0 < a < 1 d) a > 1
10) Function f(x)= logₐx is increasing on R, if
a) 0< a <1 b) a> 1 c) a < 1 d) a > 0
11) If the function f(x)= x²- Kx +5 is increasing on [2,4], then
a) k∈(2, ∞) b) k∈(-∞,2) c) k∈(4, ∞) d) k∈(-∞,4)
12 If the function f(x)= x³- 9kx²+ 27x +30 is increasing on R, then
a) -1≤ k <1 b) k< -1 or k >1 c) 0 <k < 1 d) -1< k < 0
13) The function f(x)= x⁹+ 3x⁷ + 64 is increasing on
a) R b) (-∞,0) b) (0, ∞) c) R₀
14) The interval on which the function f(x)= 2x³+ 9x²+ 12x -1 is decreasing, is
a) (-1, ∞) b) [-2,-1] c) (-∞,-2) d) [-1,1]
15) y= x(x -3)² decrease for the values of x given by
a) 1< x < 3 b) x< 0 c) x > 0 d) 0 <x < 3/2
1c 2b 3c 4b 5d 6d 7c 8b 9d 10b 11b 12a 13a 14b 15a
FILL IN THE BLANKS
1) The function f(x)= (2x²-1)/x⁴, x> 0, decreases in interval_____ .
2) The function g(x)= x + 1/x , x≠ 0 decreases in the closed interval ____ .
3) The largest open interval in which the function f(x)= 1/(1+ x²) decreases___ .
4) The set value of 'a' for which the f(x)= ax + b is strictly increasing for all real x, is ____
5) The largest interval in which f(x)= x¹⁾ˣ is strictly increasing is____ .
1) (1, ∞) 2) [-1,1] 3) (0, ∞) 4) (0, ∞) 5) (1, e)
VERY SHORT ANSWER QUESTIONS
1) What are the values of 'a' for which f(x)= aˣ is increasing on R ?
2) What are the values of 'a 0' for which f(x)= aˣ is decreasing on R ?
3) Write the set of 'a' for which f(x)= logₐx is increasing in its domain.
4) Write the set of values of 'a' for which f(x)= logₐx is decreasing function in its domain.
5) Write the set of values 'a' for which the function f(x)= ax + b is decreasing for all x ∈R.
1) a> 1 2) 0 < a < 1 3) a >1 4) 0 < a < 1 5) a ∈(-∞,0)
MAXIMA AND MINIMA
1) For the function f(x)= x+ 1/x
a) x= 1 is a point of maximum
b) x=1 is a point of minimum
c) maximum value> minimum value
d) maximum value < minimum value
2) Let f(x)=x³+ 3x²- 9x +2. Then , f(x) has
a) a maximum at x = 1
b) a minimum at x= 1
c) neither maximum nor a minimum at x= -3 d) none
3) The minimum value of f(x)=x⁴- x²- 2x +6 is
a) 6 b) 4 c) 8 d) none
4) The number which exceeds its square by the greatest possible quantity is
a) 1/2 b) 1/4 c) 3/4 d) none
5) let f(x)= (x - a)²+ (x - b)²+ (x - c)². Then, f(x) has a minimum at x=
a) (a + b + c)/3 b) 3√(abc) c) 3/(1/a + 1/b + 1/c) d) none
6) The sum of two non-zero numbers is 8, the minimum value of the sum of their reciproclas is
a) 1/4 b) 1/2 c) 1/8 d) none
7) if its lies in the internal [0,1] , then the least value of x²+ x+1 is
a) 3 b) 3/4 c) 1 d) none
8) The least value of the function f(x)= x³- 18x²+ 96x in the interval [0, 9]
a) 126 b) 135 c) 160 d) 0
9) The maximum value of f(x)= x/(4- x + x²) on [-1,1] is
a) - 1/4 b) -1/3 c) 1/6 d) 1/5
10) The point of the curve y²= 4x which is nearest to the point (2,1) is
a) (1, 2√2) b) (1,2) c) (1,- 2) d) (-2,1)
11) If x + y = 8, then the maximum value of xy is
a) 8 b) 16 c) 20 d) 24
12) The least and greatest value of f(x)= x³- 6x²+ 9x in [0,6], are
a) 3,4 b) 0, 6 c) 0, 3 d) 3, 6
13) The minimum value of (x²+ 250/x) is
a) 75 b) 50 c) 25 d) 55
14) If f(x)= x + 1/x, x> 0, then its greatest value is
a) -2 b) 0 c) 3 d) none
15) If f(x)= 1/(4x²+ 2x +1), then its maximum value is
a) 4/3 b) 2/3 c) 1 d) 3/4
16) let x, y be two variables and x> 0, xy =1, then minimum value of x + y is
a) 1 b) 2 c) 5/2 d) 10/3
17) The function f(x)= 2x³ - 15x²+ 36x +4 is maximum at x=
a) 3 b) 0 c) 4 d) 2
18) The maximum value of f(x)= x/(4+ x + x²) on [-1,1] is
a) -1/4 b) -1/3 c) 1/6 d) 1/5
19) Let f(x)= 2x³- 3x² -12x + 5 on [-2,4]. The relative maximum occurs at x=
a)- 2 b) -1 c) 2 d) 4
20) The minimum value of x logₑx is equals to
a) e b) 1/e c) -1/e d) 2e e) - e
21) The minimum value of the function f(x)= 2x³- 21x²+ 36x -20 is
a)- 128 b) -126 c) -120 d) none
22) If x is real, the minimum value of x²- 8x +17 is
a) -1 b) 0 c) 1 d) 2
23) The maximum value of (1/x)ˣ is
a) e b) eᵉ c) e¹⁾ᵉ d) (1/e)¹⁾ᵉ
24) The function f(x)= xˣ has a stationary point at
a) x= e b) x = 1/e c) x=1 d) x =√e
25) Maximum slope of the curve y= - x³+ 3x²+ 9x - 27 is
a) 0 b) 12 c) 16 d) 32
26) The function f(x)=2x³- 3x²- 12x +4, has
a) two points of local maximum
b) two points of local minimum
c) one maximum and one minimum
d) no maximum no minimum.
1d 2b 3b 4a 5a 6b 7c 8d 9c 10b 11b 12a 13a 14d 15a 16b 17d 18c 19b 20c 21a 22c 23c 24b 25b 26c
FILL IN THE BLANKS
1) The positive real number x when added to its reciprocal gives the minimum value of the sum when, x_____ .
2) The real number which must exceed its cube is ____ .
3) The function f(x)= ax + b/x, a, b, x> 0 takes on the least value at x equals to ___.
4) If y = a logx + bx²+ x has its extreme values at x=1 and x= 2, then (a, b)= ____
5) The maximum value f(x)= x e⁻ˣ is _____
6) If the function f(x)= x⁴- 62x²+ ax +9 attains a local maximum at x=1, then a= ____
7) If the sum of two non zero numbers is 4, then the minimum value of the sum of their reciprocals is ______.
8) If x and y are two real numbers such that x> 0 and xy =1. then the minimum value of x+ y is _____
9) The number that exceeds its square by the greatest amount is____
10) if m and M respectively denote the minimum and maximum values of f(x)= (x -1)²+ 3 in the interval [3,1], then the order pair (m, M)=_____
11) The minimum value of f(x)= x²+ 250/x is _____
12) The maximum slope of the curve y= - x³+ 3x²+ 9x -27 is_______
13) The function f(x)= x/2 + 2/x has a local minimum at x= _____
14) The least value of the function f(x)= ax + b/x (a>0, b>0, x>0) is____
1) 1 2) 1/√3 3) √(a/b) 4) (-2/3,-1/6) 5) 1/e 6) 120 7) 1 8) 2 9) 1/2 10) (3,19) 11) 75 12) 12 13) 2 14) √(ab)
VERY SHORT ANSWER QUESTIONS
1) write necessary condition for a point x= c to be an extreme point on the function f(x).
2) Write sufficient conditions for a point x= c to be a point of local maximum.
3) If f(x) attains a local minimum at x= c, then write the value of f'(c) and f"(c).
4) Write the minimum value of f(x)= x + 1/x, x> 0.
5) Write the maximum value of f(x)= x + 1/x, x< 0.
6) Write the point where f(x)= x logₑx attains minimum value.
7) Find the least value of f(x)= ax + b/x, where a> 0, b>0 and x >0.
8) Write the minimum value of f(x)=xˣ.
9) Write the maximum value of f(x)=x¹⁾ˣ.
10) Write the maximum value of f(x)= (logx)/x, if it exists.
1) f'(c)=0 2) f'(c)=0 and f''(c)<0 3) f'(c)=0 and f''(c)>0 4) 2 5) -2 6) (1/e, -1/e) 7) 2√(ab) 8) e⁻¹⁾ᵉ 9) e¹⁾ᵉ 10) 1/e
APPLICATION OF DERIVATIVES IN COMMERCE AND ECONOMICS
Marginal cost for an output x of a product of given as 3 + 2x 5 - 4x + 3x and if the fixed cost is 0 find the profit function in the profit one output is a physical to the marginal revenue and the marginal cost of a product approximated as 16x - x square 81 - 28 + 2x respectively the fifth cost is zero determine the profit and maximizing output and the total purple at the optimal output
A travel agent arrange a tour from Delhi to Shimla and back s66 in spherical bachat booking amount or 450% provided all seats will be occupied however for a bri increase or 15 in the booking amount one should will remain vacant you also plans to provide a mineral water bottle and snapchatting 60% find the relationship
INDEFINITE INTEGRALS
1) ∫ (x -1)e⁻ˣ dx is= ?
a) - xeˣ+ C b) eˣ+ C c) - xe⁻ˣ+ C d) xe⁻ˣ+ C
2) ∫ 2¹⁾ˣ/x² dx = k 2¹⁾ˣ+ C, then k is
a) - 1/logₑ2 b) - logₑ2 c) -1 d) 1/2
3) ∫ | x|³ dx =?
a) -x⁴/4+ c b) | x|⁴+ c c) x⁴/4+ c d) none
4) ∫ (x +3)eˣ/(x + 4)² dx =
a) eˣ/(x + 4) + C b) eˣ/(x + 3)+ c c) 1/(x + 4)²+ C d) eˣ/(x + 4)² + C
5) ∫ 2/(eˣ + e⁻ˣ) dx =
a) - e⁻ˣ/(eˣ + e⁻ˣ) + c b) - 1/(eˣ + e⁻ˣ) + c c) - 1/(eˣ + 1)² + c d) 1/(eˣ - e⁻ˣ) + c
6) The primitive of the function f(x)= (1- 1/x²). ₐ(x + 1/x), a> 0 is
a) (1/logₑa). ₐ(x + 1/x) b) (logₑa). ₐ(x + 1/x) c) (1/x)(1/logₑa). ₐ(x + 1/x) d) x (1/logₑa). ₐ(x + 1/x)
7) The value of ∫1/(x + x logx) dx
a) 1+ logx b) x + logx c) x log(1+ logx) d) log(1+ logx)
8) ∫ eˣ{f(x)+ f'(x)} dx =
a) eˣ f(x)+ c) b) eˣ + f(x)+ C c) 2eˣf(x)+ c d) eˣ - f(x) + C
9) ∫ x⁹/(4x²+ 1)⁶ dx
a) (1/5x)(4+ 1/x²)⁻⁵ + C b) (1/5)(4+ 1/x²)⁻⁵ + C c) (1/10x)(4+ 1/x²)⁻⁵+ C d) (1/10)(4+ 1/x²)⁻⁵ + C
10) ∫ x³ dx/√(1+ x²)= a√(1+ x²)³+ b√(1+ x²)+ C, then
a) a= 1/3, b= 1 b) a= - 1/3, b= 1 c) a= -1/3, b= - 1 d) a= 1/3, b= -1
11) ∫ x³ dx/(1+ x)=?
a) x+ x²/2 + x³/3 - log|1- x| + c
b) x+ x²/2 - x³/3 - log|1- x| + c
c) x- x²/2 - x³/3 - log|1+ x| + c
d) x- x²/2 + x³/3 - log|1+ x| + c
12) ∫ (3eˣ - 5e⁻ˣ)/(4eˣ + 5e⁻ˣ) dx= ax + b log|(4eˣ + 5e⁻ˣ)|+ c
a) a= -1/8, b= 7/8 b) a= 1/8, b= 7/8 c) a= -1/8, b= - 7/8 d) a= 1/8, b= 7/8
13) ∫ eˣ{(1- x)/(1+ x²)}² dx.
a) eˣ/(1+ x²) + c
b) - eˣ/(1+ x²) + c
c) eˣ/(1+ x²)² + c
d) - eˣ/(1+ x²)² + c
1c 2a 3d 4a 5a 6a 7d 8a 9d 10d 11d 12c 13c
FILL IN THE BLANKS
1) The value of the integral ∫ e²ˡᵒᵍˣ + eˣˡᵒᵍˣ dx is _______
2) ∫ eˣ{(x+3)/(x +4)²} dx= ______
3) ∫ eˣ{x/(x +1)²} dx= ______
4) ∫ {(x²+1)/(x² -1} dx= ______
5) ∫ (e⁵ˡᵒᵍˣ - e⁴ˡᵒᵍˣ)/(e³ˡᵒᵍˣ - e²ˡᵒᵍˣ) dx is _______
6) ∫ (x⁴+ x²+1)/(x²- x +1) dx =________
7) ∫ x/(ₑ3x² dx=________
8) ∫ aˣ(1+ logx)dx= ______
9) ∫ eˣdx/(eˣ +1)=_____
10) ∫ (x +3)(x²+ 6x +10)⁹ dx =_____
1) x³/3 + 2ˣ/log 2
2) eˣ/(x +4)
3) eˣ/(x+1)
4) x + log{(x-1)/(x+1)}
5) x³/3
6) x³/3 + x²/2 + x
7) -(ₑ-3x²)/6
8) xˣ
9) log|eˣ +1|
SHORT QUESTIONS
1) ∫ x⁴ e³ˡᵒᵍˣ dx. x⁸/8+ c
2) ∫ (log xⁿ)/x dx. (n/2)(logx)2+ c
3) ∫ (log x)ⁿ/x dx. (Logx)ⁿ⁺¹/(n +1) + c
4) ∫ 1/(1+ eˣ) dx. - log(1+ e⁻ˣ)+ c
5) ∫ 1/(1+ 2eˣ) dx. - log(2+ e⁻ˣ)+ c
6) ∫ logₑx dx. x(logx -1)+ c
7) ∫ aˣeˣ dx. (aeˣ)/log(ae) + c
8) ∫ ₑ(2x²+ln x) dx. (1/4) ₑ2x²+ c
9) ∫ ₑ(xlogₑa) + ₑ(alogₑx)dx. aˣ/logₑa + xᵃ⁺¹/(a+1)+ c
10) ∫ aˣ/(3+ aˣ) dx. 1/(log a) log(3+ aˣ)+ c
11) ∫ (1+ logx)/(3+ x logx) dx. Log(3+ x logx)+ c
12) ∫ dx/{x(log x)ⁿ}. (Logx)¹⁻ⁿ/(1- n)
13) ∫ eˣ(1/x - 1/x²)dx. eˣ/x + c
14) ∫ eᵃˣ{a f(x)+ f'(x)} dx. eᵃˣ f(x)+ c
15) ∫ √(9- x²) dx. (x/2) √(9- x²)+ (9/2) log|x + √(9+ x²)|+ c
16) ∫ √(x²- 9) dx. (x/2) √(x²- 9) -(9/2) log|x + √(x²- 9)|+ c
17) ∫ x²/(1+ x³) dx. (1/3) log|1+ x³|+ c
18) ∫ (x²+ 4x)/(x³+ 6x²+5) dx. (1/3) log|x³+ 6x²+5|+ c
19) ∫ (1+ log x)²/x dx. (1+ logx)³/3+ c
20) ∫ (logx)/x dx. (Logx)²/2 + c
21) ∫ 2ˣ dx. 2ˣ/log2 + c
22) ∫ (x³- 1)/x² dx. x²/2 + 1/x + c
23) ∫ (x³- x²+ x -1)/(x -1) dx. x³/3 + x + c
24) ∫ (1- x)√x dx. 2 √x³/3 - 2√x⁵/5 + c
25) ∫ eˣ{(x -1)/x²} dx = f(x)eˣ + C, then write the value of f(x). 1/x
26) Write the anti derivative of (3 √x + 1/√x). 2(√x³+ √x)+ c
27) ∫ dx/{x(1+ logx)}. Log(1+ logx)+ c
FINANCIAL MATHEMATICS - I
Annuity, Sinking fund etc
1) How much should a company set aside at the end of each year, if it has to buy a machine expected to cost Rs200000 at the end of 6 years and rate of interest is 10% per annum compounded annually ? (Given (1.1)⁶= 1.771). 25,940.33
2) A sinking fund is created for the redemption of debentures of Rs100000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund, if the investment can earn interest 4% per annum ? (Given= (1.04)²⁵= 2.6658). 47,938.64
3) A company intended to create a sinking fund to replace at the end of 20th year assets costing Rs500000. Calculate the amount to be retained out of profits every year if the intrest rate is 5%. Given (1.05)²⁰= 2.6532. 15122.18.
4) A firm anticipates a capital expenditure of Rs50000 for new equipment in 5 years. How much should be deposited quarterly in a sinking fund carrying 12% per annum compounded quarterly to provide for the purpose ? Given (1.03)²⁰= 1.8061. 1860.81
5) A person has setup a sinking fund in order to have Rs100000 after 10 years for his children's college education. How much amount should be set aside bi-annually into an account paying 5% per annum compounder half-yearly ? Given (1.025)²⁰= 1.6386. 3914.81
6) A machine costs a company Rs52000 and its effective life is estimated to be 25 years. A sinking fund is created for replacing the machine by a new model at the end of its lifetime, when its scrap realizes a sum of Rs2500 only. The price of the new model is estimated to be 25% more than the price of present one. Find what amount should be set aside at the end of each year out of the profit for the sinking fund, if it accumulates at 3.5% per annum compound ? Given (1.035)²⁵= 2.3632. 1604.68
Perpetuity
1) Find the present value of a proprituality of Rs5000 payable at the end of each year, if money is worth 5% compounded annually. 100000
2) Find the present value of a sequence of a payments of Rs8000 made at the end of 6 months and containing forever, if money is worth 4% compoundedquarterly. 400000
3) At 6% converted quarterly, find the present value of a perpetuity of Rs4500 payable at the end of each quarter . 300000
4) Robin sir a perpetual bond that generates and annual return of Rs50,000 each year. He believes that the borrower is credit worthy and that an 8% interest rate will be suitable for this bond. Compute the present value of this perpetuity. 625000,
5) Your grandfather is retiring at the end of next year. He would like to ensure that his heirs receive payments of Rs75000 a year forever, starting when he retires, How much does he need to invest in the beginning of this year to produce the desired cash flow, If money is worth 6% compounded annually. 1250000
6) If money is worth 5% compare the present value of perpetuty od Rs2000 payable at end of each year with that of an ordinary annuity of Rs2000 per year for 100 years. Given (1.05)⁻¹⁰⁰= 0.0076. 39696
7) Find the present value of a perpetuity of Rs3120 Payable at the beginning of each year, if money is or 6% effective. Rs55120
8) How much money needed to endure a series of lectures costing Rs2500 at the beginning of each year indifinitely, if the money is worth 5% compounded annually? 52500
9) If the cash equivalent of a perpetuity of Rs1200 payable at the end of each quarter is Rs96000, find the rate of interest convertible quarterly. 5%
10) The present value of a perpetual income of Rs R at the end of each 6 months is Rs14400. Find the value of R if money is worth 6% compound semi-annually. 4320
BONDS
1) A company ABC Ltd has issued a bond having a face value of Rs10000 paying annual dividend at 8.5%. The bond will be redeemed at par at the end of 10 years. Find the purchase price of this bond if the investors wishes a yield rate of 8%. Given (1.08)⁻¹= 0.46319349. 10335.50
2) A company has issued a bond having face value of Rs100000, carrying an annual dividend rate of 7% and maturing in 15 years. If the prevailing market rate of interest is 9% and the bond is redeem at par, find the bond value. Given (1.09)⁻¹=0.27453804. 83878.62
3) A company has issued a bond having the face value of Rs100000 carrying a coupon rate of 8% to be paid semi-annually and maturing in 5 years. If the prevailing market rate of interest is 7%, find the bond value. Given (1.035)⁻¹⁰= 0.70891881. 104158.30
4) A bond has a face value of Rs10000 and matures in 15 years at part. The nominal interest is 7%. What is the price of the bond that will yield an effective intrest of 8% ? Given (1.08)⁻¹⁵= 0.31524170. 9144.05
5) Find the purchase price of a Rs20000, redeemable at the end 10 years at 110, and paying annual dividends at 4% , if the yield rate is to be 5% effective . Given (1.05)⁻¹⁰= 0.61391325. 19683.48
6) Find the purchase price of a Rs50000, 6% bond, dividend payable semi-annually, redeemable at par in 10 years, if the yield rate is to be 5% compounded semi-annually. Given (1.025)⁻²⁰= 0.61027094. 53897.29
7) Find the purchase price of Rs10000, redeemable at the end of 10 years at Rs11000 and paying annual dividends at 4%, if the yield rate is to be 5% per annum effective. (1.05)⁻¹⁰= 0.61391325. 9841.74
EFFECTIVE RATE OF INTEREST
1) Find the effective rate that is equivalent to anominal rate of 8% compounded :
a) semi-annually. 0.0816
b) quarterly . 0.08243216
c) continuously. 0.0833
2) A money lender charges 'interest' at the rate of 10 rupees per hundred rupees per half year, payable in advance. What is the effective rate of interest does he charge per annum . 23.45%
3) A money lender charges 'interest' at the rate of 10 paise per rupee per month, payable in advance. What effective rate of interest does he charge per annum? 25.45%
4) Which gives better yields : 9.1% compounded semi annually on 9% compounded monthly ? Second is better
5) To what sum will Rs6000 accumulate in 8 years if invested at an effective rate of 8% ? Rs 11105.58
6) Find the amount to which Rs12000 will accumulate at the effective rate of 3% for 10 years, 4% for 4 years and 5% for 2 years. 20800.10
7) How many years will it take for a sum of money to triple at the effective rate of 4% ? 28.06 years
8) Find the force of interest corresponding to the effectivat rate 8%. 7.69%
9) What annual rate compounded continuously is equivalent to an effective rate of 10% ? 9.53%
10) A banker credits the fixed deposit account of a depositor on a continuous basis. As a result, the effective rate of interest earned by a depositor is 9.3%. Find out the rate of interest that is allowed by the banker. What is the effective rate of interest if it is compounded on quarterly basis ? 9.3%
Nominal Rate Of Returns
1) Ronit made an investment of Rs225000 in a no-fee fond for one year. At the end of the year the value of the investment increases to 250000. Find the nominal rate of return percent on his investment . 11.11%
2) Mr Robin purchases 100 shares of a company that cost Rs250 each . After 1 year the price of each share to Rs300. Assuming that there no trading costs and no dividends, Find the nominal rate of the return of investment . 20%
3) Mira takes a loan of Rs300000 at an intrest of 10% compounded annually for a period of 3 years. Find her EMI by using flate rate method. 1083.33.
4) A man borrowed a home loan amount of Rs5000 from a bank at an interest rate 12% per annum for 30 years. Find the monthly installment amount Aman has to pay to the bank. given (1.01)⁻³⁶⁰= 0.02781668. 51430.63
5) Pragya takes a loan of Rs50000 from a bank at an interest rate of 6% per annum for 10 years. She wants to pay back the loan in equated monthly installments . Find her EMI by using
a) flat rate method.
b) reduced balance method.
Given (1.005)⁻¹²⁰= 0.5496327334. 6666.67, 5551.02
6) Rohit buys a car for which he makes down payment of Rs150000 and the balance is to be paid in 2 years by monthly installment of Rs25448 each. If the financer charges interest at the rate of 20%p.a., find the actual price of the car given (61/60)⁻²⁴ = 0.6725335725. 650002
7) Komal purchased a house from a company for Rs700000 and made a down payment of Rs150000. He repays the balance in 25 years by monthly installments at 9%. Compound monthly
a) What are the monthly payments ?
b) what is the total interest payment ? Given (1.0075)⁻³⁰⁰= 0.1062878338. 4615.58, 834674
8) A person buys a house for which he agrees to pay Rs5000 at the end of each month for 8 years. If money is worth 12% converted monthly, what is the cash price of the house ? Given (1.01)⁻⁹⁶= 0.3847229701. 307638.51
DEPRECIATION
1) A Machine costing Rs50000 has a useful life of 4 years. The estimated scrap value is Rs10000. Using the straight line method, find the annual depreciation and construct a schedule for depreciation. Also, find the depreciation rate percent. 25%
FINANCIAL MATHEMATICS - II
1) Rohan invested Rs 4455 in Rs10 shares quoted at Rs8.25. If the rate of dividend be 12%, his annual income is
a) Rs648 b) Rs668 c) Rs655.60 d) Rs534.60
2) Mr. X boys Rs50 shares in a company which pays 10% dividend. If he gets 12.5% on his investment p, at what price did he buy the shares ?
a) Rs52 b) Rs48 c) Rs40 d) Rs42
3) Roman invests Rs 5508 in 4% stock at 102. He afterward sells out at 105 and reinvest in 5% stock at 126. The change in his income is
a) Rs7 b) Rs9 c) Rs10 d) Rs20
4) At what price should I buy a share the value of which is Rs100, paying a dividend of 8% so that my yield is 11% ?
a) Rs84 b) Rs75 c) Rs70 d) RsRs72.72
5) A person has deposited Rs13200 in a bank which pays 14% intrest . He withdraws the money and invests in Rs100 stock at Rs110 which dividend of 15%. How much does he gain or lose ?
a) losses Rs48 b) losses Rs312 c) gains Rs48 d) gains Rs132
6) In order to obtain an income of Rs650 from 10% stock at Rs96, one must make an investment of
a) Rs3100 b) Rs 6240 c) Rs6500 d) Rs9600
7) Mr X invested Rs913 partly in 4% stock at Rs97 and partly in 5% stock at Rs107. if his income from both is equal, the amount of his investment in each stock is
a) Rs485, Rs428 b) Rs475, Rs438 c) Rs495, Rs 418 d) Rs505, Rs408
8) A man invested Rs14400 in Rs100 shares of a company at 20% premium . If the company declared 5% dividend at the end of the year, then how much does he got ?
a) Rs500 b) Rs600 c) Rs650 d) Rs720
9) By investing Rs3450 in a 9/2% stock, a man obtain an income of Rs150. The market price of the stock is.
a) Rs 110 b) Rs105 c) Rs103.50 d) Rs107.50
10) A man buys Rs 25 shares in a company which pays 9% dividend . The money invested is such that it gives 10% on investment. The price at which he bought the shares is
a) Rs22.50 b) Rs 22 c) Rs 45 d) Rs20.50
LINEAR PROGRAMMING
1) The solution set of the inquation 2x + y > 5 is
a) half plane that contains origin
b) open half plane not containing the origin
c) whole xy plane except the points lying on the line 2x + y = 5. d) none
2) Ibjective function of a LPP is
a) a constant b) a function to be optimised c) a relation between the variables d) none
3) Which of the following sets are convex ?
a) {(x,y): x²+ y²≥ 1}
b) {(x,y): y²≥ x}
c) {(x,y): 3x²+ 4y²≥ 25}
d) {(x,y): y ≥ 2, y≤ 4}
4) Let A and B are optimal solution of LPP , than
a) X= λA+ (1- λ)B, λ ∈ R is also an optimal solution.
b) X= λA+ (1- λ)B, 0≤λ ≤ 1 gives an optimal solution.
c) X= λA+ (1+ λ)B, 0≤ λ ≤ 1 give an optimal solution.
d) X= λA+ (1+ λ)B, λ ∈ R gives an optimal solution.
5) The maximum value of Z= 4x + 2y subject to the constraints 2x + 3 y≤ 18, x + y≥ 10 ; x, y ≥ 0 is
a) 36 b) 40 c) 20 d) none
6) The optimal value of the objective function is attained at the points.
a) given by intersection of inequations with the axes only
b) given by intersection of inequations with x-axis only.
c) given by corner points of the feasible region. d) none
7) The maximum value of Z= 4x + 3y subjected to the constraints 3x + 2y≥ 160, 5 x + 2y≥ 200 0, x + 2 y ≥ 80 ; x, y ≥ 0 is
a) 320 b) 300 c) 230 d) none
8) Consider o LPP given by ; minimum Z=6x+ 10y ; subject to x≥ 6; y≥2; 2x + y≥ 10; x, y ≥ 0
Redundant constraints in this LPP are
a) x≥ 0, y≥ 0 b) x≥ 6, 2x +y≥ 10 c) 2x + y≥ 0, y≥ 10 d) none
9) The objective function Z= 4x + 3y can be maximized subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y≤ 6; x, y≥ 0
a) at only one point b) at 2 points only c) at an infinite number of points d) none
10) If the constant in a linear programming problem are changed
a) the problem is to be re-evaluated
b) solution is not defined
c) the objective function has to be modified
d) the change is constraints is ignored
11) which of the following statements is correct ?
a) every LPP admits an optimal solution
b) A LPP admits unique optimal solution
c) if LPP admits two optimal solutions it has an infinite number of optimal solutions
d) the set of all feasible solutions of LPP is not a converse set
12) Which of the following is not a convex set ?
a) {(x,y): 2x + 5y< 7}
b) {(x,y): x² + y²≤ 4}
c) {x: |x| = 5}
d) {(x,y): 3x² + 2y²≤ 6}
13) By graphical method, the solution of linear programming problem
Maximize: Z= 3x + 5y
Subject 3x + 2y≤ 18; x≤ 4 ; x ≤6 , x≥ 0, y≥ 0 is
a) x=2, y= 0, Z= 6
b) x=2, y= 6, Z= 36
c) x=4, y= 3, Z= 27
d) x=4, y= 6, Z= 42
14) The region represented by the inequation system x, y ≥ 0, y≤ 6, x+ y ≤ 3 is
a) unbounded in first quadrant
b) unbounded in first and second quadrants
c) bounded in first quadrant d) none
15) The point at which the maximum value of x + y, subject to the constraints x + 2y≤ 70, 2x + y≤ 95, x, y ≥ 0 is obtained is
a) (30, 25) b) (20, 35) c) (35,20) d) (40,15)
16) The value of objective function is maximum under linear constraints
a) at the centre of feasible region
b) at (0,0)
c) at any vertex of feasible region
d) the vertex which is maximum distance from (0,0)
17) The corner points of the feasible region determined by the following system of linear inequalities: 2x + y≤ 10 ; x + 3y≤ 15, x, y≥ 0 are (0,0),(5,0),(3,4) and (0,5). Let Z= px + qy, where p,q> 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is
a) p= q b) p= 2q c) p= 3q d) 3p= q
18) The corner points on the feasible region determined by the system of linear constants are:
(0, 10),(5,5), (15 ,15),(0,20). Let Z= px + qy, where p,q > 0 Condition on p and q so that the maximum of z occurs at both the points (15,15) and (0,20) is
a) p= q b) p= 2q c) 2p= q d) 3p= q
19) Corner points of the feasible region determined by the system of linear constraints (0,3), (1,1) and (3,0). Let Z= px+ qy, where p,q> 0. Condition on p and q so that the minimum of Z occurs at (3,0) and (1, 1) is
a) p= 2q b) 2p= q c) p= 3q d) p= q
20) Corner points of the feasible region for an LPP are :(0,2),(3,0),(6,0),(6,8), and (0,5). Let Z= 4 x + 6y the objective function. The minimum value of z occurs at
a) (0,2) only b) (3,0) only
c) the midpoint of the line segment joining the points (0,2) and (3,0)
d) any point on the line segment joining the points (0,2) and (3,0)
21) Corner points of the feasible region for an LPP are (0,2),(3,0),(6,0) and (0,5). Let Z= 4x +6y br the objective function. Then , max. z - min z=
a) 60 b) 48 c) 42 d) 18
22) The feasible region for an LPP is shown in figure
. Let Z= 3x- 4y be the objective function. Maximum value of z is
a) 0 b) 8 c) 12 d) -18
23) The corner points of the feasible region determine by the system of linear constraints are (0,0),(0,40), (60,20),(60,0). The objective function is 4x + 3y.
Compare the quantity in column A and column B
Column A column B
Max of z 324
a) The quantity in column A is greater
b) the quantity of colum B is greater
c) the two quantities are equal
d) the relationship cannot be determined on the basis of information supplied
24) The feasible region a LPP is shown in figure.
Let Z= 3x - 4y be the objective function. Minimum of Z at
a) (0,0) b) (0,8) c) (5,0) d) (4,10 0)
25) In question number 24, Maximum of z occurs at
a) (5,0) b) (6,5) c) ( 6,8) d) (4,10)
26) In question number 24, (Maximum value of z+ Minimum value of z) is equals to
a) 13 b) 1 c) -13 d) -17
FILL in the blanks
1) In LPP, the objective function is always_____
2) The feasible region for an LPP is always a _____ polygon.
3) If the feasible region for an LPP is _____, then the optimal value of the objective function z= ax + by may or maynot exit.
4) A feasible region of a system of linear inequalities is said to be ____, if it can be enclosed within a circle.
5) A corner point of a feasible region is a point in the region which is the____ of two boundary lines.
6) The maximum value of z= 4x + 2y :0; subject to the constraints 2x + 3y≤ 18, x + y≤ 10, x≥ 0, y≥ 0, is_____
7) The point which provide the optimal solution of the linear programming problem
Max z= 45x + 55y, 6x + 4y ≤ 120; 3x + 10y ≤ 180; x, y≥ 0.
has the coordinates_____
8) The maximum value of z= 3x + 4y subject to the constraints x + y≤ 40, x + 2y≤ 60, x, y≥ 0 is______
9) The minimum of the objective function z= 2x + 10y for linear constraints x - y≥ 0, x - 5y ≤ -5, x, y≥ 0 is_____
10) The coordinates of the point for minimum value of z= 7x -8y subject to the conditions x + y≤ 20; y≥ 5, x, y≥ 0 are_____
11) In a LPP , if the linear inequalities or restriction on the variables are called____
12) In a LPP , if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same ____value.
13) In a LPP , the linear function which has to be maximized or minimised is called a linear____ function .
14) The common region determined by all the linear constraints of a LPP is called the ______ region.
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