SET THEORY (2)
1) For any Set A, (A')' is equals to
a) A' b) A c) θ d) none
2) Let A and B be two sets in the same universal set, then, A - B=
a) A ∩B b) A' ∩B c) A ∩ B' d) none
3) The number of subsets of a set containing n elements is
a) n b) 2ⁿ -1 c) n² d) 2ⁿ
4) For any two sets A and B, A ∩ B A∩( A∪ B)=
a) A B) B c) θ d) none
5) if A={1, 3, 5}, B={2,4}, then
a) 4 ∈ A B) {4}⊂ A c) B ⊂ A d) none
6) The symmetric difference of A and B is
a) (A - B) ∩ (B - A)
b) (A - B) ∪ (B - A)
c) (A ∪ B) - (A ∩ B)
d) {(A ∪B) - A} ∪ {(A ∪ B) - B}
7) The symmetric difference of A={1,2,3} and B={3, 4,5} is
a) {1,2} b) {1,2,4,5} c) {4,3} d) {2,5, 1, 4, 3}
8) For any two sets A and B, (A - B) ∪ (B - A) =
a) (A - B) ∪ A
b) (B - A) ∪ B
c) (A ∪B) - (A ∩B)
d) (A ∪B) ∩(A ∩ B)
9) Which of the following statement is false:
a) A - B= A ∩ B'
b) A - B = A - (A ∩B)
c) A - B = A - B'
d) A - B = (A ∪B) - B
10) For any three sets A, B and C
a) A ∩(B - C)= (A ∩B) - (A ∩C)
b) A ∩(B - C)= (A ∩B) - C
c) A ∪(B - C) = (A∪B) ∩ (A ∪C')
d) A ∪(B - C) = (A ∪ B) - (A ∪ C)
11) Let a={x: x ∈ R, x ≥ 4} and B={x ∈ R; x < 5}. Then, A ∩ B=
a) (4,5) b) (4,5) c) (4,5) d) (4,5)
12) Let U be the universal set containing 700 elements. if A and B are sub-sets of U such that n(A)= 200, n(B)= 300 and n(A∩B)= 100. Then , n (A' ∩ B')=
a) 400 b) 600 c) 300 d) none
13) Let A and B be two sets such that n(A)= 16, n(B)= 14, n(A∪B)= 25. Then n(A∩B) is equal to
a) 30 b) 50 c) 5 d) none
14) If A={1,2,3,4,5}, then the number of proper subsets of A is
a) 120 b) 30 c) 31 d) 32
15) In set builder method the null set is represented by
a) { } b) ∅ c) {x : x ≠ x} d) {x : x=x}
16) If A and B are two disjoint sets, then n(A∪B) is equals to
a) n(A)+ n(B)
b) n(A)+ n(B) - n(A ∩ B)
c) n(A)+ n(B)+ n(A ∩B)
d) n(A) n(B)
e) n(A) - n(B)
17) For two sets A ∪ B= A iff
a) B ⊆ A B) A ⊆ B c) A≠ B d) A= B
18) If A and B are two set such that n(A)= 70, n(B)= 60, n(A∪B)= 110 , then n(A ∩B) is equal to
a) 240 b) 50 c) 40 d) 20
19) If A and B are two apgiven sets , then A ∩ (A∩ B)ᶜ is equals to
a) A b) B c) ∅ d) Aᶜ ∩ Bᶜ
20) If A={x : x is a multiple of 3} and, B={x: x is a multiple of 5}, then A - B is
a) A ∩B b) A∩B' c) A' ∩B' d) (A ∩B)'
21) In the city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, Person travelling by car or bus is
a) 80% b) 40% c) 60% d) 70%
22) An investigator interviewed 100 students to determine the performance of three drinks; milk, coffee and tea. The investigator reported the 10 students take all the three drinks milk, coffee and tea; 20 students take milk and coffee; 25 students take milk and tea; 12 students take milk only; 5 student take coffee only and 8 students take tea only; then the number of students who did not take any of three drinks is
a) 10 b) 20 c) 25 d) 30
23) Two finite sets have m and n elements . The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set. Then, the values of m and n are:
a) 7, 6 b) 6, 3 c) 6, 4 d) 7, 4 e) 3, 7
24) In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70, Chemistry 40; mathematics and physics 30; mathematics and chemistry 28; physics and chemistry 23; mathematics, physics, chemistry 18. How many students have offered mathematics alone ?
a) 35 b) 48 c) 60 d) 22 e) 30
RELATIONS-(2)
1) If A={1,2,4}, B={2,4,5}, C={2,5}, then (A - B) x (B - C) is
a) {(1,2),(1,5),(2,5)}
b) {(1,4)} c) (1,4) d) none
2) If R is a relation on the set A={1,2,3,4,5,6,7,8,9} given by x R y <=> y = 3x, then R=
a) {(3,1),(6,2),(8 ,2),(9,3)}
b) {(3,1),(6,2),(9,3)}
c) {(3,1),(2, 6),(3,9) d) none
3) Let A={1,2,3}, B={1,3,5}, if relation R from A to B is given by R={(1,3),(2,5),(3,3)}. Then , R⁻¹ is
a) {(3,3),(3,1),( 5,2)}
b) {(1,3),(2,5),(3,3)}
c) {(1,3),(5,2)} d) none
4) if A={1,2,3}, B={1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is
a) {1,4,6,9} b) {4,6,9} c) {1} d) none
5) if R = {(x,y): x, y ∈Z, x²+ y²≤ 4} is a relation on Z, then domain of R is
a) {0,1,2} b) {0, -1,-2} c) {-2,-1, 0,1,2} d) none
6) A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y <=> is relatively prime to y. Then , domain of R is
a) {2,3,5} b) {3,4} c) {2, 3,4} d) {2,3,4,5}
7) A relation Φ from C to R is defined by x Φ y <=> |x |= y. Which one is correct ?
a) (2 + 3i) Φ 13 b) 3Φ (-3) c) (1+ i)Φ2 d) i Φ 1.
8) Let R be a relation on N defined by x + 2y = 8. The domain of R is
a) {2,4,8} b) {2,4,6,8} c) {2,4,6} d) {1,2,3,4}
9) R is a relation from {11, 12, 13} to {8, 10,12} defined by y = x - 3. Then R⁻¹ is
a) {(8,11), (10,13)} b) {(11,8),(13,10)} c) {(10,13),(8,11),(12,10)} d) none
10) If the set A has p elements, B has q elements , then the number of elements in A x B is
a) p+ q b) p+ q+1 c) pq d) p²
11) Let R be a relation from a set A to a set B, then
a) R= A ∪B b) R= A ∩B c) R⊆ A x B d) R ⊆ B x A
12) If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is
a) 2ᵐⁿ b) 2ᵐⁿ -1 c) 2mn d) mⁿ
13) If R is a relation on a finite set having n elements, then the number of relations on A is
a) ₂n b) ₂n² c) n² d) nⁿ.
1) n arithmetic means are inserted between 1 and 41. If the sum of the means and the two numbers be 189, find the value of n.
2) If (b + c)/a, (c + a)/b, (a+ b)/c are in AP and a+ b + c ≠ 0, show that 1/am 1/b, 1/c are in AP.
3) If b²+ bc + c², c²+ ca + a², a²+ ab + b² are in AP, show that a, b, c are in AP.
4) If a, b, c are in AP show that (1/a) (1/b + 1/c), (1/b)(1/c + 1/a), (1/c) (1/a + 1/b) are in AP.
5) If a,b,c are in AP, show that ab + cd + ad = 3bc.
6) A farmer undertakes to pay off a debt of Rs 2700 by monthly installments. He pays Rs 200 as the first instalment and increases every subsequent installment by Rs 25 over the immediate previous instalment. In how many installment his debt will be cleared up ?
7) The sum of first n terms of an AP is denoted by Sₙ. If Sₙ = n²p and Sₘ = m²p, m≠ n, then show that, Sₚ = p³.
8) There are 100 terms in an AP. If the sum of the terms occupying the even places is 1600 and the sum of the terms occupying the odd places is 1300. Find the two middle terms of the AP.
9) If the sum of first p terms of an AP be P and the sum of first q terms be Q, show that the first term of the progression will be {Pq(q -1) - Qp(p -1)}/{pq(q - p)}.
10) If the first and the last terms of an AP are a and l respectively and the sum of the terms is S, show that the common difference is (l² - a²)(2S - (l + a)).
11) The sum of three numbers in GP is 7 and the sum of their squares is 21, find the sum of their cubes.
12) The sum of three positive numbers in GP is 65 and the product of the first and the third number is 225. Find the numbers.
13) Find, without assuming the summation formula, the sum of the GP
2+4+8+.....to 20 terms.
14) If (a + bx)/(a - bx)= (b + cx)/(b - cx)= (c + dx)/(c - dx) (x ≠ 0), prove that a, b, c, d are in GP.
15) If a,b,c,d are in GP, show that, 1/(a² + b²) , 1/(b² + c²), 1/(c² + d²) are in GP.
16) The 5th term of a GP is 8 and the 13th term is 128. Find the sum upto 12th term.
17) The sum of three consecutive terms of an AP is 15; if 1 is subtracted from the second term, the resulting numbers form a GP. Find the terms.
18) The second, third and sixth terms of an AP are three consecutive terms of a GP. Find the common ratio of the GP.
19) Find the sum of:
a) 1+11+111+1111+ .....to n terms.
b) 0.8+0.88+0.888+ .....to n terms.
c) 1+ 4+ 10+22+ ....to n terms.
d) 1+ (1+2) + (1+2+2²)+.....to n terms.
e) 1+ (1+ x)+ (1+ x + x²)+.....to n terms.
f) 1/2+ 3/2²+ 5/2³+ ......+ (2n -1)/2ⁿ.
20) If the first, second and last terms of a GP are a, b and l respectively show that the sum of the terms is (bl - a²)/(l - a).
21) If u₁, u₂, u₃, .... are in GP whose common ratio is k, then express the value of u₁u₂ + u₂u₃ + .....+ uₙuₙ₊₁ in terms of k and u₁.
22) If pth, qth , rth terms of an AP are in GP, then show that the common ratio of the GP is (q - r)/(p - q).
23) If the pth, qth, rth and sth terms of an AP are in GP, then show that p - q, q - r, r - s are in GP.
24) Solve: 1+ 3+ 3²+......3ˣ = 1093.
25) If a, b, c, d are in AP and a, c, d are in GP, show that a² - d² = 3(b² - ad).
26) The product of three numbers in GP is 512. If 8 be added to the first and 6 to the second, the resulting numbers with the third number from form an AP. Find the numbers.
27) If a, b, c are in AP and a, b- a, c - a are in GP, show that a= b/3= c/5.
28) If (a - x)/px = (a - y)/qy = (a - z)/rz and p, q, r are three consecutive terms of an AP, show that 2/y = 1/x + 1/z.
29) There are even number of terms in a GP. If the sum of the progression is thrice the sum of the odd terms, then find the common ratio of the GP.
30) Prove that the product of n geometric means between a and b is (ab)ⁿ⁾².
31) The arithmetic mean of two numbers is A and their geometric mean is G; find the numbers.
32) Find the least value of n for which 1+ 3+ 3² + ....+ 3ⁿ⁻¹ > 900.
33) If tₙ be the nth term of a GP and t₁ = 2, find the minimum value of 2t₂ + 3t₃. Find also the value of the common ratio for which the expression will be minimum.
34) A man saves Rs 4 in the first month. Now from the second month he saves in every month twice the savings of the preceding month. Thus after 8 months, from the 9th month he saves in every month Rs 4 less than the savings of the immediate previous month. What will be his total savings in 16 months.
35) If logᵧx, logᵥy, logₓv are in GP and also in AP, show that, x= y = z.
36) Simplify:
a) 20/(√3 - √-2) + 30/(3√-2 - 2√3) - 14/(2√3 - √-2).
b) (cos30°+ i sin30)⁴/(cos60 + i sin 60)². 1
37) If {(1+ i)/(1- i)}³ - {(1- i)/(1+ i)}³ = p + iq, find (p, q).
38) If x= 2+ 3i and y= 2- 3i, find the value of (x³ - y³)/(x³ + y³).
38) If x= 1+ 3i, find the value of x⁴- 5x³ + 18x² - 34x +2.
39) If x= - 5+ 2√-4, find the value of x⁴+ 9x³+ 35x² - x +4.
40) If x= 2+ i, find the value of x³ - 5x²+ 9x -5.
----------------+++++++------------------+-+--
MATHS- 2
(Attempt all questions from section A and all questions Either from section B or Section C
Section - A (80 Marks)
1) (10 x 2 =20)
a) Let T{x: (x+5)/(x -7) - 5 = (4x -40)/(13- x)}. is T an empty set?
b) Find the domain and the range of the function
f(x)= 1/√(5- x).
c) Show: tan50= tan40+ 2 tan 10.
d) In a triangle ABC, a= 1, b=√3 and c= 7/6. Find the other two angles.
e) If (2+ 3i)/(3- 4i)= a + ib, find the value of a and b.
f) If m,n be the roots of the equation ax²+ bx + c= 0 form an equation whose roots are 1/m. 1/n.
g) Find n if ⁿC₅ = ⁿC₇.
Or cos2A/(1+ sin2A)= tan(π/4 - A).
h) Find dy/dx of (xeˣ + x⁷+ (x +1)√x)/x.
i) Evaluate: lim ₓ→₀ (sinx - 2 sin3x + sin5x)/x.
j) While shuffling a pack of 52 cards, 2 cards are accidentally dropped. Find the probability that the missing cards are of different colours.
2) a) If f(x)= y= (ax - b)/(cx - a), then show that f(y)= x where x≠ a/c.
b) Find the term containing x¹⁰ in expansion of (2x²- 3/x)¹¹. (2+2)
3) a) Show that √{(1+ sinx)/(1- sinx)}+ tan(π/4+ x/2).
b) Solve for x: 4 sin⁴x + cos⁴x = 1. (2+2)
4) Using mathematical induction show that 5ⁿ⁺¹ + 4. 6ⁿ - 9 is divisible by 20 and n ∈ N.
Or
If (z -i)/(z -1) is purely imaginary show that the point z lies on the circle whose centre is the point (1/2) (1+ i) and radius is 1/√2. (4)
5) 7 candidates are to be examined, 2 in mathematics and the remaining in different subjects. In how many ways can they be seated in a row so that the two examinees in mathematics may not sit together.
Or
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if---
i) They can be of any colour
ii) two must be white and two must be red. (4)
6) Determine whether the expansion of (x²- 2/x)¹⁸ will contain a term containing x¹² ? If it contains also find out the term. (4)
7) Find the equation of the lines through the point (3,2) which make and angle of 45° with the line x - 2y = 3. (4)
8) Find the equation of the circle passing through the point (7,3) having 3 units and whose centre lies on the line y= x -1. (4)
Or
Find the equation of tangents to the circle x²+ y²- 2x - 4y - 20 =0 through the point P(8,1).
9) Differentiate the function (x³+ 2x) by first principal of differentiation. (4)
10)a) If the equation 3x²+ px +1=0 and 2x²+ ax +1=0 have a common root, show that 2p²+ 3q²- 5pq +1= 0.
b) Show that for all real values of x the expression (x²- 2x +4)/(x²+ 2x +4) has the greatest value 3 and least value 1/3. (3+3)
11)a) 1/(x + y), 1/2y, 1/z are in AP, then show that y is the geometric mean between x and z.
Or
Find the sum of the series to the n terms 1 +3 +7+ 15 +31+... to n terms. 6
12) The diameter of circle (in mm) drawn in a design are given below:
Diameter (in mm) no of circles
33-36 15
37-40 17
41-44 21
45-48 22
49-52 25
Calculate the standard deviation and mean diameter of the circles . 6
Section - B
13) (3x2=6)
a) Find the equation of the parabola with focus (6,0) and directrix x= -6.
b) Construct the table: ~ (p v ~ q)
c) Using contrapositive method proved that is n² is an even integer the x is also an even integer the x is also an even integer .
14) a) Find the equation of the ellipse if its foci are (±2,0) and the length of the latus rectum is 10/3.
Or
Find the equation of the hyponovaloids whose foci are (0,±13) and length of the conjugate axis is 24. (4)
15) Find the coordinates of the point P which is five-sixth of the way from A(-2,0,6) to B(10,-6,-12).
Or
The centroid of ∆ ABC is at the point (1,1,1). if the co-ordinates of A and B are (3,-5,7) and (-1,7,-6) respectively , find the co-ordinate of the point C. 6
16) A double olimate of the parabola y²= 4ax is the length 8a. Prove that the lines from the vertex to its ends are at right angles.
Section - C. (20 Marks)
16)a) Let mean monthly salary paid to all employees of a company is Rs 8300. The mean monthly salary paid to male and female employees was Rs 8000 and Rs 9000 respectively. Determine the percentage of males and females employed by the company. (2)
b) Find the values of a and b from the following data: (4)
Marks. No of students
00-05 7
05-10 a
10-15 25
15-20 30
20-25 b
Total. 100 given that the third decile is 1.1 (4)
Or
Calculate the mode of the following data:
C-I frequency
05-15 6
15-25 11
25-35 21
35-45 23
45-55 14
55-65 5
17) a) If in a sample of n observation given that ∑ d²= 55 and rank correlation r= 2/3. Then find the value of n. (2)
b) The mathematical aptitude score of 10 computer programmers with their job performance is given below: (4)
Person math(sc) job rating
A 7 8
B 5 16
C 1 8
D 4 9
E 3 5
F 0 4
G 2 3
H 6 8
I 8 17
J 9 12
Calculate spearman's rank correlation.
Or
Find Karl Perason's coefficient of correlation between X and Y for the following data:
X: 5 4 3 2 1
Y: 4 2 10 8 6
18) Find the consumer price index for 2007 on the basis of 2005 from the following data using weighted average of price relative method. (4)
Items: Food Rent cloth fuel
Price ('05): 200 100 150 50
Price ('07): 280 200 120 100
Weight: 30 20 20 10
19) Obtain the three year moving average for the following series of observations.
Year: Sales(Rs '0000)
1995 3.6
1996 4.3
1997 4.3
1998 3.4
1999 4.4
2000 5.4
2001 3.4
2002 2.4
Represent these graphically. (4)
H. W-1
1) If the pth, qth and rth terms of an AP are respectively a⁻¹, b⁻¹, and c⁻¹ show that, (q- r)bc + (r - p)ca + (p - q)ab =0.
2) How many terms of the series 1/2+ 1/3+ 1/6+ ..... must be taken so that the sum may be (-3/2) ?
3) Find the sum of 1- 3 + 5 - 7 + 9 - 11+ .....to n terms.
4) How many even numbers are there between 15 and 115 ? Find the sum of all these numbers.
5) Find the sum of all the numbers between 200 and 300 which are multiples of 7.
6) If (p +1)th term of an AP be a, find the sum of first (2p +1) terms of the AP.
7) If the 11th term of an AP be 25, find the sum of first 21 terms of the AP.
8) There are (2n +1) terms in an AP. Show that the ratio of the sum of odd terms and the sum of even terms is (n +1) : n.
9) Find the 99th term of the series 2+ 7+14+23+34+....
10) How many terms are there in the series 1+ 3+6+10+15+21+...+5050?
11) The sum of four numbers in AP is 20 and the sum of their squares is 120, find the numbers.
12) The sum of six numbers in AP is 345 and the difference between the first and the sixth is 55, find the numbers.
13) The fourth term of an AP is thrice the first term and the seventh term exceeds twice the third term by 2. Find the sum of first ten terms of the AP.
14) If the sum of first n terms of an AP is 40, the common difference is 2 and the last term is 13, find the value of n.
15) The sum of first n terms of two AP's are in ratio (3n +1) : (3n -1). Find the ratio of their tenth terms.
16) Prove that the sum of n arithmetic means between two numbers is n times the arithmetic mean of those two numbers.
____&&&______________&&&&&&_____
FUNCTIONS - 1
1) Let A={1, 2,3}, B={2,3,4} , then which of the following is a function from A to B?
a) {(1,2),(1,3),(2,3),(3,3)}
b) {(1,3),( 2,4)}
c) {(1, 3),(2,2) ( 3,3}
d) {(1,2),(2,3),(3,2),(3,4)}
2) If f: Q--> Q is defined as f(x)= x², then f(9) is equal to
a) 3 b) -3 c) {-3,3} d) φ
3) Which one of the following is not a function?
a) {(x,y): x, y ∈ R, x²= y}
b) {(x,y): x, y ∈ R, x= y²}
c) {(x,y): x, y ∈ R, x= y³}
d) {(x,y): x, y ∈ R, x³= y}
4) If f(x)= cos(log x), then
f(x²) f(y²)- (1/2){ f(x²/y²}+ f(x²y²)} has the value
a) -2 b) -1 c) 1/2 d) none
5) If cos(log x), then f(x)f(xy) - (1/2){f(x/y)+ f(xy)} has the value
a) -1 b) 1/2 c) -2 d) none
6) Let f(x)= |x -1|. Then
a) f(x²)={f(x)}²
b) f(x+ y)=f(x)f(xy)
c) f(|x|)= |f(x)| d) none
7) The range of f(x)= cos[x], for -π/2< x<π/2 is
a) {-1,1,0} b) {cos 1, cos2, 1} c) {cos1, - cos 1, 1} d) [-1,1]
8) Which of the following are functions ?
a) {(x,y): y²= x, x, y ∈ R}
b) {(x,y): y = |x|, x, y ∈ R}
c) {(x,y): x²+ y²= xp1, x, y ∈ R}
d) {(x,y): x²- y²= 1, x, y ∈ R}
9) If f(x)= log{(1+ x)/(1- x)} and g(x)= (3x + x³)/(1+ 3x²), then f(g(x)) =
a) f(3x) b) f(x)}³ c) 3f(x) d) - f(x)
10) If A={1,2,3}, B={x,y}, then the number of functions that can be defined from A into B is
a) 12 b) 8 c) 6 d) 3
11) If f(x)= log{(1- x)/(1- x)}, then f{(2x)/(1+ x²)} =
a) (f(x))² b) (f(x))³ c) 2f(x) d) 3f(x)
12) If f(x)=cos(logx), then value of
f(x)f(4) - (1/2) {f(x/4)+ 4f(x)} is
a) 1 b) -1 c) 0 d) ±1
13) If f(x)= (2ˣ + 2⁻ˣ)/2, then f(x+ y)f(x - y) is equal to
a) (1/2) [f(2x)+ f(2y)]
b) (1/2) [f(2x)- f(2y)]
c) (1/4) [f(2x)+ f(2y)]
d) (1/4) [f(2x) - f(2y)]
14) If f(x) - 3 f(1/x)=x²(x ≠ 0), then f(2) is equal to
a) -7/4 b) 5/2 c) -1 d) none
15) Let f: R--> R be defined by f(x)= 2x + |x|. Then f(2x)+ f(- x)- f(x)=
a) 2x b) 2|x| c) -2x d) -2|x|
16) If f(x)=log{(1+ x)/(1- x)} thenf{2x/(1+ x²)} is equal to
a) (f(x))² b) (f(x))³ c) 2f(x) d) 3f(x)
17) If x≠ 1 and f(x)= (x+1)/(x-1) is real function, then f(f(f(2))) is
a) 1 b) 2 c) 3 d) 4
18) If f(x)= cos(logx), then f(1/x)f(1/y) - (1/2) {f(xy)+ f(x/y)} is equal to
a) cos(x - y) b) log(cos(x - y)) c) 1 d) cos(x + y)
19) Let f(x)=x, g(x)= 1/x and fph(x)= f(x)g(x). Then, h(x)= 1
a) x ∈R b) x ∈Q c) x ∈R - Q d) x ∈R, x ∈Q
20) If f(x)=(sin⁴x + cos²x)/(sin²x + cos⁴x) for x ∈R , then f(2002)=
a) 1 b) 2 c) 3 d) 4
21) The function f: R--> R is defined by f(x)=cos²x + sin⁴x. Then f(R)=
a) [3/4,1) b) (3/4,1] c) [3/4,1] d) (3/4,1)
22) Let A={x ∈R: x≠ 0, -4≤ x ≤4} and f: A ∈ R be defined by f(x)= |x|/x for x ∈ A. Then [1,-1] b) x: 0≤ x ≤4] c) [1] d) {x: -4≤ x≤0}
23) If f: R--> R and g: R-->R are defined by f(x)=2x +3 and g(x)= x²+7, Then the value of x such that g(f(x)) = 8 are
a) 1,2 b) -1,2 c) -1,-2 d) 1,-2
26/7/26
RELATIONS - 1
1) If A={1, 2,4}, B={2,4,5}, C={2,5}, the pn (A- B) x (B - C) is
a) {(1,2),(1,5),(2,5)} b) {(1,4)} c) (1,4) d) {(1,4)}
2) If R is a relation on the set A={1,2,3,4,5,6,7,8,9} given by x R y <=> y= 3x, then R=
a) {(3,1),(6,2),(8,2),(9,3)}
b) {(3,1),(6,2),(9,3)}
c) {(3,1),(2,6),(3,9)} d) none
3) Let A={1,2,3}, B={1,3,5}, if relation R from A and B is given by R={(1,3),(2,5),(3,3)}. Then, R⁻¹ is
a) {(3,3),(3,1),(5,2)}
b) {(1,3),(2,5),(3,3)}
c) {(1,3),(5,2)} d) none
4) If A={1,2,3}, B={1,4,6,9} and R is a relation from A to B defined by 'x' is greater than y. The range of R is
a) {1,4,6,9} b) {4,6,9} c) {1} d) none
5) If R={(x,y: x, y∈ Z, x²+ y²≤ 41} is a relation on Z, then domain of R is
a) {0,1,2} b) {0,-1,-2} c) {-2,-1,0,1,2} d) none
6) A relation R is defined from {2,3,4,5} to {3,6,7,10} by : x R y <=> x relatively prime to y. Then , domain of R is
a) {2, 3,5} b) {3,5} c) {2,3,4} d) {2,3,4,5}
7) a relation φ from C to R is refined by x φ y <=> |x|= y. Which one is correct ?
a) (2+ 3i) φ 13 b) 3φ (-3) c) (1+ i) φ 2 d) iφ 1.
8) Let R be a relation on N defined by x+2= 8. The domain of R is
a) {2,4,8} b) {2,4,6,8} c) {2,4,6} d) {1,2,3,4}
9) R is a relation from {11, 12, 13} to { 8,10,12} defined by y= x -3. Then, R ⁻¹ is
a) {(8,11),( 10, 13)}
b) {( 11, 8),( 13, 10)}
c) {10, 13),(8, 11),(12, 10)} d) none
10) if the set A has p elements , B has q elements, then the number of elements in A x B is
a) p+ q b) p+ q +1 c) pq d) p²
11) Let R be a relation from a set A to a set B, then
a) R= AUB b) R= A ∩B c) R ⊆ A x B d) R⊆ B x A
12) If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is
a) 2ᵐⁿ b) 2ᵐⁿ -1 c) 2mn d) mⁿ
13) If R is a relation on a finite set having n elements, then the number of relations on A is
a) 2ⁿ b) ₂n² c) n² d) nⁿ.
19/7/26
COMPLEX NUMBER - 2
1) The value of (1+ i)(1+ i²)(1+ i³)(1+ i⁴) is
a) 2 b) 0 c) 1 d) i
2) If (3+ 2i sinθ)/(1- 2i sin θ) is a real number and 0 <θ< 2π, then θ =
a) π b) π/2 c) π/3 d) π/6
3) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a + ib, then 2x 5x 10 x......x (1+ n²) is equal to
a) √(a²+ b²) b) √(a² - b²) c) a²+ b² d) a² - b² e) a+ b
4) If √(a + ib)= x + iy, then possible of √(a - ib) is
a) x² + y² b) √(x² + y²) c) x + iy d) x - iy e) √(x² - y²)
5) z= cos(π/4) + sin(π/6), then
a) |z|= 1, arg(z)=π/4
b) |z|= 1, arg(z)=π/6
c) |z|= √3/2, arg(z)=5π/24
d) |z|= √3/2, arg(z)= tan⁻¹(1/√2)
6) The polar form is (i²⁵)³ is
a) cos(π/2) + i sin (π/2)
b) cos π+ i sinπ
c) cos π- i sinπ
d) cos (π/2) - i sin(π/2)
7) If i² = -1, then the sum i + i² + i³ + ..... upto 1000 terms is equal to
a) 1 b) -1 c) i d) 0
8) If z= -2/(1+ i √3), then the value of arg(z) is
a) π b) π/3 c) 2π/3 d) π/4
9) If a= cosθ + i sinθ, then (1+ a)/(1- a) =
a) cot(θ/2) b) cot θ c) i cot(θ/2) d) i tan(θ/2)
10) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a + ib, then 2,5,10,17....(1+ n²)=
a) a - ib b) a² - b² c) a² + b² d) none
11) If (a² +1)²/(2a - i) = x + it, then x² + y² is equal to
a) (a²+1)⁴/(4a² +1)
b) (a+1)²/(4a² +1)
c) (a² -1)²/(4a² - 1)² d) none
12) The principal value of the amplitude of (1+ i) is
a) π/4 b) π/12 c) 3π/4 d) π
13) The principal positive integer n such that {2i/(1+ i)}ⁿ is a positive integer, is
a) 16 b) 8 c) 4 d) 2
14) If z is a non-zero complex number, then | |conjugate z|²/z. Conjugate z| is equal to
a) |conjugate z/z|
b) |z| c) |conjugate z| d) none
15) If a= 1+ i, then a² equals
a) 1- I b) 2i c) (1+ i)(1- i) d) i - 1
16) If (x + iy)¹⁾³= a + ib , Then x/a + y/b =
a) 0 b) 1 c) -1 d) none
17) (√-2)(√-3) is equal to
a) √6 b) -√6 c) i √6 d) none
18) The argument of (1- i √3)/(1+ i √3) is
a) 60° b) 120° c) 210° d) 240°
18/7/26
SETS - 1
1) For any set A, (A'b)' is equal to
a) A'b b) A c) φ d) none
2) Let A and B be two sets in the same universal set. Then A - B=
a) A∩B b) A'∩B c) A∩B' d) none
3) The number of subsets of a set containing n elements is
a) n b) 2ⁿ -1 c) n² d) 2ⁿ
4) For any two sets A and B, A ∩(A U B)=
a) A b) B c) φ d) none
5) If A={1,3,5,B} and B={2,4}, then
a) 4 ∈ A b) {4} ⊂ A c) B ⊂ A d) none
6) The symmetric difference of A and B is
a) (A- B) ∩(B - A)
b) (A- B) U(B - A)
c) (A U B) - ( A∩B)
d) {(A U B) - A} U {(AUB) - B}
7) The symmetric difference of A={1,2,3} and B={3,4,5} is
a) {1,2} b) {1,2,4,5} c) {4,3} d) {2,5,1,4,3}
8) For any two sets A and B, (A- B) U (B - A)=
a) (A- B) U A b) (B- A) U B c) (AU B) - (A∩B )
9) Which of the following statements is false:
a) A- B = A ∩B' b) A- B= A - (A ∩ B)
c) A- B= A - B' d) A- B = (AU B) - B
10) For any three sets A, B and C
a) A ∩(B - C)= (A∩B) - (A ∩C)
b) A ∩(B - C)= (A ∩B) - C
c) AU(B - C)= (AU B) ∩ (A U C')
d) AU(B - C)= (AU B) - (A U C).
11) Let A={x: x ∈R, x≥ 4} and B={x ∈ R: x < 5}, then A ∩B=
a) (4,5) b) (4,-5) c) (-4,5) d) (-4,-5)
12) Let U be the universal set containing 700 elements. If A, B are subsets of U such that n(A)= 200, n(B)= 300 and n(A∩B)= 100. Then n(A' ∩ B')=
a) 400 b) 600 c) 300 d) none
12/7/26
Complex number
1) Simplify: 1+ i² + i⁴+ i⁶.
2) Show that: [i³⁷ - (1/i)⁴¹]³= 8i.
3) Find the conjugate of (2+ 3i)².
4) Find x and y if (3x -7)+ 5iy = 2y +3 -4(1- x)i.
5) Find the modulus of -12+ 5i.
6) Express the reciprocal of the complex number 3+ i √5 in the form of a+ ib.
7) Find the modulus of (1+ i)/(1- i) + (1- i)/(1+ i).
8) Find the solution of the equation |1- i|ˣ= 2ˣ.
9) Show that the points representing the complex numbers (3+ 3i), (-3-3i) and (-3√3+ 3√3 i) are the vertices of an equilateral triangle.
10) Express (3+ i)/(-5- 4i) in the standard form a+ ib.
11) Find the modulus of (2+ 3i)/(3+ 2i).
12) Prove that the representative points of the complex numbers 1+ 4i, 2+ 7i, 3+ 10i are collinear.
13) Find the modulus of
{(3+2i)(1+ i)(2+ 3i)}/{(3+ 4i)(4+ 5i)}.
14) If x+ iy = √{(a+ ib)/(c + id)}, show that (x²+ y²)²= (a²+ b²)/(c²+ d²).
Short Questions Answer Type
i) Find the conjugate of the complex number (2+ 3i)/(2 - 3i). -5/13 -12i/13
ii) Find the modulus. of (1 + i)³/(1 - i³). 2
iii) Find the amplitude of -3-3i. -3π/4
iv) Resolve into factors: a²+ ab + b². (a - bω)(a - bω²)
v) Find the square root of -i . ±(1/√2) (1- i)
vi) Evaluate: (1- ω²)(1- ω⁴)(1- ω⁸)(1- ω¹⁶). 9
vii) If z= x + iy and |z - 2| = |2z - 1|, prove that x²+ y²=1.
viii) Find the smallest positive integer n for which {(1+ i)/(1- i)}ⁿ=. 1. 4
ix) Find the square root of q+ √(q²- 1), 0< q < 1. ±(1/√2) {√(1+ q)+ i √(1- q)}
x) If x = a+ b, y= aω + bω², z= aω²+ bω , find xyz. a³+ b³
xi) Solve: |z| - z = 1+ 2i. (z = x+ iy). z= 3/2 - 2i
xii) The sum and the product of the two complex numbers are respectively 6 and 25. Find the numbers. 3+4i, 3 -4i
xiii) Determine the three cube roots of i. - i, (i+√3)/2, (i - √3)/2
Choose the correct option
i) The absolute value of (-i/2) is
A) i/2 B) 1/2 C) 1/√2 C) none
ii) The value of √-4 . √-9 is
A) 6 B) -6 C) -6 or 6 D) none
iii) The value of ω⁴+ ω⁸+ ω⁻¹. ω⁻² is
A) ω B) ω² C) -1 D) 0
iv) The argument of ia (a < 0), is
A) 0 B) π/2 C) 3π/2 D) π
v) The value of (1- ω+ω²)⁵ + (1+ ω- ω²)⁵.
A) -1 B) 1 C) -32 D) 32
vi) The amplitude of (a + ib)² is
A) tan⁻¹(b/a) B) 2tan⁻¹(b/a) C) 2 tan⁻¹(a/b) D) tan⁻¹(a/b)
vii) If z = x + iy, the value of (amp z + amp of Conj of z) is
A) 0 B) π/2 C) π D) none
viii) The quantity, whose cube root is (1/2) (√3 + i), is
A) -1 B) 1 C) - i D) i.
ix) The value of (1+ ω)(1+ω²)(1+ ω⁴)(1+ ω⁸)......2n factors, is
A) 1 B) 2ⁿ C) -1 D) ωⁿ
x) For any complex number z, the minimum value of |z|+|z -1| is
A) 0 B) 1/2 C) 1 D) 3/2
xi) The real part of (2- i)²/(2+ i) is
A) -2/5 B) -6/5 C) -11/5 D) none
i b ii b iii d iv c v d vi b vii a viii d ix a x c xi d
General Questions
1) Show that, (1- ω+ω²)(1-ω²+ ω⁴)(1- ω⁴ +ω⁸)+ ..... to 2n factors = 2²ⁿ.
2) If X + iY be one of the cube roots of x + iy, prove that, 4(X²- Y²)= x/X + y/Y.
3) If x= a + b, y= aα + bβ, z= aβ + bα, where α, β are complex cube roots of unity, show that, x³+ y³+ z³= 3(a³+ b³).
4) If x= ω²- ω -2, evaluate x⁴+ 3x³+ 2x² --11x -4. 3
5) If x = 2- i √3, find the value of K from the equation 2x⁴- 5x³- 3x²+ 41x + K=0. -35
6) If x+ iy =√{(a+ ib)/(c + id)}, prove that (x²+ y²)²= (a²+ b²)/(c²+ d²).
7) If a = cos α + i sin α, b= cosβ+ i sinβ, c= cosγ + i sinγ and a+ b +c=0, show that a²+ b²+ c²=0.
8) If z = x + iy and (z -i)/(z -1)= ib, prove that (x - 1/2)²+ (y - 1/2)²= 1/2.
9) If (1+ x + x²)ⁿ = a₀ + a₁x + a₂x² + a₃x³+ ......+ a₂ₙx²ⁿ , then show that a₀ + a₃ + a₆ + .......= 3ⁿ⁻¹.
10) Prove that (a+ bω+ cω²)³+ (a+ bω²+ cω)³ = (2a- b - c)(2b - c - a)(2c - a - b) and 27abc, if a+ b + c =0.
11) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a+ ib, then show that 2.5.10......(1+ n²)= a²+ b².
12) If If cos k + i sin k and 1+ √(1- a²) = na, prove that (a/2n) (1+ nx)(1+ n/x) = 1+ a cos k.
13) If z = x + iy and arg{(z -1)/(z +1)}=π/4 , show that the locus of (x,y) is a circle.
14) If ω be a Complex cube root of unity, find the simplified value of (a+ bω + cω²)/(c + aω + bω²) + (a+ bω + cω²)/(b + cω + aω²) . -1
15) Prove that the expression x³ᵖ + x³ᑫ⁺¹ + x³ʳ⁺² , where p, q, r are integers, is divisible by x²+ x+1.
16) Show that the points 2+ 3i, 0 ai 1/(-2+ 3i) are collinear.
17) Express a+ ib in the form pω + qω²). (b/√3 - a)ω + (-b/√3 - a)ω² or (- b/√3 - a)ω + (b)√3 - a)ω²
18) Show that the sum and the product of two complex numbers are real if and only if they are conjugate of each other.
19) Solve: z²+ Conj of z=0. (z= x + iy). 0, -1, 1/2 ± i√3/2
20) If a²+ b²+ c²=1 and b + ic = (1+ a)z, then prove that (1+ iz)/(1- iz) = (a+ ib)/(1+ c).
11/7/26
1995
1) Find a quadratic equation whose one root is a square root of -47 + 8√-3.
2) Prove that the expression (x² - yz)³+ (y² - zx)³+ (z² - xy)³ - 3(x² - yz)(y² - zx)(z² - xy) is a perfect square and find its square root.
4) Prove that tan20 tan40 tan80=√3.
5)) Show that tan225 cot405 + tan765 cot675= 0
6) If (r, θ) denotes the polar coordinates then the equation r cos²(θ/2)= 1 represents
a) a circle b) a parabola c) an ellipse d) none
7) If x lies in the interval [0,1] then the minimum value of x²+ x+ 1 is
a) 3/4 b) 1 c) 3 d) none
10//7/26
1) If p= a+ b+ c, q= a+ wb+ w²c, r= w²b+ wc where w is a nonreal cube root of 1, show that p³+ q³+ r³ - 3pqr= 27abc.
2) Solve: x¹⁾³ + (2x -3)¹⁾³ = {12(x -1)}¹⁾³.
3) In an arithmetic progression of n terms (n is even) the two middle terms are p- q, p+ q respectively. Show that the sum of the squares of all the terms of the progression is n[p² + (n² -1)q²/3].
4) Show that 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π - x)] is independent of x.
.5) If 3ˣ - 3ˣ⁻² = 8, find the value of xˣ.
6) If cosα + cosβ = cos(3π/7) and sinα + sinβ= sin(3π/7) find cos²{(α -β)/2}.
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