28/10/24
SET THEORY (1)
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∩( 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 c) 22 d) 30
26/10/24
1) If α , β be the roots of the equation ax²+ bx+ c= 0 and γ, δ those of the equation px²+ qx + r= 0, show that ac/pr = b²/q², if αδ = βγ .
2) If the sum of the first 2n terms of a GP is twice the sum of the reciprocals of the terms, then show that the continued product of the terms is equal to 2ⁿ.
3) How many numbers of four digits can be formed from the numbers 1,2,3, 4? Find the sum of all such numbers (digits being used once only). 24, 66660
4) If 9α=π, find the value of sinα sin2α sin3α sin4α. 3/16
5) If tan θ = (tanα - tan β)/(1- tanα tan β), then show that sin2θ = (sin2α - sin2 β)/(1- sin2α sin2β).
6) If 8R²= a²+ b²+ c² (or, cos²A+ cos²B + cos²C= 1), then show that the triangle ABC is right angled.
7) Solve: cos³ θ cos3 θ+ sin³ θsin3 θ = 1/8. nπ± π/6
8) If {m tan (x- y)}/cos²y= n tan y/cos²(x - y), show that y= (1/2)[x - tan⁻¹{n - m)/(n+ m)} tan x].
30/4/24
1) T
he rth term of an AP is n and it's nth term is r; show that its mth term is r+ n- m.
2) If the 1st term of an AP is 34 and 6th term is 48, then find 2nd, 3rd, 4th and 5th term of the AP. 36.8,39.6,42.4,45.2
3) Find the middle term (or terms) and the sum of the following arithmetic series:
3+7+11+15+....+95. 1176
4) The fifth term of an AP is 30 and its twelfth term is 65, find the sum of its 20 terms. 1150
5) The sum of n terms of an AP is 3n²+ 5n. Find the number of the term which is equal to 152. 25
6) How many terms of the series {22+ 18+ 14+ 10 +....} must be added to get the sum 64 ? Explain the double answer. 4 or 8
7) The nth term of an AP is p; show that , the sum of its first (2n -1) terms is (2n -1)p.
8) Insert 7 arithmetic mean between 1 and 41. 6,11,16,21,26,31,36
9) Find the sum of all natural number between 500 and 1000 divisible by 13. 28405
10) a,b,c,d be respectively the sums of p, q, r terms of an AP , show that, a/p (q -r)+ b/q (r - p)+ c/r (p - q)=0
TEST
(TOTAL 62 MARKS). Time 1:30 minutes
(Attempt all: 1x 10)= 1)
1) There are ____ method of describing a set
a) 1 b) 2 c) 3 d) 4
2) If A ⊆ B, then B is called a ____ of A.
a) subset b) superset c) proper subset d) equal
3) A relation which is reflexible, symmetric and transitive, is called an ___
a) Domain b) range c) co-domain d) equivalence
4) Let f(x)= x² and g(x)=(3x+2) be two real functions. Then, find f/g)(x)
5) simplify: iⁿ + iⁿ⁺¹ + iⁿ⁺² + iⁿ⁺³
a) 1 b) 0 c) i d) -1
6) if x²+5=0 then x is
a) ±√5 b) 5 c) ±√5 d) none
7) If (n+2)!= 2550 (n!) then n is
a) 4 b) 9 c) 49 d) 94
8) Which term of the AP 64,60,56,.......is 0?
a) 15th b) 16th c) 17th d) 18th
9) Which term of the GP 1/4, -1/2, 1, .....is -128?
a) 10th b) 11th c) 12th d) 13th
10) Value of sin(31π/3)
a) √3 b) 1/2 c) √3/2 d) 1
Attempt any 6: 6 x 2= 12)
11) Find the conjugate of 1/(4+ 3i).
12) (x +7)/(x+4) > 1, x ∈ R
13) solve graphically: 2y + 5> 0
14) Find the sum of 32 terms of an AP whose third term is 1 and the 6th term is -11.
15) The 5th, 8th and 11th terms of a GP are a, b c, c respectively. Show that b²= ac.
16) Prove sin(40+x) cos(10+x) - cos(40+ x) sin(10+ x)=1/2
17) Show that sin10 sin50 sin60 sin70= √3/16
Attempt any 10 : 10x 4= 40
18) In an Examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.
19) Let A={1,2,3,4,5} and B={1,4,5}.
Let R be a relation 'is less than ' from A to B.
a) List the elements of R
b) Find the domain, co-domain and range of R.
c) Depict the above relation by an arrow diagram.
20) Find the domain and range of f(x)= √(x²-4).
21) If ³√(x + iy)= (a + ib), show that (x/a + y/b)= 4(a²- b²).
22) In how many ways can 5 children be arranged in a line such that
a) two of them, Ram and Shyam, are always together;
b) two of them, Ram and Shyam, are never together ?
23) How many numbers greater than a million can be formed with the digits 2,3,0,3,4,2,3?
24) Prove that: ²ⁿCₙ = {2ⁿ. (1.3.5........(2n-1))}/n!
25) Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
26) The sum of n terms of two Arithmetic progression are in the ratio (7n -5): (5n +17). Show their 6th terms are equal.
27) Find the sum of the series 0.7+ 0.77+ 0.777+......to n terms.
28) Show that √[2+ √{2+ 2 cos4x}]= 2 cosx.
29) sin(π/5) sin(2π/5) sin(3π/5) sin(4π/5)= 16
Test-1 (24/6/24)
1) Find the value of
a) sin 1755°.
b) cot(-870).
c) cot 660+ tan(-1050).
2) Value sec(-1680) sin330
a) 0 b) 1 c) -1 d) none
3) Value of tan130 tan 140
a) 0 b) 1 c) -1 d) 2
4) value of m² sin(π/2)- n² sin(3π/2)+ 2mn secπ.
a) m- b) m+ n c) (m+n)² d) (m - n)²
5) If 6k=11π, the value of 2 cosk + 3 tank is
a) 1 b) 0 c) √3 d) 2√3
6) If cotx= cos60+ sin30, the value of cosx + cos(x - 90) is
a) 1 b) √2 c) 1/√2 d) 0
7) If 3sin²x+ 5 cos²x = 4 and π/2< x <π, value of sin2x.
a) 0 b) 1 c) -1 d) none
8) If tanx= -4/3, find the value of sinx.
a) 4 b) 5 c) ±5/4 d) ±4/5
9) If sinx= -2/3 and 270< x <360, then find the value of sin(x -270) tan(360- x).
a) 2 b) 3 c) 3/2 d) 2/3
10) If A,B,C,D are four angles of a quadrilateral, show that, sin(A+ B)+ sin(C+ D)= 0.
12) If A,B,C are the angles of a triangle, show that tan (C- A)/2= cot(A+ B/2).
13) If A+ B= 60°, show that, sin(120°- A)= cos(30° - B).
14) Find the value of tan(nπ+ π/4).
15) If x= 100, determine the sign of the expression sinx + cosx.
a) Positive b) negative c) both a and b d) either a or b
16) If sin(A - B)=√3/2 and sin A= 1/√2, find the least value of B.
a) 75° b) 60° c) 30° d) 15°
17) Show that: tan1+ tan2 + tan3.........tan87 tan88 tan89= 1.
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