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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1151. |
Write the set `A={1,2,3,4,5,6,7}` in the set- builder form. |
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Answer» Clearly, `A=` Set of all natural numbers less than 8. Thus, in the set- builder form, we write it as `A={x:x in N and xlt 8}`. |
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| 1152. |
Write the set `B={1,2,3,4,7,14, 28}` in the set builder form. |
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Answer» Clearly, `B=` Set of all factors of 28. Thus, in the set-builder form, we write it as `B={x:x in N and x " is a factor of "28}` |
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| 1153. |
Show that the following four conditions are equivalent:(i)`Asub B` (ii) `A B =varphi` (iii) `A uuB = B` (iv) `A nnB = A` |
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Answer» 1 and 2) A-B`!= phi` x`in`A,x`notin`B not possible A`subset`B x`in`B A-B=`phi` 1 and 3) `x in AUB` `x in A or x in B` AUB=B 1 and 4) `x in A nn B` `x in A and x in B` `A= A nn B` `A subset B`. |
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| 1154. |
If B is contained in A and C is contained in B, then relation between A and C is |
| Answer» Correct Answer - `C sub A` | |
| 1155. |
If `A = {1,2}, B = {1,2,3}` and `c = {1,2,3,4,5}` then the relation between A, B and C? |
| Answer» Correct Answer - `A sub B, A sub C, B sub C` | |
| 1156. |
If `n (A Delta B) = 12` and `n(A nn B) = 3`, then find the greatest possible value of `n(A xx B)`. |
| Answer» Correct Answer - `81` | |
| 1157. |
Given that `A = {2,4,6,7,8}` and `B = {1,2,3,4}` Find `A Delta B`. The following are the steps involved in sovling the above problem. Arrange them in sequential order. (A) `A Delta B = (A - B) uu (B - A)` (B) `A - B = {2,4,6.8} - {1,2,3,4}` and `B-A = {1,2,3,4} - {2,4,6,7,8}` (C ) `A Delta B = {6,7,8} uu {1,3} = {1,3,6,7,8}` (D) `A - B = {6,7,8}` and `B - A = {1,3}`A. BCADB. BACDC. BDACD. BADC |
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Answer» Correct Answer - C (B), (D), (A) and (C ) is the required sequential order. |
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| 1158. |
If `n(A) = 6` and `n(B) = 8` and `n(A nn B) = 4`, then find `n(A Delta B)`. The following are the step involved in solving the above problem . Arrange them in sequnential order. (a) `n (A uu B) = n(A) + n(B) - n(A nn B)` (b) `n(A uu B) = 10` (c )`n(A uu B) =6+8-4` (d) `n(A Delta B) = 10-4= 6` (e) `n(A Delta B) = n(A uu B) - n(A nn B)`.A. ABCDEB. ACBEDC. ACEBDD. AEBCD |
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Answer» Correct Answer - B (A), (C), (B), (E ) and (D) is the required sequential order. |
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| 1159. |
If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R – Q)? |
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Answer» Given; R is the set of all real numbers and Q is the set of all rational numbers. Then (R – Q) is the set of all irrational numbers. |
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| 1160. |
If A ⊂ B, then A – B = (A) B (B) Φ (C) A (D) B – A |
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Answer» Correct option is (B) Φ |
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| 1161. |
If A ⊂ B then (A ∪ B) – A = ……………A) B B) Φ C) A D) B – A |
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Answer» Correct option is (D) B – A |
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| 1162. |
If A = {x/x∈N, 3 ≤ x ≤ 6}, then A = (A) {3, 4, 5} (B) {3, 4, 5, (6} C) {4, 5} (D) {4, 5, 6} |
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Answer» Correct option is (B) {3, 4, 5, 6} |
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| 1163. |
If A ⊂ B, then A ∪ B = A) A B) μ C) Φ D) B |
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Answer» Correct option is D) B |
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| 1164. |
If n(A) = 15, n(B) = 10, n(A ∩ B) = 5, then n(A ∪ B) = (A) 5 (B) 15 (C) 20 (D) 25 |
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Answer» Correct option is (C) 20 |
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| 1165. |
If A ⊂ B and B ⊂ C, then (A ∩ B) ∪ C = A) Φ B) A C) B D) C |
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Answer» Correct option is (D) C |
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| 1166. |
`f:R->R` is defined as `f(x)=2x+|x|` then `f(3x)-f(-x)-4x=`A. f(x)B. `-f(x)`C. `f(-x)`D. 2f(x) |
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Answer» Correct Answer - D `f : R rarr R` `f(x) = 2x + |x|` When `x ge 0`, then f(x) = 2x + x = 3x When `x lt 0`, then f(x) = 2x - x = x Now, when `x ge 0` `f(3x)-f(-x)-4x=3x-(-3x)-4x=2x=2f(x)` |
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| 1167. |
If `f(y) = (y)/(sqrt(1-y^(2))), g(y) = (y)/(sqrt(1+y^(2)))`, then (fog) y is equal toA. `(y)/(sqrt(1-y^(2)))`B. `(y)/(sqrt(1+y^(2)))`C. yD. `((1-y^(2)))/(sqrt(1-y^(2)))` |
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Answer» Correct Answer - C `f(y)=(y)/(sqrt((1-y^(2)))),g(y)=(y)/(sqrt((1+y^(2))))` and `(fog)y=f(g(y))=f((y)/(sqrt((1+y^(2)))))=((y)/(sqrt((1+y^(2)))))/(sqrt(1-(y^(2))/((1+y^(2)))))=((y)/(sqrt((1+y^(2)))))/((1)/(sqrt((1+y^(2)))))=y` |
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| 1168. |
If `f(x)=(a-x)/(a+x)`, the domain of `f^(-1)(x)` containsA. `(-oo,oo)`B. `(-oo,-1)`C. `(-1,oo)`D. `(0,oo)` |
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Answer» Correct Answer - B::C::D Let `y=f(x)=(a-x)/(a+x)impliesay+xy=a-x` `therefore x=(a(1-y))/((1+y))=f^(-1)(y)impliesf^(-1)(x)=(a(1-x))/((1+x))` `therefore f^(-1)(x)` is not defined for x = - 1. Domain of `f^(-1)(x)` belongs to `(-oo,-1)uu(-1,oo)`. Now, for a = - 1, given function f(x) = - 1, which is constant. Then, `f^(-1)(x)` is not defined. `therefore ane-1` |
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| 1169. |
A = {1, 3, 2, 7}, then write any three subsets of A |
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Answer» Three subsets of A: i. B = {3} ii. C = {2, 1} iii. D= {1, 2, 7} |
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| 1170. |
Two sets A and B are disjoint sets, if and only if A) A ∩ B = B B) A – B = A C) A – B = Φ D) A ∪ B = Φ |
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Answer» Correct option is B) A – B = A |
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| 1171. |
Which of the following sets has only one subset ? A) {0, 1} B) {0} C) {1} D) { } |
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Answer» Correct option is A) {0, 1} |
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| 1172. |
i. Write the subset relation between the sets.P is the set of all residents in Pune.M is the set of all residents in Madhya Pradesh.I is the set of all residents in Indore.B is the set of all residents in India. H is the set of all residents in Maharashtra.ii. Which set can be the universal set for above sets ? |
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Answer» i. a. The residents of Pune are residents of India. ∴ P ⊆ B b. The residents of Pune are residents of Maharashtra. ∴ P ⊆ H c. The residents of Madhya Pradesh are residents of India. ∴ M ⊆ B d. The residents of Indore are residents of India. ∴ I ⊆ B e. The residents of Indore are residents of Madhya Pradesh. ∴ I ⊆ M f. The residents of Maharashtra are residents of India ∴ H ⊆B ii. The residents of Pune, Madhya Pradesh, Indore and Maharashtra are all residents of India.
∴ B can be the Universal set for the above sets. |
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| 1173. |
If n(A) = 5, then n[p(A)] = A) 0 B) 5 C) 32 D) 25 |
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Answer» Correct option is C) 32 |
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| 1174. |
If A is any set such that n[p(A)] = 64, then n(A) = A) 6 B) 16 C) 32 D) 8 |
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Answer» Correct option is A) 6 |
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| 1175. |
If A = {1, 2, 3}, B = {2, 3, 4}, then A ∪ B = A) {4} B) {2, 3} C) {1, 2, 3, 4} D) {1} |
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Answer» Correct option is C) {1, 2, 3, 4} |
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| 1176. |
Let A = {1,2,4,5}, B = {2,3,5,6}, C = {4,5,6,7} verify the following identity A ∪ (B∩C) = [(A ∪ B) ∩ (A ∪ C)] |
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Answer» L.H.S = A ∪ (B ∪ C) = {1,2,3,4,5} ∪ [{2,3,5,6} ∩ {4,5,6,7}] = {1,2,4,5} ∪ {5,6} = {1,2,4,5,6} R.H.S = (A ∪ B) ∩ (A ∪ C) = [{1,2,4,5} ∪ {2,3,5,6}] ∩ [{1,2,4,5} ∪ {4,5,6,7}] = {1,2,3,4,5,6} ∩ {1,2,4,5,6,7} = {1,2,4,5,6} ∴ L.H.S = R.H.S |
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| 1177. |
If A = {1, 2, 3}, B = {2, 3, 4}, then A ∩ B = A) {1, 2, 3, 4} B) {4} C) {1} D) {2, 3} |
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Answer» Correct option is D) {2, 3} |
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| 1178. |
If A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 8}, then A∩B∩C = A) {4} B) {2, 4} C) μ D) Φ |
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Answer» Correct option is A) {4} |
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| 1179. |
If `S=R,A={x:-3lexlt7} and B={x:0ltxlt10}`, the number of positive integers in `ADeltaB` is |
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Answer» Correct Answer - 3 Here, `A=[-3,7),B=(0,10)` `S=(-oo,oo)` `therefore A-B=[-3,0]andB-A=[7,10)` `therefore ADeltaB=(A-B)uu(B-A)=[-3,0]uu[7,10)` `therefore` Positive integers are 7, 8, 9. Number of positive integers = 3 |
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| 1180. |
Which one of the following is correct?A. `AuuP(A)=P(A)`B. `AnnP(A)=A`C. `A-P(A)=A`D. `P(A)-{A}=P(A)` |
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Answer» Correct Answer - A `AuuP(A)=P(A)` is correct. Since A is a subset of its power set. |
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| 1181. |
Let `A={x in R|-9lexlt4},B={x inR|-13ltxle5} and C={x inR|-7lexle8}`. Then, which one of the following is correct?A. `-9in(AnnBnnC)`B. `-7in(AnnBnnC)`C. `4in(AnnBnnC)`D. `5in(AnnBnnC)` |
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Answer» Correct Answer - B Given sets in set builder form are : `A={x inR:-9lexlt4}` `B={x in R: -13ltx5}` `and C={x inR: -7lex le 8}` `impliesAnnBnnC={x inR: -7lex lt4}` Hence, `-7in(AnnBnnC)` |
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| 1182. |
The difference of two numbers 10001100 and 1101101 in binary system is expressed in decimal system by which one of the following?A. 27B. 29C. 31D. 33 |
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Answer» Correct Answer - C The given binary number `10001100=1xx2^(7)+1xx2^(3)+1xx2^(2)` =128+8+4=140 (decimal numbers) and `1101101=1xx2^(6)+1xx2^(5)+1xx2^(3)+1xx2^(2)+1xx2^(0)` `=64+32+8+4+1=109` their difference = 140-109=31 |
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| 1183. |
What is the value of `log_9(27)+log_8(32)`A. `(7)/(2)`B. `(19)/(2)`C. 4D. 7 |
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Answer» Correct Answer - B `log_(9)27+log_(8)32` `=log_(9)3^(3)+log_(8)2^(5)` `=3log_(9)3+5log_(8)2` `3log_((3^(""^(2))))3+5log_((2^(""^(3))))2` `(3)/(2)log_(3)3+(5)/(3)log_(2)2` `(3)/(2)+(5)/(3)=(19)/(6)` |
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| 1184. |
A survey was conducted among 300 students. If was found that 125 students like play cricket, 145 students like to play football and 90 students like to play tennis, 32 students like to play exactly two games out of the three games. How many students like to play exactly only one game?A. 196B. 228C. 254D. 268 |
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Answer» Correct Answer - C Exactly one `=|A|+|B|+|C|-` `2[|AnnB|+|BnnC|+|AnnC|]+3|AnnBnnC|` `=125+145+90-2[32+3xx14]+3xx14` `=360-106=254` |
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| 1185. |
Out of 25 members in a family, 12 like to take tea, 15 like to take coffee and 7 like to take coffee and tea both. How many like (i) at least one of the two drinks. (ii) only tea but not coffee, only coffee but not tea. (iv) neithertes nor cofee. |
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Answer» Given that n(T) = 12 n(C)= 15 n(T ∩ C) = 7 (i) n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 12 + 15 – 7 n(T ∪ C) = 20 20 members like at least one of the two drinks. (ii) Only tea but not coffee n(T) − n(T ∩ C) = 12 – 7 = 5 (iii) Only coffee but not tea = n(c) − n(T ∩ C) = 15 – 7 = 8 (iv) Neither tea nor coffee = n(u) − n(T ∪ C) = 25 – 20 = 5 Given that n(T) = 12n(C)= 15 n(T ∩ C) = 7 (i) n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 12 + 15 – 7 n(T ∪ C) = 20 20 members like at least one of the two drinks. (ii) Only tea but not coffee n(T) − n(T ∩ C) = 12 – 7 = 5 (iii) Only coffee but not tea = n(c) − n(T ∩ C) = 15 – 7 = 8 (iv) Neither tea nor coffee = n(u) − n(T ∪ C) = 25 – 20 = 5 |
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| 1186. |
From 50 Students taking examination in Mathematics, Physics and chemistry, each of the student has passed in at least one of the subject, 37 passes Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, atmost 29 Mathematics and Chemist and at most 20 Physics and Chemistry. What is the largest possible number that could have passes in all the three subjects? |
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Answer» Let M, P and C denote the students of Mathematics, Physics and Chemistry, respectively. Given, n(M ∪ P ∪ C) = 50 No. of students passed in Mathematics, n(M) = 37 No, of students passed in Physics, n(P) = 24 No. of students passed in Chemistry, n(C) = 43 No. of students passed in Mathematics and Physics, n(M ∩ P) = 19 No. of students passed in Mathematics and Chemistry, n(M ∩ C) = 29 No. of students passed in Physics and chemistry, n(P ∩ C) = 20 Using identity n(M ∪ P ∪ C) = n(M) + n(P) + n(C) − n(M ∩ P) − n(M ∩ C) − n(P ∩ C) + n(M ∩ P ∩ C) ∴ n(M ∪ P ∪ C) = n(M ∪ P ∪ C) − n(M) − n(P) − n(C) + n(M ∩ P) + n(M ∩ C) + n(P ∩ C) = 50 – 37 – 24 – 43 +19 + 29 + 20 = 50 – (37 + 24 + 43) + (19 + 29 + 20) = 50 – 104 + 68 = 50 – 36 = 14 |
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| 1187. |
For all sets A, B and C, A – (B – C) = (A – B) – C |
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Answer» For all sets A, B and C, A – (B – C) = (A – B) – C Answers is False |
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| 1188. |
If `n(A) = 44, n(B) = 28` and `n(A uu B) = 56`, then find `n(B - A) =`A. 12B. 16C. 28D. None of these |
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Answer» Correct Answer - A `n(B - A) = n(B) - n(A nn B)`. |
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| 1189. |
A and B are non-empty sets A - B = A and B - A = B. Then which of the following is true ?A. `A sub B`B. `B sub A`C. A and B are disjointD. A = B |
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Answer» Correct Answer - C `A - B= A rArr A` and B have no common elements. |
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| 1190. |
If `B = {***, Delta, ?, !}`, then `n[P(B)] = "____"`. |
| Answer» Correct Answer - 16 | |
| 1191. |
If `A = {1,2,4,5}` and `B={1,4,6}`, then `A Delta B = ?`A. `{2,6}`B. `{2,4,5}`C. `{2,4,6}`D. `{2,5,6}` |
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Answer» Correct Answer - D `A Delta B = (A uu B) - (Ann B)`. |
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| 1192. |
If `A sub B`, then the illustration of `A - B` in a Venn diagram isA. B. C. D. None of these |
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Answer» Correct Answer - B A-B represents the region of set A which does not belong to B. |
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| 1193. |
Which of the following sets are disjoint. A = {Multiples of 3} B = {Multiple of 5} C = {Multiple of 7}A. A and BB. B and CC. A and CD. None of these |
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Answer» Correct Answer - D Disjoint sets has no element is common. |
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| 1194. |
If A = {a, b, c, d, e}, B = {c, d, e, f}, C = {b, d}, D = {a, e}, then which of the following statements are true and which are false?i. C ⊆ 3ii. A ⊆ D iii. D ⊆ Biv. D ⊆ A V. B ⊆ A vi. C ⊆ A |
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Answer» i. C = {b, d}, B = {c, d, e ,f} C ⊆ B False Since, all the elements of C are not present in B. ii. A = {a, b, c, d, e}, D = {a, e} A ⊆ D False Since, all the elements of A are not present in D. iii. D = {a, e}, B = {c, d, e, f} D ⊆ B False Since, all the elements of D are not present in B. iv. D = {a, e}, A = {a, b, c, d, e} D ⊆ A True Since, all the elements of D are present in A v. B = {c, d, e, f}, A = {a, b, c, d, e} B ⊆ A False Since, all the elements of B are not present in A. vi. C = {b, d}, A= {a, b, c, d, e} C ⊆A True Since, all the elements of C are present in A. |
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| 1195. |
If E is the set of equilateral triangle and I is the net of isoscales triangle then find `I - E`A. `cancelcirc`B. lC. ED. None of these |
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Answer» Correct Answer - D (i) Recall the properties of triangles. (ii) Every equilateral triangle is an isosceles. |
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| 1196. |
Let g(x) be a function defined on [-1,1]. If the area of the equilateral triangle with two of its vertices at `(0,0)` and `(x,g(x))` is `sqrt(3)/4`.then the function g(x) is: |
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Answer» side length(a)=`sqrt((x-0)^2+(g(x)-0)^2` `a=sqrt(x^2+g(x)^2` Area of equilateral triangle=`sqrt3/4a^2` `sqrt3/4a^2=sqrt3/4` `a^2=1` `x^2+g(x)=1` `g(x)^2=1-x^2` `g(x)=pmsqrt(1-x^2`. |
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| 1197. |
using Mathematical induction , prove that `3^(2n)+7` is divisible by 8. |
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Answer» `3^(2n)+7` divisible by 8 put n=1 `3^2+7=16` is divisible by 8 put n=2 `3^4+7=88` is divisible by 8 `3^n+7` is also divisible by 8. |
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| 1198. |
`lim_(x->0){tan(pi/4+x)}^(1/x)` |
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Answer» `lim_(x->0)((tan(pi/4)+tanx)/(1-tan(pi/4)tanx))^(1/x)` `lim_(x->0)((1+tanx)/(1-tanx))^(1/x)` `lim_(x->0)(1+(2tanx)/(1-tanx))^(1/x)` `lim_(x->0)(2tanx)/x*(1/(1-tanx))` `e^(2*1*(1/(1-0))` `e^2`. |
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| 1199. |
Find the range of `f(theta) = 5 costheta + 3 cos(theta + pi/3) + 3 ` |
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Answer» `f(theta) = 5costheta+3cos(theta+pi/3)+3` `=>f(theta) = 5costheta + 3(costhetacospi/3-sinthetasinpi/3)+3` `=>f(theta) = 5costheta + 3(costheta(1/2)-sintheta(sqrt3/2))+3` `=>f(theta) = 5costheta + 3/2costheta-(3sqrt3)/2sintheta+3` `=>f(theta) = 13/2costheta-(3sqrt3)/2sintheta+3` We know, for a function `f(x) = acosx+bsinx+c`, `f(x)_max = c + sqrt(a^2+b^2)` `f(x)_min = c - sqrt(a^2+b^2)` Here, `a = 13/2, b = (3sqrt3)/2 and c = 3` `:. f(x)_max = 3 + sqrt((13/2)^2+((3sqrt3)/2)^2) = 3+sqrt49 = 3+7 = 10` `:. f(x)_min = 3 - sqrt((13/2)^2+((3sqrt3)/2)^2) = 3-sqrt49 = 3-7 = -4` So, range of `f(theta)` is between `-4` and `10`. `f(theta) in [4,10]`. |
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| 1200. |
List all the elements of the following sets: A = {x:x2≤ 10, xZ} |
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Answer» First of all, x is an integer hence it can be positive and negative also. x2 ≤ 10 x≤ √10 x = ±1,±2,±3 A = {±1,±2,±3} |
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