Bright Zoom Maths

  Linear Algebra Symbols Symbol Symbol Name Meaning / definition Example · dot scalar product a  ·  b × cross vector product a  ×  b A ⊗ B tensor product tensor product of A and B A  ⊗  B inner product [ ] brackets matrix of numbers ( ) parentheses matrix of numbers |  A  | determinant determinant of matrix A det( A ) determinant determinant of matrix A ||  x  || double vertical bars norm A T transpose matrix transpose ( A T ) ij  = ( A ) ji A † Hermitian matrix matrix conjugate transpose ( A † ) ij  = ( A ) ji A * Hermitian matrix matrix conjugate transpose ( A * ) ij  = ( A ) ji A  -1 inverse matrix A A -1  =  I rank( A ) matrix rank rank of matrix A rank( A ) = 3 dim( U ) dimension dimension of matrix A rank( U ) = 3

  Algebra symbols Symbol Symbol Name Meaning / definition Example x x variable unknown value to find when 2 x  = 4, then  x  = 2 ≡ equivalence identical to ≜ equal by definition equal by definition := equal by definition equal by definition ~ approximately equal weak approximation 11 ~ 10 ≈ approximately equal approximation sin (0.01) ≈ 0.01 ∝ proportional to proportional to y  ∝  x  when  y  =  kx, k  constant ∞ lemniscate infinity symbol ≪ much less than much less than 1 ≪ 1000000 ≫ much greater than much greater than 1000000 ≫ 1 ( ) parentheses calculate expression inside first 2 * (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 { } braces set ⌊ x ⌋ floor brackets rounds number to lower integer ⌊4.3⌋ = 4 ⌈ x ⌉ ceiling brackets rounds number to upper integer ⌈4.3⌉ = 5 x ! exclamation mark factorial 4! = 1*2*3*4 = 24 |  x  | single vertical bar absolute value | -5 | = 5 f  ( x ) function ...

  Geometry symbols Symbol Symbol Name Meaning / definition Example ∠ angle formed by two rays ∠ABC = 30° measured angle ABC = 30° spherical angle AOB = 30° ∟ right angle = 90° α = 90° ° degree 1 turn = 360° α = 60° deg degree 1 turn = 360deg α = 60deg ′ prime arcminute, 1° = 60′ α = 60°59′ ″ double prime arcsecond, 1′ = 60″ α = 60°59′59″ line infinite line AB line segment line from point A to point B ray line that start from point A arc arc from point A to point B  = 60° ⊥ perpendicular perpendicular lines (90° angle) AC ⊥ BC | | parallel parallel lines AB | | CD ≅ congruent to equivalence of geometric shapes and size ∆ABC≅ ∆XYZ ~ similarity same shapes, not same size ∆ABC~ ∆XYZ Δ triangle triangle shape ΔABC≅ ΔBCD | x - y | distance distance between points x and y |  x - y  | = 5 π pi constant π  = 3.141592654... is the ratio between the circumference and diameter of a circle c  =  π · d  = 2· π · r rad radians radians angle unit 360° = 2π rad c ...

  Basic math symbols Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+3 5 is equal to 2+3 ≠ not equal sign inequality 5 ≠ 4 5 is not equal to 4 ≈ approximately equal approximation sin (0.01) ≈ 0.01, x  ≈  y  means  x  is approximately equal to  y > strict inequality greater than 5 > 4 5 is greater than 4 < strict inequality less than 4 < 5 4 is less than 5 ≥ inequality greater than or equal to 5 ≥ 4, x  ≥  y  means  x  is greater than or equal to  y ≤ inequality less than or equal to 4 ≤ 5, x ≤ y  means  x  is less than or equal to  y ( ) parentheses calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)×(1+5)] = 18 + plus sign addition 1 + 1 = 2 − minus sign subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 ± minus - plus both minus and plus operations 3 ± 5 = -2 and 8 * asterisk m...

Logical Reasoning skills  Logical Reasoning -Analogy  1. Complete analogous pair: Automobile : Petrol :: ? A. fire : fuel B. plane : propeller C. diesel : gas D. man : food Answer: D. man : food Explanation: An automobile needs petrol to function. Similarly, man needs food in order to function. 2. Complete analogous pair: Calf : cow :: Puppy : ? A. cub B. kitten C. dog D. snake Answer: C. dog Explanation: A calf's parent is cow and puppy's parent is dog. 3. Complete analogous pair: Tungsten : Filament :: Tree : ? A. graphite B. paper C. wool D. fabric Answer: B. paper Explanation: The first term is used in the production of the second. 4.Complete analogous pair: Letter : Word :: A. homework : school B. club : people C. product : factory D. page : book Answer: D. page : book Explanation: A group of letters form a word. Similarly, a group of pages combined together form a book. 5. Complete analogous pair. Brick : Clay :: Ruby : ? A. graphite B. corundum C. diamond D. pearl Answe...

 Logical Reasoning skills  Logical Reasoning -Solved Problems  1. Arrange the given words in a meaningful sequence. 1. Sentence, 2. Chapter, 3. Word, 4. Paragraph, 5. Book A. 2, 3, 1, 5, 4. B. 3, 2, 5, 4, 1. C. 5, 2, 4, 1, 3. D. 5, 3, 2, 4, 1. Answer: C. 5, 2, 4, 1, 3. Explanation: The correct order is: Book (5) - Chapter (2) - Paragraph (4) - Sentence (1) - Word (3). 2. In a row of 54 persons, A is 35th from the left side of the row and B is 22nd from the right side of the row. Find the no. of persons sitting between A and B? A. 1 B. 2 C. 3 D. 4 Answer: A. 1 Explanation:  Here Sum of positions of A & B from opposite ends = 35 + 22 = 57 > Total no. of persons ∴ No. of persons between A & B = (Position of A from left + Position of B from right) - Total no. of students - 2 No. of persons between A & B = (35+22) - 54 - 2  = 57 - 54 - 2 = 1. 3. Choose the pair in which the words are differently related. A. Cow : Fodder B. Crow : Carrion C. Poultry : F...

Logical Reasoning skills Logical Reasoning - Syllogism  1. Direction (1-7) : In each of the questions below three statements are given followed by two conclusions numbered I and II. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts. 1. Statements: Some buildings are white. All whites are oranges. Some oranges are boxes. Conclusions: I. Some boxes are buildings. II. Some oranges are buildings. Answer: If only conclusion II follows. Explanation: Some buildings are white + All whites are oranges = I + A = I Some buildings are oranges - conversion - some oranges are buildings (I). Hence II follows. Some buildings are oranges + Some oranges are boxes = I + I = No Conclusions. Hence I does not follow.. 2. Statements: All desks are mirrors. All mirrors are chairs. Some chairs are...

Logical Reasoning skills  Logical Reasoning - Artificial Language  1. Here are some of the words translated from an artificial language. If creektulo means treeplantation; creekfuos means treemutation; fuosseed means mutationsapling; Which word means sapling? A. Creek B. Tulo C. Fuos D. Seed Answer: D. Seed Explanation: tree means creek; mutation means fuos; sapling means seed. 2. Here are some words translated from an artificial language. jalkamofti means happy birthday; moftihoze means birthday party; mentogunn means goodness; Which word could mean "happiness"? A. jalkagunn B. mentohoze C. moftihoze D. hozemento Answer: A. jalkagunn Explanation: Jalka means happy; mofti means birthday; hoze means party; mento means good; and gunn means ness; Thus, jalkagunn means happiness. 3. Here are some of the words translated from an artificial language. If robbhood means nextgeneration; hoodgough means generationproud; goughspur means proudnation; Which word means generation? A. robb ...

Logical Reasoning skills  Logical Reasoning - Arithmetic Signs : Q: 1. Find out the two signs to be interchanged for making following equation correct: 5 + 3 x 8 - 12 / 4 =3 A. + and - B. - and / C. + and x D. + and / Answer: B. - and / Explanation: On interchanging - and /, we get the equation  as 5 + 3 x 8 / 12 - 4 = 3 5 + 3 x (2/3) - 4 = 3 3 = 3, which is true. Q: 2. If 3, 2 = 13; 4, 8 = 80; then 1, 5 =? A. 14 B. 26 C. 25 D. 50 Answer: B. 26 Explanation: The sum of the square of the number 32 + 22 = 9 + 4 = 13 42 + 82 = 16 + 64 = 80 12 + 52 = 1 + 25 = 26. Q: 3. If 1 = 17 2 = 33 3 = 49 4 = 65 then 5 = ? A. 80 B. 81 C. 89 D. 84 Answer: B. 81 Explanation: Multiply by 16 and add 1 (1 x 16) + 1 = 17 (2 x 16) + 1 = 33 (3 x 16) + 1 = 49 (4 x 16) + 1 = 65 (5 x 16) + 1 = 81. Q: 4. If 2 + 3 = 13 3 + 4 = 25 4 + 5 = 41 then 5 + 6 = ? A. 60 B. 61 C. 51 D. 63 Answer: B. 61 Explanation: Step 1 :  First multiply to the given number Step 2 :   Multiply by 2 Step 3 :  add 1...

Formulas : Problems on Trains  1. km/hr to m/s conversion: a km/hr = {(a) x(18 /5)} m/s. 2. m/s to km/hr conversion: a m/s = {(a) x(18 /5)} km/hr. 3. Formulas for finding Speed, Time and Distance 4. Time taken by a train of length l metres to pass a pole or standing man or a signal post is equal to the time taken by the train to cover l metres. 5. Time taken by a train of length l metres to pass a stationery object of length b metres is the time taken by the train to cover (l + b) metres. 6. Suppose two trains or two objects bodies are moving in the same direction at u m/s and v m/s, where u > v, then their relative speed is = (u - v) m/s. 7. Suppose two trains or two objects bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s. 8. If two trains of length a metres and b metres are moving in opposite directions at u m/s and v m/s, then: The time taken by the trains to cross each other = (a + b) / (u + v) sec.   ...

Non Verbal Logical Reasoning 1. Choose the figure which is different from the rest. Answer :  4 Explanation :  In all other figures, the dot appears in the same corner of the square as the line outside it. 2. Select a suitable figure from the Answer Figures that would replace the question mark (?). Answer : D Explanation :  The figure rotates through 135° AntiClockWise.

Logical Reasoning -Solved Problems  1. Statement: P < Q ≤ R < S > T Conclusions: I. T < R II. S > P III. R > T IV. Q < S A. Only I follows B. Only II follows C. Both II and III follows D.All I, II and III follows E. Only I and II follows Answer: B. Only II follows Explanation: Here, we can see that R is either greater or lesser or equal to T because no specific relation between them is defined. So, we cannot conclude anything about part I. But in part II, we can see that S > P. So, it is true. 2. Arrange the given words in alphabetical order and select one that comes in the middle. A. Hilt B. History C. Hillock D. Heaven E. Hawker Answer: C. Hillock Explanation: The correct alphabetical order of the given words is: Hawker, Heaven, Hillock, Hilt, History. 3. 1.If 'ish lto inm' stands for 'neat and tidy' ; 'qpr inm sen' stands for 'small but neat' ; and 'hsm sen rso' stands for 'good but erratic', what would '...

Logical Reasoning skills  Non Verbal Logical Reasoning 1. Select a suitable figure from the Answer Figures that would replace the question mark (?). Problem Figures: Answer Figures: Answer : 1  Explanation :  The figure gets vertically inverted and enlarged. 2. Choose the figure which is different from the rest. Answer : 5  Explanation :  All others are punctuation marks.

Aptitude - Numbers closer and greater than 60 !! Example 1: 61 * 66 Both the numbers are greater than 60. Step 1: 61 is 1 greater than 60 and 66  is 6 greater than 60 Step 2: ( +1 ) * ( +6 ) = 06 Step 3: ( 61 + 6 ) or ( 66 + 1 ) = 67  Then multiply 67 by 60 67 * 60 = 4020 Step 4: Add step2 and step3 answers. 4020 + 06 = 4026 Final answer = 4026 Example 2: 62 * 68 Both the numbers are  greater than 60. Step 1: 62 is 2 greater than 60 and 68 is 8 greater than 60 Step 2: ( +2 ) * ( +8 ) = 16 Step 3: ( 62 + 8 ) or ( 68 + 2 ) = 70 Then multiply 70 by 60 70 * 60 = 4200 Step 4: Add step2 and step3 answers. 4200 + 16 = 4216 Final answer = 4216 Example 3: 64 * 67 Both the numbers are  greater than 60. Step 1: 64 is 4 greater than 60 and 67  is 7 greater than 60 Step 2: ( +4 ) * ( +7 ) = 28 Step 3: ( 64 + 7 ) or ( 67 + 4 ) = 71 Then multiply 71 by 60 71 * 60 = 4260 Step 4: Add step2 and step3 answers. 4260 + 28 = 4288 Final answer = 4288