The equation xy=0 in three-dimensional space represents
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Equation of planes
Consider what xy=0 means
Factorize equation
Recognize as x=0 or y=0
Thinking in 2D only
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Think about 3D interpretation
Equation of planes
Consider what xy=0 means
Factorize equation
Recognize as x=0 or y=0
Thinking in 2D only
The plane containing point (3,2,0) and line x-3/1=y-6/5=z-4/4 is
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Plane equation through point and line
Use point and line to get plane
Form equation using direction ratios
Verify point lies on plane
Not using point-direction form correctly
😢 Uh oh! Incorrect answer, Try again
Use point-direction form
Plane equation through point and line
Use point and line to get plane
Form equation using direction ratios
Verify point lies on plane
Not using point-direction form correctly
Coordinates of foot of perpendicular from origin to plane 2x-3y+4z=29 are
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Foot coordinates = λ(a,b,c)
Use formula for foot of perpendicular
Substitute in plane equation
Verify coordinates
Not verifying final answer
😢 Uh oh! Incorrect answer, Try again
Remember foot of perpendicular formula
Foot coordinates = λ(a,b,c)
Use formula for foot of perpendicular
Substitute in plane equation
Verify coordinates
Not verifying final answer
If A and B are independent events with P(A)=0.75, P(A∪B)=0.65, and P(B)=x, find x
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Independence: P(A∩B)=P(A)P(B)
Use addition formula for independent events
Substitute given probabilities
Solve for x
Not using independence property
😢 Uh oh! Incorrect answer, Try again
Use P(A∪B)=P(A)+P(B)-P(A)P(B)
Independence: P(A∩B)=P(A)P(B)
Use addition formula for independent events
Substitute given probabilities
Solve for x
Not using independence property
A die is thrown 10 times. Probability that odd number will come up at least once is
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P(at least one) = 1-P(none)
P(at least one) = 1-P(none)
Use binomial probability
Calculate 1-(1/2)¹⁰
Not using complement method
😢 Uh oh! Incorrect answer, Try again
Use complement rule
P(at least one) = 1-P(none)
P(at least one) = 1-P(none)
Use binomial probability
Calculate 1-(1/2)¹⁰
Not using complement method
Find the mean number of heads in three tosses of a fair coin
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E(X) = Σ(x·P(x))
List all possible outcomes: {0,1,2,3} heads
Find probability for each: {1/8,3/8,3/8,1/8}
Calculate E(X) = Σ(x·P(x)) = 1.5
Not considering all outcomes
😢 Uh oh! Incorrect answer, Try again
Use probability distribution for mean
E(X) = Σ(x·P(x))
List all possible outcomes: {0,1,2,3} heads
Find probability for each: {1/8,3/8,3/8,1/8}
Calculate E(X) = Σ(x·P(x)) = 1.5
Not considering all outcomes
If A and B are two events such that P(A)=1/2, P(B)=1/3 and P(A/B)=1/4, then P(A'∩B') is
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P(A'∩B') = 1 - P(A∪B)
Use conditional probability formula
Find P(A∩B) using P(A/B)
Use complement rule for A'∩B'
Not using complement correctly
😢 Uh oh! Incorrect answer, Try again
Consider complement of union
P(A'∩B') = 1 - P(A∪B)
Use conditional probability formula
Find P(A∩B) using P(A/B)
Use complement rule for A'∩B'
Not using complement correctly
The degree measure of π/32 is equal to
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Degree = radian × 180°/π
Convert π/32 to degrees: 180°/32
Convert decimal degrees to DMS
Simplify to get 5°37'30"
Not converting properly to DMS
😢 Uh oh! Incorrect answer, Try again
Use π radians = 180°
Degree = radian × 180°/π
Convert π/32 to degrees: 180°/32
Convert decimal degrees to DMS
Simplify to get 5°37'30"
Not converting properly to DMS
The value of sin(5π/12)sin(π/12) is
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sin A sin B = ½[cos(A-B) - cos(A+B)]
Use product to sum formula
Substitute angles
Simplify trigonometric expression
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Use sin A sin B formula
sin A sin B = ½[cos(A-B) - cos(A+B)]
Use product to sum formula
Substitute angles
Simplify trigonometric expression
Not using correct formula
The trigonometric function y=tanx in II quadrant
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tan x = sin x/cos x
Recall behavior of tanx in II quadrant
Note that x is between π/2 and π
Determine direction of change
Not considering quadrant properly
😢 Uh oh! Incorrect answer, Try again
Consider sign changes
tan x = sin x/cos x
Recall behavior of tanx in II quadrant
Note that x is between π/2 and π
Determine direction of change
Not considering quadrant properly
The equation of the line joining the points (-3,4,11) and (1,-2,7) is
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(x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Calculate direction ratios by finding differences: (1-(-3), -2-4, 7-11) = (4,-6,-4)
Simplify direction ratios: (4,-6,-4) = -2(2,-3,2)
Apply line equation: (x+3)/-2 = (y-4)/3 = (z-11)/2
Not simplifying direction ratios
😢 Uh oh! Incorrect answer, Try again
Always simplify direction ratios first
(x-x₁)/a = (y-y₁)/b = (z-z₁)/c
Calculate direction ratios by finding differences: (1-(-3), -2-4, 7-11) = (4,-6,-4)
Simplify direction ratios: (4,-6,-4) = -2(2,-3,2)
Apply line equation: (x+3)/-2 = (y-4)/3 = (z-11)/2
Not simplifying direction ratios
The angle between the lines whose direction cosines are (3/4, 1/4, √3/2) and (3/4, 1/4, -√3/2) is
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cosθ = l₁l₂ + m₁m₂ + n₁n₂
Calculate dot product of direction cosines
Use cosθ = l₁l₂ + m₁m₂ + n₁n₂
Solve to get θ = π/3
Forgetting to normalize vectors
😢 Uh oh! Incorrect answer, Try again
Direction cosines are normalized vectors
cosθ = l₁l₂ + m₁m₂ + n₁n₂
Calculate dot product of direction cosines
Use cosθ = l₁l₂ + m₁m₂ + n₁n₂
Solve to get θ = π/3
Forgetting to normalize vectors
If a plane meets the coordinate axes at A, B and C in such a way that the centroid of triangle ABC is at the point (1,2,3) then the equation of the plane is
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x/a + y/b + z/c = 1
Use centroid formula: G = (A+B+C)/3
Substitute given point (1,2,3)
Find intercepts and form equation
Not using centroid properties correctly
😢 Uh oh! Incorrect answer, Try again
Centroid divides each median in ratio 2:1
x/a + y/b + z/c = 1
Use centroid formula: G = (A+B+C)/3
Substitute given point (1,2,3)
Find intercepts and form equation
Not using centroid properties correctly
A random variable X has probability distribution: X:0,1,2; P(X):25/36,k,1/36. If mean is 1/3, then variance is
🥳 Wohoo! Correct answer
Variance = E(X²)-[E(X)]²
Find k using Σp=1: 25/36+k+1/36=1
Use mean=1/3 to get k=5/18
Calculate variance using E(X²)-[E(X)]²
Not verifying probability sum
😢 Uh oh! Incorrect answer, Try again
Use probability distribution formulas
Variance = E(X²)-[E(X)]²
Find k using Σp=1: 25/36+k+1/36=1
Use mean=1/3 to get k=5/18
Calculate variance using E(X²)-[E(X)]²
Not verifying probability sum
Given that A and B are two events such that P(B)=3/5, P(A/B)=1/2 and P(A∪B)=4/5 then P(A) =
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P(A∪B) = P(A) + P(B) - P(A∩B)
Use P(A/B) = P(A∩B)/P(B)
Apply addition formula P(A∪B) = P(A) + P(B) - P(A∩B)
Solve for P(A)
Not using conditional probability correctly
😢 Uh oh! Incorrect answer, Try again
Use Venn diagram for visualization
P(A∪B) = P(A) + P(B) - P(A∩B)
Use P(A/B) = P(A∩B)/P(B)
Apply addition formula P(A∪B) = P(A) + P(B) - P(A∩B)
Solve for P(A)
Not using conditional probability correctly
If A, B and C are three independent events such that P(A)=P(B)=P(C)=P then P(at least two of A, B, C occur) =
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P(at least 2) = P(A∩B) + P(B∩C) + P(A∩C) - P(A∩B∩C)
List all favorable outcomes
Use addition principle
Subtract triple counting
Not considering all cases
😢 Uh oh! Incorrect answer, Try again
Use principle of inclusion-exclusion
P(at least 2) = P(A∩B) + P(B∩C) + P(A∩C) - P(A∩B∩C)
List all favorable outcomes
Use addition principle
Subtract triple counting
Not considering all cases
Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6 the probability of getting a sum as 3 is
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P(E) = n(E)/n(S)
List all possible outcomes with sum<6
Count favorable outcomes (sum=3)
Divide favorable by total
Not considering given condition
😢 Uh oh! Incorrect answer, Try again
Make sample space diagram
P(E) = n(E)/n(S)
List all possible outcomes with sum<6
Count favorable outcomes (sum=3)
Divide favorable by total
Not considering given condition
The value of cos1200° + tan1485° is
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cos(360°n + θ) = cos θ
Convert angles to standard form
Simplify using periodic properties
Add the results
Not reducing angles first
😢 Uh oh! Incorrect answer, Try again
Use angle conversion formulas
cos(360°n + θ) = cos θ
Convert angles to standard form
Simplify using periodic properties
Add the results
Not reducing angles first
The value of tan1°tan2°tan3°....tan89° is
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tan θ × tan(90°-θ) = 1
Pair complementary angles
Use tan θ × tan(90°-θ) = 1
Multiply all pairs
Missing complementary pairs
😢 Uh oh! Incorrect answer, Try again
Look for complementary pairs
tan θ × tan(90°-θ) = 1
Pair complementary angles
Use tan θ × tan(90°-θ) = 1
Multiply all pairs
Missing complementary pairs
The cost and revenue functions of a product are given by c(x)=20x+4000 and R(x)=60x+2000 respectively where x is the number of items produced and sold. The value of x to earn profit is
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P(x)=R(x)-c(x)
Write profit function P(x)=R(x)-c(x)
Set P(x)>0
Solve inequality
Not considering fixed costs
😢 Uh oh! Incorrect answer, Try again
Profit occurs when revenue>cost
P(x)=R(x)-c(x)
Write profit function P(x)=R(x)-c(x)
Set P(x)>0
Solve inequality
Not considering fixed costs
If random variable X follows binomial distribution with n=5, p and P(X=2)=9P(X=3), then p equals
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P(X=r)=ⁿCᵣp^r(1-p)^(n-r)
Use binomial probability formula
Set up P(X=2)=9P(X=3) equation
Solve for p=1/10
Not using binomial formula correctly
😢 Uh oh! Incorrect answer, Try again
Compare consecutive terms
P(X=r)=ⁿCᵣp^r(1-p)^(n-r)
Use binomial probability formula
Set up P(X=2)=9P(X=3) equation
Solve for p=1/10
Not using binomial formula correctly
If P(A)=0.59, P(B)=0.30 and P(A∩B)=0.21 then P(A'∩B') =
🥳 Wohoo! Correct answer
P(A'∩B') = 1 - [P(A) + P(B) - P(A∩B)]
Use complement rule: P(A'∩B') = 1 - P(A∪B)
Use P(A∪B) = P(A) + P(B) - P(A∩B)
Substitute values and calculate
Not using DeMorgan's laws correctly
😢 Uh oh! Incorrect answer, Try again
Draw Venn diagram
P(A'∩B') = 1 - [P(A) + P(B) - P(A∩B)]
Use complement rule: P(A'∩B') = 1 - P(A∪B)
Use P(A∪B) = P(A) + P(B) - P(A∩B)
Substitute values and calculate
Not using DeMorgan's laws correctly
cos[cot⁻¹(-√3)+π/6] =
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cos(A+B)=cosAcosB-sinAsinB
Convert cot⁻¹(-√3) to standard angle
Add π/6
Find cosine of sum
Not using correct angle addition formula
😢 Uh oh! Incorrect answer, Try again
Use inverse trigonometric formulas
cos(A+B)=cosAcosB-sinAsinB
Convert cot⁻¹(-√3) to standard angle
Add π/6
Find cosine of sum
Not using correct angle addition formula
The point (1, -3, 4) lies in the octant
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• Octant classification rules
Check sign of x-coordinate
Check sign of y-coordinate
Check sign of z-coordinate
Not knowing octant classifications
😢 Uh oh! Incorrect answer, Try again
Remember octant numbering system
• Octant classification rules
Check sign of x-coordinate
Check sign of y-coordinate
Check sign of z-coordinate
Not knowing octant classifications
Distance of point (1, 2, -4) from line (x-3)/2 = (y-3)/3 = (z+5)/6 is
🥳 Wohoo! Correct answer
• Distance formula for point from line
Find direction ratios of line
Use distance formula
Simplify result
Not using correct distance formula
😢 Uh oh! Incorrect answer, Try again
Use perpendicular distance formula
• Distance formula for point from line
Find direction ratios of line
Use distance formula
Simplify result
Not using correct distance formula
If line makes angle of π/3 with each of x and y-axis, then acute angle made by z-axis is
🥳 Wohoo! Correct answer
• Direction cosine relations
Use direction cosines relation
Apply cos²α + cos²β + cos²γ = 1
Solve for angle with z-axis
Not using angle relations correctly
😢 Uh oh! Incorrect answer, Try again
Consider sum of direction cosines
• Direction cosine relations
Use direction cosines relation
Apply cos²α + cos²β + cos²γ = 1
Solve for angle with z-axis
Not using angle relations correctly
A die is thrown 10 times, probability that odd number will come up at least once is
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• P(at least one) = 1 - P(none)
Find P(no odd number)
Subtract from 1
Simplify fraction
Not using complement method
😢 Uh oh! Incorrect answer, Try again
Use complement rule
• P(at least one) = 1 - P(none)
Find P(no odd number)
Subtract from 1
Simplify fraction
Not using complement method
If A and B are two events such that P(A)=1/3, P(B)=1/2 and P(A∩B)=1/6, then P(A'/B) is
🥳 Wohoo! Correct answer
• P(A'/B) = P(B∩A')/P(B) • P(A') = 1 - P(A)
Express P(A'/B) using conditional probability formula
Use P(A'/B) = P(B∩A')/P(B)
Simplify using given probabilities
Not using conditional probability correctly
😢 Uh oh! Incorrect answer, Try again
Use complement rule for A'
• P(A'/B) = P(B∩A')/P(B) • P(A') = 1 - P(A)
Express P(A'/B) using conditional probability formula
Use P(A'/B) = P(B∩A')/P(B)
Simplify using given probabilities
Not using conditional probability correctly
Events E₁ and E₂ form partition of sample space S. A is any event such that P(E₁)=P(E₂)=1/2, P(E₂/A)=1/2 and P(A/E₂)=2/3, then P(E₁/A) is
🥳 Wohoo! Correct answer
• Bayes' theorem • Total probability theorem
Use Bayes' theorem
Apply partition property
Solve for P(E₁/A)
Not using Bayes' theorem correctly
😢 Uh oh! Incorrect answer, Try again
Use total probability theorem
• Bayes' theorem • Total probability theorem
Use Bayes' theorem
Apply partition property
Solve for P(E₁/A)
Not using Bayes' theorem correctly
Probability of solving problem by three persons A, B and C independently is 1/2, 1/4 and 1/3 respectively. Then probability of problem solved by any two of them is
🥳 Wohoo! Correct answer
• P(A∩B) = P(A)P(B) for independent events
Find P(exactly two solve)
Use addition principle
Simplify expression
Not considering all combinations
😢 Uh oh! Incorrect answer, Try again
Consider all possible pairs
• P(A∩B) = P(A)P(B) for independent events
Find P(exactly two solve)
Use addition principle
Simplify expression
Not considering all combinations
The value of sin²51° + sin²39° is
🥳 Wohoo! Correct answer
• sin²A + cos²A = 1 • sin(90°-x) = cos(x)
Use sin²A + cos²A = 1
Apply complementary angle property
Simplify
Not recognizing complementary angles
😢 Uh oh! Incorrect answer, Try again
Remember complementary angles
• sin²A + cos²A = 1 • sin(90°-x) = cos(x)
Use sin²A + cos²A = 1
Apply complementary angle property
Simplify
Not recognizing complementary angles
If tanA + cotA = 2, then value of tan⁴A + cot⁴A is
🥳 Wohoo! Correct answer
• tan²A·cot²A = 1 • (tanA + cotA)² = tan²A + cot²A + 2
Square both sides
Use identity tan²A + cot²A = (tanA + cotA)² - 2
Find tan⁴A + cot⁴A
Not using identities correctly
😢 Uh oh! Incorrect answer, Try again
Consider tan²A·cot²A = 1
• tan²A·cot²A = 1 • (tanA + cotA)² = tan²A + cot²A + 2
Square both sides
Use identity tan²A + cot²A = (tanA + cotA)² - 2
Find tan⁴A + cot⁴A
Not using identities correctly
If A,B,C are mutually exclusive and exhaustive events with P(A)=2P(B)=3P(C), then P(B) is
🥳 Wohoo! Correct answer
• Mutually exclusive: Sum = 1 • Given ratios
Use P(A)+P(B)+P(C)=1
Express P(A),P(C) in terms of P(B)
Solve for P(B)
Not using probability sum correctly
😢 Uh oh! Incorrect answer, Try again
Consider given ratios
• Mutually exclusive: Sum = 1 • Given ratios
Use P(A)+P(B)+P(C)=1
Express P(A),P(C) in terms of P(B)
Solve for P(B)
Not using probability sum correctly
The value of cos(sin⁻¹(π/3)+cos⁻¹(π/3)) is
🥳 Wohoo! Correct answer
• sin⁻¹x + cos⁻¹x = π/2 • cos(A+B) formula
Convert sin⁻¹ and cos⁻¹ to angles
Add angles inside cosine
Simplify using trigonometric formulas
Not using correct inverse trig properties
😢 Uh oh! Incorrect answer, Try again
Consider complementary angles
• sin⁻¹x + cos⁻¹x = π/2 • cos(A+B) formula
Convert sin⁻¹ and cos⁻¹ to angles
Add angles inside cosine
Simplify using trigonometric formulas
Not using correct inverse trig properties
cos[cos(2sin⁻¹(3/4))+cos⁻¹(3/4)] =
🥳 Wohoo! Correct answer
Trigonometric identities and formulas
Simplify inner expressions first
Use trigonometric identities
Evaluate step by step
Not simplifying systematically
😢 Uh oh! Incorrect answer, Try again
Break into manageable parts
Trigonometric identities and formulas
Simplify inner expressions first
Use trigonometric identities
Evaluate step by step
Not simplifying systematically
If a+π/2 < 2tan⁻¹x+3cot⁻¹x < b then 'a' and 'b' are respectively
🥳 Wohoo! Correct answer
Ranges of inverse trigonometric functions
Find range of tan⁻¹x
Find range of cot⁻¹x
Combine inequalities
Not considering function ranges properly
😢 Uh oh! Incorrect answer, Try again
Consider ranges of inverse trig functions
Ranges of inverse trigonometric functions
Find range of tan⁻¹x
Find range of cot⁻¹x
Combine inequalities
Not considering function ranges properly
√3 cosec 20° - sec 20° =
🥳 Wohoo! Correct answer
Trigonometric ratios and identities
Convert to sin and cos
Use standard angle values
Simplify expression
Not converting consistently
😢 Uh oh! Incorrect answer, Try again
Use trigonometric ratios for 20°
Trigonometric ratios and identities
Convert to sin and cos
Use standard angle values
Simplify expression
Not converting consistently
Foot of the perpendicular drawn from the point (1,3,4) to the plane 2x-y+z+3=0 is
🥳 Wohoo! Correct answer
Distance formula from point to plane
Use perpendicular distance formula
Find direction cosines of normal
Calculate foot of perpendicular
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Use perpendicular projection formula
Distance formula from point to plane
Use perpendicular distance formula
Find direction cosines of normal
Calculate foot of perpendicular
Not using correct formula
Acute angle between the line (x-5)/(2)=(y+1)/(-1)=(z+4)/(1) and the plane 3x-4y+z+5=0 is
🥳 Wohoo! Correct answer
Angle between line and plane formula
Find direction vector of line
Find normal vector of plane
Use angle formula between line and plane
Not finding acute angle
😢 Uh oh! Incorrect answer, Try again
Consider acute angle
Angle between line and plane formula
Find direction vector of line
Find normal vector of plane
Use angle formula between line and plane
Not finding acute angle
The distance of the point (1,1,1) from the line (x-1)/2 = (y-2)/1 = (z-3)/2 is
🥳 Wohoo! Correct answer
Distance from point to line formula
Convert line to parametric form
Use distance formula
Simplify result
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Use vector approach
Distance from point to line formula
Convert line to parametric form
Use distance formula
Simplify result
Not using correct formula
XY-plane divides the line joining points A(2,3,-5) and B(-1,-2,-3) in ratio
🥳 Wohoo! Correct answer
Section formula in 3D
Find where line intersects XY-plane
Use section formula
Compare ratios
Not considering external division
😢 Uh oh! Incorrect answer, Try again
Consider z=0 plane intersection
Section formula in 3D
Find where line intersects XY-plane
Use section formula
Compare ratios
Not considering external division
Two letters chosen from 'EQUATIONS'. Probability one is vowel and other consonant is
🥳 Wohoo! Correct answer
P(E) = favorable outcomes/total outcomes
Count vowels and consonants
Calculate favorable outcomes
Divide by total outcomes
Not considering all possible combinations
😢 Uh oh! Incorrect answer, Try again
Consider order of selection
P(E) = favorable outcomes/total outcomes
Count vowels and consonants
Calculate favorable outcomes
Divide by total outcomes
Not considering all possible combinations
If |x + 5| ≥ 10 then
🥳 Wohoo! Correct answer
|x| ≥ a ⟺ x ≥ a or x ≤ -a
Take |x + 5| ≥ 10
Split into x + 5 ≥ 10 or x + 5 ≤ -10
Solve to get x ≥ 5 or x ≤ -15
Forgetting negative case
😢 Uh oh! Incorrect answer, Try again
Break absolute value into two cases
|x| ≥ a ⟺ x ≥ a or x ≤ -a
Take |x + 5| ≥ 10
Split into x + 5 ≥ 10 or x + 5 ≤ -10
Solve to get x ≥ 5 or x ≤ -15
Forgetting negative case
If P(A)=0.5 and P(B)=0.3 are mutually exclusive, P(neither A nor B) =
🥳 Wohoo! Correct answer
P(neither) = 1 - P(A∪B)
Use P(A∪B) = P(A) + P(B) - P(A∩B) where P(A∩B) = 0 (mutually exclusive)
P(A∪B) = 0.5 + 0.3 = 0.8
P(neither) = 1 - P(A∪B) = 1 - 0.8 = 0.2
Not subtracting from 1
😢 Uh oh! Incorrect answer, Try again
Mutually exclusive means no overlap
P(neither) = 1 - P(A∪B)
Use P(A∪B) = P(A) + P(B) - P(A∩B) where P(A∩B) = 0 (mutually exclusive)
P(A∪B) = 0.5 + 0.3 = 0.8
P(neither) = 1 - P(A∪B) = 1 - 0.8 = 0.2
Not subtracting from 1
In simultaneous throw of pair of dice, P(total > 7) =
🥳 Wohoo! Correct answer
P(E) = n(E)/n(S)
List all possible outcomes where sum > 7
Count favorable outcomes: 15 cases
Divide by total outcomes (36) = 15/36 = 5/12
Missing some combinations
😢 Uh oh! Incorrect answer, Try again
List systematically to avoid missing cases
P(E) = n(E)/n(S)
List all possible outcomes where sum > 7
Count favorable outcomes: 15 cases
Divide by total outcomes (36) = 15/36 = 5/12
Missing some combinations
If A,B mutually exclusive, P(A)=3/5, P(B)=1/5, then P(A or B) =
🥳 Wohoo! Correct answer
P(A∪B) = P(A) + P(B)
For mutually exclusive events, P(A∩B) = 0
Apply P(A∪B) = P(A) + P(B)
P(A∪B) = 3/5 + 1/5 = 4/5 = 0.8
Subtracting probabilities
😢 Uh oh! Incorrect answer, Try again
Mutually exclusive means add probabilities
P(A∪B) = P(A) + P(B)
For mutually exclusive events, P(A∩B) = 0
Apply P(A∪B) = P(A) + P(B)
P(A∪B) = 3/5 + 1/5 = 4/5 = 0.8
Subtracting probabilities
If sin⁻¹x + cos⁻¹y = 2π/5, then cos⁻¹x + sin⁻¹y =
🥳 Wohoo! Correct answer
sin⁻¹x + cos⁻¹x = π/2
Use relation sin⁻¹x + cos⁻¹x = π/2
Substitute and solve
Get cos⁻¹x + sin⁻¹y = π - 2π/5 = 3π/5
Sign errors in substitution
😢 Uh oh! Incorrect answer, Try again
Inverse trig function relations
sin⁻¹x + cos⁻¹x = π/2
Use relation sin⁻¹x + cos⁻¹x = π/2
Substitute and solve
Get cos⁻¹x + sin⁻¹y = π - 2π/5 = 3π/5
Sign errors in substitution
Value of tan[(1/2) cos⁻¹(2/√5)] =
🥳 Wohoo! Correct answer
cos 2θ = 2cos²θ - 1
Let θ=(1/2)cos⁻¹(2/√5)
Use double angle formula for cos 2θ
Get tan θ = √5-2
Formula application errors
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Double angle formulas help
cos 2θ = 2cos²θ - 1
Let θ=(1/2)cos⁻¹(2/√5)
Use double angle formula for cos 2θ
Get tan θ = √5-2
Formula application errors
Image of point (1,6,3) in the line x/1 = (y-1)/2 = (z-2)/3 is
🥳 Wohoo! Correct answer
Line equation forms
Form vector equation of line
Find direction cosines
Use projection formula to get (1,0,7)
Projection formula errors
😢 Uh oh! Incorrect answer, Try again
Direction ratios method
Line equation forms
Form vector equation of line
Find direction cosines
Use projection formula to get (1,0,7)
Projection formula errors
Angle between lines 2x=3y=-z and 6x=-y=-4z is
🥳 Wohoo! Correct answer
cos θ = |a·b|/(|a||b|)
Convert to direction ratios (2:3:-1) and (6:-1:4)
Use dot product formula for angle
Get cos θ = 0, so θ = 90°
Direction ratio errors
😢 Uh oh! Incorrect answer, Try again
Direction cosines formula
cos θ = |a·b|/(|a||b|)
Convert to direction ratios (2:3:-1) and (6:-1:4)
Use dot product formula for angle
Get cos θ = 0, so θ = 90°
Direction ratio errors
Value of k where line (x-4)/1 = (y-2)/1 = (z-k)/2 lies on plane 2x-4y+z=7 is
🥳 Wohoo! Correct answer
Plane equation
Put point (4,2,k) in plane equation
Solve 2(4)-4(2)+k=7
Get k=7
Substitution errors
😢 Uh oh! Incorrect answer, Try again
Point lies on plane
Plane equation
Put point (4,2,k) in plane equation
Solve 2(4)-4(2)+k=7
Get k=7
Substitution errors
Locus represented by xy+yz=0 is
🥳 Wohoo! Correct answer
Normal vectors perpendicular
Factor as y(x+z)=0
Get y=0 or x+z=0 (two planes)
Check perpendicularity using normal vectors
Factoring errors
😢 Uh oh! Incorrect answer, Try again
Factor the equation
Normal vectors perpendicular
Factor as y(x+z)=0
Get y=0 or x+z=0 (two planes)
Check perpendicularity using normal vectors
Factoring errors
A bag contains 17 tickets numbered from 1 to 17. A ticket is drawn at random, then another ticket is drawn without replacing the rst one. The probability that both the tickets may show even numbers is
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Conditional probability
Count even numbers (8)
Use P(E₁)·P(E₂|E₁)
Get (8/17)(7/16)=7/34
Counting errors
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Without replacement
Conditional probability
Count even numbers (8)
Use P(E₁)·P(E₂|E₁)
Get (8/17)(7/16)=7/34
Counting errors
A ashlight has 10 batteries out of which 4 are dead. If 3 batteries are selected without replacement and tested, then the probability that all 3 are dead is
🥳 Wohoo! Correct answer
Combination formula
Use combinations formula
Calculate (⁴C₃)/(¹⁰C₃)
Simplify to 1/30
Combination errors
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Without replacement
Combination formula
Use combinations formula
Calculate (⁴C₃)/(¹⁰C₃)
Simplify to 1/30
Combination errors
The value of cos²45° - sin²15° is
🥳 Wohoo! Correct answer
cos²A - sin²B = cos(A+B)cos(A-B)
Convert cos²45° using cos 45° = 1/√2
Convert sin²15° using sin 15° = (1-cos 30°)/2
Substitute and simplify to get √3/4
Not recognizing standard angles
😢 Uh oh! Incorrect answer, Try again
Use standard angle values and double angle formulas
cos²A - sin²B = cos(A+B)cos(A-B)
Convert cos²45° using cos 45° = 1/√2
Convert sin²15° using sin 15° = (1-cos 30°)/2
Substitute and simplify to get √3/4
Not recognizing standard angles
If |x-2| ≤ 1, then
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|x-a| ≤ b ⟺ a-b ≤ x ≤ a+b
Convert modulus inequality: -1 ≤ x-2 ≤ 1
Add 2 to all parts: 1 ≤ x ≤ 3
Therefore x ∈ [1,3]
Not handling modulus correctly
😢 Uh oh! Incorrect answer, Try again
Draw number line to visualize
|x-a| ≤ b ⟺ a-b ≤ x ≤ a+b
Convert modulus inequality: -1 ≤ x-2 ≤ 1
Add 2 to all parts: 1 ≤ x ≤ 3
Therefore x ∈ [1,3]
Not handling modulus correctly
If cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π, then x(y+z) + y(z+x) + z(x+y) equals
🥳 Wohoo! Correct answer
cos⁻¹x + cos⁻¹y = cos⁻¹(xy-√(1-x²)(1-y²))
Take cos of both sides
Use addition formulas
Simplify to get 6
Not using addition formulas
😢 Uh oh! Incorrect answer, Try again
Use inverse trig properties
cos⁻¹x + cos⁻¹y = cos⁻¹(xy-√(1-x²)(1-y²))
Take cos of both sides
Use addition formulas
Simplify to get 6
Not using addition formulas
The perpendicular distance of point P(6,7,8) from XY-plane is
🥳 Wohoo! Correct answer
Distance from XY-plane = |z|
Distance from XY-plane is |z-coordinate|
For point (6,7,8), z=8
Therefore distance = 8
Confusing coordinate planes
😢 Uh oh! Incorrect answer, Try again
Remember plane definitions
Distance from XY-plane = |z|
Distance from XY-plane is |z-coordinate|
For point (6,7,8), z=8
Therefore distance = 8
Confusing coordinate planes
If 2sin⁻¹x-3cos⁻¹x=4,x∈[-1,1] then 2sin⁻¹x+3cos⁻¹x equals
🥳 Wohoo! Correct answer
sin⁻¹x + cos⁻¹x = π/2
Let sin⁻¹x + cos⁻¹x = π/2
Use this to solve equations
Get (6π-4)/5
Not using complementary relation
😢 Uh oh! Incorrect answer, Try again
Use complementary property
sin⁻¹x + cos⁻¹x = π/2
Let sin⁻¹x + cos⁻¹x = π/2
Use this to solve equations
Get (6π-4)/5
Not using complementary relation
If tanx + tany = 4π/5, then cot⁻¹x + cot⁻¹y equals
🥳 Wohoo! Correct answer
tan⁻¹x + cot⁻¹x = π/2
Use tanx + tany relation
Convert to cot⁻¹ form
Simplify to get π/5
Wrong inverse conversion
😢 Uh oh! Incorrect answer, Try again
Use tan⁻¹ and cot⁻¹ relation
tan⁻¹x + cot⁻¹x = π/2
Use tanx + tany relation
Convert to cot⁻¹ form
Simplify to get π/5
Wrong inverse conversion
Reflection of point (α,β,γ) in XY plane is
🥳 Wohoo! Correct answer
Reflection formulas in coordinate planes
Reflection in XY plane only changes z-coordinate
z-coordinate becomes -z
Therefore point becomes (α,β,-γ)
Changing wrong coordinate
😢 Uh oh! Incorrect answer, Try again
Visualize reflection in plane
Reflection formulas in coordinate planes
Reflection in XY plane only changes z-coordinate
z-coordinate becomes -z
Therefore point becomes (α,β,-γ)
Changing wrong coordinate
Plane 2x-3y+6z-11=0 makes angle sin⁻¹(α) with x-axis. Value of α is
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sin²α+cos²α=1
Normal vector to plane is n=2i-3j+6k
Angle between normal and x-axis is complementary to required angle
Calculate sin(α)=√2/3
Wrong angle relationship
😢 Uh oh! Incorrect answer, Try again
Use direction cosines
sin²α+cos²α=1
Normal vector to plane is n=2i-3j+6k
Angle between normal and x-axis is complementary to required angle
Calculate sin(α)=√2/3
Wrong angle relationship
Probability that out of 5 pens drawn with replacement at most one is defective from box with 10 defective out of 100 is
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P(X≤1)=P(X=0)+P(X=1)
P(defective)=1/10, P(good)=9/10
Use binomial probability for exactly 0 or 1 defective
Add probabilities
Not considering both cases
😢 Uh oh! Incorrect answer, Try again
Use complement method
P(X≤1)=P(X=0)+P(X=1)
P(defective)=1/10, P(good)=9/10
Use binomial probability for exactly 0 or 1 defective
Add probabilities
Not considering both cases
Two events A and B will be independent if
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P(A∩B)=P(A)P(B) for independence
For independence, P(A∩B)=P(A)P(B)
This implies P(A'∩B')=(1-P(A))(1-P(B))
Check other options are false
Confusing independence with mutual exclusion
😢 Uh oh! Incorrect answer, Try again
Use multiplication rule
P(A∩B)=P(A)P(B) for independence
For independence, P(A∩B)=P(A)P(B)
This implies P(A'∩B')=(1-P(A))(1-P(B))
Check other options are false
Confusing independence with mutual exclusion
The value of sin⁻¹(cos(53π/5)) is
🥳 Wohoo! Correct answer
sin⁻¹(cos θ) = π/2 - θ (in standard range)
Understand that sin⁻¹(cos θ) = π/2 - θ if -π/2 ≤ θ ≤ π/2
Simplify 53π/5 to get equivalent angle in standard range
Calculate π/2 - (3π/5) = -π/10
Not reducing angle to standard range
😢 Uh oh! Incorrect answer, Try again
Remember the relationship between inverse sine and cosine
sin⁻¹(cos θ) = π/2 - θ (in standard range)
Understand that sin⁻¹(cos θ) = π/2 - θ if -π/2 ≤ θ ≤ π/2
Simplify 53π/5 to get equivalent angle in standard range
Calculate π/2 - (3π/5) = -π/10
Not reducing angle to standard range
If 3 tan⁻¹x + cot⁻¹x = π then x equal to
🥳 Wohoo! Correct answer
tan⁻¹a + tan⁻¹b = tan⁻¹((a+b)/(1-ab))
Convert cot⁻¹x to tan⁻¹(1/x)
Equation becomes 3tan⁻¹x + tan⁻¹(1/x) = π
Use formula for sum of arctangent to get x = 1
Not converting cot⁻¹ to tan⁻¹
😢 Uh oh! Incorrect answer, Try again
Use the relationship between tan⁻¹ and cot⁻¹
tan⁻¹a + tan⁻¹b = tan⁻¹((a+b)/(1-ab))
Convert cot⁻¹x to tan⁻¹(1/x)
Equation becomes 3tan⁻¹x + tan⁻¹(1/x) = π
Use formula for sum of arctangent to get x = 1
Not converting cot⁻¹ to tan⁻¹
The simplified form of tan⁻¹[x/y] - tan⁻¹[(x-y)/(x+y)] is equal to
🥳 Wohoo! Correct answer
tan⁻¹a - tan⁻¹b = tan⁻¹((a-b)/(1+ab))
Convert to single tan⁻¹ using difference formula
Simplify using algebraic manipulation
Get result as π/4
Complex algebraic manipulation errors
😢 Uh oh! Incorrect answer, Try again
Use difference formula for inverse tangent
tan⁻¹a - tan⁻¹b = tan⁻¹((a-b)/(1+ab))
Convert to single tan⁻¹ using difference formula
Simplify using algebraic manipulation
Get result as π/4
Complex algebraic manipulation errors
The general solution of cotθ + tanθ = 2 is
🥳 Wohoo! Correct answer
tan²θ + 1 = sec²θ
Express in terms of sin and cos: (cosθ/sinθ + sinθ/cosθ) = 2
Simplify to get cos²θ + sin²θ = 2sinθcosθ
Solve to get θ = nπ/2 + (-1)^n π/4
Not considering all possible solutions
😢 Uh oh! Incorrect answer, Try again
Convert to sin and cos ratio
tan²θ + 1 = sec²θ
Express in terms of sin and cos: (cosθ/sinθ + sinθ/cosθ) = 2
Simplify to get cos²θ + sin²θ = 2sinθcosθ
Solve to get θ = nπ/2 + (-1)^n π/4
Not considering all possible solutions
The value of tan(π/8) is equal to
🥳 Wohoo! Correct answer
tan(A/2) = (1-cosA)/sinA
Use formula tan(A/2) = (1-cosA)/sinA
Put A = π/4
Simplify to get 1/(√2+1)
Not using half angle formula correctly
😢 Uh oh! Incorrect answer, Try again
Double angle formula helps
tan(A/2) = (1-cosA)/sinA
Use formula tan(A/2) = (1-cosA)/sinA
Put A = π/4
Simplify to get 1/(√2+1)
Not using half angle formula correctly
Two cards are drawn at random from a pack of 52 cards. The probability of these two being "Aces" is
🥳 Wohoo! Correct answer
P(E) = (favorable outcomes)/(total outcomes)
Calculate total possible outcomes: ⁵²C₂
Calculate favorable outcomes: ⁴C₂
Divide to get 1/221
Not using correct combination formula
😢 Uh oh! Incorrect answer, Try again
Use combination formula
P(E) = (favorable outcomes)/(total outcomes)
Calculate total possible outcomes: ⁵²C₂
Calculate favorable outcomes: ⁴C₂
Divide to get 1/221
Not using correct combination formula
If sin⁻¹x + sin⁻¹y = π/2 then x² is equal to
🥳 Wohoo! Correct answer
sin(A+B) = sinAcosB + cosAsinB
Take sin of both sides: sin(sin⁻¹x + sin⁻¹y) = 1
Use sin(A+B) formula
Solve to get x² = 1-y²
Not using correct trigonometric formula
😢 Uh oh! Incorrect answer, Try again
Use sin(A+B) formula
sin(A+B) = sinAcosB + cosAsinB
Take sin of both sides: sin(sin⁻¹x + sin⁻¹y) = 1
Use sin(A+B) formula
Solve to get x² = 1-y²
Not using correct trigonometric formula
Two dice are thrown simultaneously, the probability of obtaining a total score of 5 is
🥳 Wohoo! Correct answer
P(E) = n(E)/n(S)
List favorable outcomes: (1,4), (4,1), (2,3), (3,2)
Calculate total possible outcomes: 6 × 6 = 36
Probability = 4/36 = 1/9
Missing some favorable outcomes
😢 Uh oh! Incorrect answer, Try again
Count favorable and total outcomes
P(E) = n(E)/n(S)
List favorable outcomes: (1,4), (4,1), (2,3), (3,2)
Calculate total possible outcomes: 6 × 6 = 36
Probability = 4/36 = 1/9
Missing some favorable outcomes
Find the co-ordinates of the foot of the perpendicular drawn from the origin to the plane 5y + 8 = 0
🥳 Wohoo! Correct answer
Foot lies on plane and line from origin
Convert to normal form: y = -8/5
Point lies on y-axis
Coordinates are (0, -8/5, 0)
Not using shortest distance concept
😢 Uh oh! Incorrect answer, Try again
Use perpendicular property
Foot lies on plane and line from origin
Convert to normal form: y = -8/5
Point lies on y-axis
Coordinates are (0, -8/5, 0)
Not using shortest distance concept
The value of sin1° + sin2° +...+ sin359° is equal to
🥳 Wohoo! Correct answer
sin(360°-x) = -sin(x)
Pair terms: sin1° + sin359°, sin2° + sin358°, etc.
Use sin(360°-x) = -sin(x)
All pairs sum to zero
Not pairing complementary angles
😢 Uh oh! Incorrect answer, Try again
Look for symmetric properties
sin(360°-x) = -sin(x)
Pair terms: sin1° + sin359°, sin2° + sin358°, etc.
Use sin(360°-x) = -sin(x)
All pairs sum to zero
Not pairing complementary angles
Equation of line through (2,3,1) parallel to intersection of planes x - 2y - z + 5 = 0 and x + y + 3z = 6
🥳 Wohoo! Correct answer
• Cross product for direction • Line through point formula
Find direction ratios of intersection
Write line equation through point
Verify parallel condition
Wrong direction ratios
😢 Uh oh! Incorrect answer, Try again
Use direction of intersection
• Cross product for direction • Line through point formula
Find direction ratios of intersection
Write line equation through point
Verify parallel condition
Wrong direction ratios
Foot of perpendicular from origin to plane 2x - 3y + 4z = 29
🥳 Wohoo! Correct answer
• r·n = p • Foot formula
Use perpendicular formula
Substitute in plane equation
Find foot coordinates
Wrong normal vector
😢 Uh oh! Incorrect answer, Try again
Use shortest distance formula
• r·n = p • Foot formula
Use perpendicular formula
Substitute in plane equation
Find foot coordinates
Wrong normal vector
Two dice thrown simultaneously, probability sum more than 5
🥳 Wohoo! Correct answer
• P(A) = n(A)/n(S) • Complement rule
List favorable outcomes
Count total outcomes
Divide favorable by total
Missing combinations
😢 Uh oh! Incorrect answer, Try again
Consider complement method
• P(A) = n(A)/n(S) • Complement rule
List favorable outcomes
Count total outcomes
Divide favorable by total
Missing combinations
Man takes step forward prob 0.4, backward 0.6, probability after 11 steps one step from start
🥳 Wohoo! Correct answer
• Binomial probability • P(X=k) formula
Use binomial probability
Consider net displacement 1
Multiply probabilities
Wrong combinations
😢 Uh oh! Incorrect answer, Try again
Consider possible paths
• Binomial probability • P(X=k) formula
Use binomial probability
Consider net displacement 1
Multiply probabilities
Wrong combinations
Probability distribution X:0,1,2,3 P(x):0.2,k,k,2k find k
🥳 Wohoo! Correct answer
• ΣP(x) = 1 • Probability properties
Use ΣP(x) = 1
Substitute values
Solve for k = 0.1
Wrong equation setup
😢 Uh oh! Incorrect answer, Try again
Consider probability axioms
• ΣP(x) = 1 • Probability properties
Use ΣP(x) = 1
Substitute values
Solve for k = 0.1
Wrong equation setup
A box contains 6 red marbles (1-6) and 4 white marbles (12-15), P(white and odd)
🥳 Wohoo! Correct answer
• P(A) = n(A)/n(S) • Counting principle
Count favorable outcomes
Count total outcomes
Divide = 2/10 = 1/5
Wrong outcome counting
😢 Uh oh! Incorrect answer, Try again
List all white odd numbers
• P(A) = n(A)/n(S) • Counting principle
Count favorable outcomes
Count total outcomes
Divide = 2/10 = 1/5
Wrong outcome counting
Value of tan(10°) + tan(89°)
🥳 Wohoo! Correct answer
• tan(90°-x) = cot(x) • tan sum formula
Use complementary angle property
Apply tan addition formula
Simplify to get 2/sin2°
Wrong angle properties
😢 Uh oh! Incorrect answer, Try again
Consider angle relationships
• tan(90°-x) = cot(x) • tan sum formula
Use complementary angle property
Apply tan addition formula
Simplify to get 2/sin2°
Wrong angle properties
If α ≤ 2sin⁻¹x + cos⁻¹x ≤ β, then
🥳 Wohoo! Correct answer
• sin⁻¹x + cos⁻¹x = π/2 • Range of inverse trig
Use inverse function relations
Apply range conditions
Get α=0, β=π
Wrong range calculation
😢 Uh oh! Incorrect answer, Try again
Consider function ranges
• sin⁻¹x + cos⁻¹x = π/2 • Range of inverse trig
Use inverse function relations
Apply range conditions
Get α=0, β=π
Wrong range calculation
If sinx + siny = 1/2 and cosx + cosy = 1, then tan(x+y) =
🥳 Wohoo! Correct answer
• tan(A+B) formula • sin/cos sum formulas
Use addition formulas
Convert to tan form
Simplify to 4/3
Wrong identity application
😢 Uh oh! Incorrect answer, Try again
Use trigonometric identities
• tan(A+B) formula • sin/cos sum formulas
Use addition formulas
Convert to tan form
Simplify to 4/3
Wrong identity application
If sin⁻¹(2a/(1+a²)) + cos⁻¹((1-a²)/(1+a²)) = tan⁻¹(2x/(1-x²)) where a,x ∈ (0,1) then the value of x is
🥳 Wohoo! Correct answer
Inverse trigonometric formulas
Let a = tanθ, x = tanϕ
Express given equation in terms of θ and ϕ
Solve to get x = 2a/(1-a²)
Wrong domain consideration
😢 Uh oh! Incorrect answer, Try again
Convert to tan form to simplify
Inverse trigonometric formulas
Let a = tanθ, x = tanϕ
Express given equation in terms of θ and ϕ
Solve to get x = 2a/(1-a²)
Wrong domain consideration
The value of cot⁻¹[(√(1-sinx) + √(1+sinx)) / (√(1-sinx) - √(1+sinx))] where x ∈ [0,π/4] is
🥳 Wohoo! Correct answer
Trigonometric identities
Simplify numerator and denominator separately
Use identity cosecx - sinx = 1/tanx
Get final answer π-x/2
Missing rationalization step
😢 Uh oh! Incorrect answer, Try again
Rationalize and use trigonometric identities
Trigonometric identities
Simplify numerator and denominator separately
Use identity cosecx - sinx = 1/tanx
Get final answer π-x/2
Missing rationalization step
The equation of the plane through the points (2,1,0), (3,2,-2) and (3,1,7) is
🥳 Wohoo! Correct answer
Plane equation: ax+by+cz+d=0
Form direction vectors using points
Use vector normal form of plane
Get 7x-9y-z+5=0
Wrong normal vector calculation
😢 Uh oh! Incorrect answer, Try again
Use vector cross product method
Plane equation: ax+by+cz+d=0
Form direction vectors using points
Use vector normal form of plane
Get 7x-9y-z+5=0
Wrong normal vector calculation
The point of intersection of the line x+1=(y+3)/3=(z+2)/2 with the plane 3x+4y+5z=10 is
🥳 Wohoo! Correct answer
Line-plane intersection
Parametric form: r = a + λd
Substitute in plane equation
Solve for λ to get (2,6,-4)
Parameter calculation error
😢 Uh oh! Incorrect answer, Try again
Use parametric equations
Line-plane intersection
Parametric form: r = a + λd
Substitute in plane equation
Solve for λ to get (2,6,-4)
Parameter calculation error
If (2,3,-1) is the foot of the perpendicular from (4,2,1) to a plane, then the equation of the plane is
🥳 Wohoo! Correct answer
Plane equation from point and normal
Use direction cosines of normal = (P-F)
Form plane equation ax+by+cz+d=0
Get 2x-y+2z+1=0
Wrong normal vector
😢 Uh oh! Incorrect answer, Try again
Normal vector is parallel to PF
Plane equation from point and normal
Use direction cosines of normal = (P-F)
Form plane equation ax+by+cz+d=0
Get 2x-y+2z+1=0
Wrong normal vector
If a line makes an angle of π/3 with each X and Y axis then the acute angle made by Z-axis is
🥳 Wohoo! Correct answer
Direction cosines formula
Use direction cosines relation cos²α+cos²β+cos²γ=1
Given cos²α=cos²β=1/4
Find cos²γ=1/2, γ=π/4
Wrong angle substitution
😢 Uh oh! Incorrect answer, Try again
Direction cosines sum to 1
Direction cosines formula
Use direction cosines relation cos²α+cos²β+cos²γ=1
Given cos²α=cos²β=1/4
Find cos²γ=1/2, γ=π/4
Wrong angle substitution
If A and B are events such that P(A)=1/4, P(A/B)=1/2 and P(B/A)=2/3 then P(B) is
🥳 Wohoo! Correct answer
P(A/B)=P(A∩B)/P(B)
Use P(A∩B)=P(A)P(B/A)
Also P(A∩B)=P(B)P(A/B)
Get P(B)=1/3
Wrong probability formula
😢 Uh oh! Incorrect answer, Try again
Use conditional probability
P(A/B)=P(A∩B)/P(B)
Use P(A∩B)=P(A)P(B/A)
Also P(A∩B)=P(B)P(A/B)
Get P(B)=1/3
Wrong probability formula
A bag contains 2n+1 coins. It is known that n of these coins have head on both sides whereas the other n+1 coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is 31/42, then the value of n is
🥳 Wohoo! Correct answer
P(total) = weighted sum of individual probabilities
For fair coins P(H)=1/2, for double-headed P(H)=1. Setup equation: (n·1 + (n+1)·1/2)/(2n+1) = 31/42
Simplify: (2n + n+1)/(2(2n+1)) = 31/42
Solve to get n=10
Not considering total outcomes
😢 Uh oh! Incorrect answer, Try again
Consider probability from both types of coins
P(total) = weighted sum of individual probabilities
For fair coins P(H)=1/2, for double-headed P(H)=1. Setup equation: (n·1 + (n+1)·1/2)/(2n+1) = 31/42
Simplify: (2n + n+1)/(2(2n+1)) = 31/42
Solve to get n=10
Not considering total outcomes
Given that a,b and x are real numbers and a
🥳 Wohoo! Correct answer
Inequality rules with negative numbers
Consider inequality a
Multiply both sides by x (note sign change as x<0)
Get a/x > b/x
Missing sign change
😢 Uh oh! Incorrect answer, Try again
Consider sign when multiplying inequality
Inequality rules with negative numbers
Consider inequality a
Multiply both sides by x (note sign change as x<0)
Get a/x > b/x
Missing sign change
The value of e ^ (log₁₀tan1°·log₁₀tan2°·log₁₀tan3°....log₁₀tan89°) is
🥳 Wohoo! Correct answer
Logarithm properties
Convert to single logarithm using log properties
Observe that tan(90°-θ)=cot(θ)=1/tan(θ)
Product equals 1
Missing angle pattern
😢 Uh oh! Incorrect answer, Try again
Use complementary angle property
Logarithm properties
Convert to single logarithm using log properties
Observe that tan(90°-θ)=cot(θ)=1/tan(θ)
Product equals 1
Missing angle pattern
If lines x-1/-3=y-2/2k=z-3/2 and x-1/3k=y-5/1=z-6/-5 are mutually perpendicular, then k is equal to
🥳 Wohoo! Correct answer
Perpendicular vectors dot product = 0
Use direction vectors
Apply perpendicularity condition
Solve for k
Wrong direction vectors
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Direction vectors must be perpendicular
Perpendicular vectors dot product = 0
Use direction vectors
Apply perpendicularity condition
Solve for k
Wrong direction vectors
The octant in which the point (2, -4, -7)
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Octant numbering: (+,+,+)=1, (+,+,-)=2, (+,-,+)=3, (+,-,-)=4, (-,+,+)=5, (-,+,-)=6, (-,-,+)=7, (-,-,-)=8
Identify signs of coordinates: x=+2 (positive), y=-4 (negative), z=-7 (negative)
Recall octant classification rules based on coordinate signs
Match sign combination (+,-,-) to octant number
Confusing octant numbering sequence
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Remember octants by sign combinations
Octant numbering: (+,+,+)=1, (+,+,-)=2, (+,-,+)=3, (+,-,-)=4, (-,+,+)=5, (-,+,-)=6, (-,-,+)=7, (-,-,-)=8
Identify signs of coordinates: x=+2 (positive), y=-4 (negative), z=-7 (negative)
Recall octant classification rules based on coordinate signs
Match sign combination (+,-,-) to octant number
Confusing octant numbering sequence