If there are two values of 'a' which make determinant Δ = [1 -2 5; 2 a -1; 0 4 2a] = 86, then sum of these numbers is
🥳 Wohoo! Correct answer
3×3 determinant expansion
Expand determinant using first row
Get quadratic in 'a'
Add roots of quadratic
Not expanding determinant correctly
😢 Uh oh! Incorrect answer, Try again
Use determinant expansion
3×3 determinant expansion
Expand determinant using first row
Get quadratic in 'a'
Add roots of quadratic
Not expanding determinant correctly
The function f(x) = log(1+x) - 2x/(2+x) is increasing on
🥳 Wohoo! Correct answer
f'(x) > 0 for increasing
Find f'(x)
Find where f'(x) > 0
Solve inequality
Not checking domain
😢 Uh oh! Incorrect answer, Try again
Check derivative sign
f'(x) > 0 for increasing
Find f'(x)
Find where f'(x) > 0
Solve inequality
Not checking domain
If A={1,2,3,...,10} then number of subsets of A containing only odd numbers is
🥳 Wohoo! Correct answer
Number of subsets = 2ⁿ
Identify odd numbers in A: {1,3,5,7,9}
Count possible subsets: 2⁵
Subtract 1 for empty set: 32-1=31
Not considering empty set
😢 Uh oh! Incorrect answer, Try again
Use power set formula
Number of subsets = 2ⁿ
Identify odd numbers in A: {1,3,5,7,9}
Count possible subsets: 2⁵
Subtract 1 for empty set: 32-1=31
Not considering empty set
In a certain two 65% families own cell phones, 15000 families own scooter and 15% families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is
🥳 Wohoo! Correct answer
n(A∪B) = n(A) + n(B) - n(A∩B)
Use Venn diagram
Apply addition principle
Solve equation
Not considering intersection properly
😢 Uh oh! Incorrect answer, Try again
Draw Venn diagram first
n(A∪B) = n(A) + n(B) - n(A∩B)
Use Venn diagram
Apply addition principle
Solve equation
Not considering intersection properly
If A and B are non-singleton sets where n(A×B) = 35 and B ⊂ A, then ⁿ⁽ᴬ⁾ C ₙ₍ʙ₎ =
🥳 Wohoo! Correct answer
n(A×B)=n(A)×n(B)
Since B⊂A, let n(B)=k and n(A)=k+r
Use n(A×B)=n(A)×n(B)=35
Solve equation k(k+r)=35
Not using subset property correctly
😢 Uh oh! Incorrect answer, Try again
Remember B is subset of A
n(A×B)=n(A)×n(B)
Since B⊂A, let n(B)=k and n(A)=k+r
Use n(A×B)=n(A)×n(B)=35
Solve equation k(k+r)=35
Not using subset property correctly
If ((1+i)/(1-i))^x = 1 then
🥳 Wohoo! Correct answer
(1+i)/(1-i) = i
Convert complex fraction to standard form
Take log of both sides
Solve for x
Not simplifying complex fraction
😢 Uh oh! Incorrect answer, Try again
Use properties of complex numbers
(1+i)/(1-i) = i
Convert complex fraction to standard form
Take log of both sides
Solve for x
Not simplifying complex fraction
A student has to answer 10 questions, choosing at least 4 from each of the parts A and B. If there are 6 questions in part A and 7 in part B, then the number of ways can the student choose 10 questions is
🥳 Wohoo! Correct answer
nCr formula
List possible combinations of questions from A and B
Use combination formula for each case
Sum all possibilities
Not considering all cases
😢 Uh oh! Incorrect answer, Try again
Make table of possible selections
nCr formula
List possible combinations of questions from A and B
Use combination formula for each case
Sum all possibilities
Not considering all cases
If the middle term of the A.P is 300 then the sum of its first 51 terms is
🥳 Wohoo! Correct answer
Sn = n/2[2a+(n-1)d]
Use middle term formula
Find first term and common difference
Apply sum formula
Not using middle term properly
😢 Uh oh! Incorrect answer, Try again
Middle term = a + (n-1)d/2
Sn = n/2[2a+(n-1)d]
Use middle term formula
Find first term and common difference
Apply sum formula
Not using middle term properly
The equation of straight line which passes through the point (acos3,asin3) and perpendicular to xsec θ+ycosec θ=a is
🥳 Wohoo! Correct answer
m₁m₂ = -1
Find slope of given line
Use perpendicular slope condition
Form equation through point
Not using perpendicular condition correctly
😢 Uh oh! Incorrect answer, Try again
Perpendicular lines have negative reciprocal slopes
m₁m₂ = -1
Find slope of given line
Use perpendicular slope condition
Form equation through point
Not using perpendicular condition correctly
The mid points of the sides of triangle are (1,5,-1) (0,4,-2) and (2,3,4) then centroid of the triangle
🥳 Wohoo! Correct answer
Centroid = (x₁+x₂+x₃)/3
Use midpoint formula to find vertices
Average the vertices
Simplify coordinates
Not using centroid formula correctly
😢 Uh oh! Incorrect answer, Try again
Centroid divides median in ratio 2:1
Centroid = (x₁+x₂+x₃)/3
Use midpoint formula to find vertices
Average the vertices
Simplify coordinates
Not using centroid formula correctly
Two finite sets have m,n elements. Total subsets of first set is 56 more than second set. Values of m,n are
🥳 Wohoo! Correct answer
Number of subsets = 2ⁿ
Use 2ᵐ-2ⁿ=56
Try values of m,n
Verify m=6,n=3 satisfies
Not considering all possibilities
😢 Uh oh! Incorrect answer, Try again
Think about powers of 2
Number of subsets = 2ⁿ
Use 2ᵐ-2ⁿ=56
Try values of m,n
Verify m=6,n=3 satisfies
Not considering all possibilities
At x=1, the function f(x)={x³-1 when 1
🥳 Wohoo! Correct answer
Left and right hand derivatives
Check left and right hand limits
Check derivatives from both sides
Compare results
Not checking both conditions separately
😢 Uh oh! Incorrect answer, Try again
Check both continuity and differentiability
Left and right hand derivatives
Check left and right hand limits
Check derivatives from both sides
Compare results
Not checking both conditions separately
If f(x)=sin⁻¹(2x/(1+x²)), then f'(√3) is
🥳 Wohoo! Correct answer
• d/dx(sin⁻¹x) = 1/√(1-x²)
Use formula for derivative of arcsin: f'(x) = 1/√(1-x²)
Substitute x=√3 into the derivative formula
Simplify to get 1/2
Not properly applying inverse function derivative formula
😢 Uh oh! Incorrect answer, Try again
Use derivative formula for inverse sine carefully
• d/dx(sin⁻¹x) = 1/√(1-x²)
Use formula for derivative of arcsin: f'(x) = 1/√(1-x²)
Substitute x=√3 into the derivative formula
Simplify to get 1/2
Not properly applying inverse function derivative formula
Real value of α for which (1-isinα) / (1+2isinα) is purely real is
🥳 Wohoo! Correct answer
z is real if Im(z)=0
Rationalize denominator
Set imaginary part to zero
Solve for α=nπ
Not rationalizing properly
😢 Uh oh! Incorrect answer, Try again
Complex number is real if Im=0
z is real if Im(z)=0
Rationalize denominator
Set imaginary part to zero
Solve for α=nπ
Not rationalizing properly
d/dx[cos²{cot⁻¹(√(2+x)/√(2-x))}] is
🥳 Wohoo! Correct answer
Chain Rule, d/dx(cot⁻¹x)=-1/(1+x²)
Let u=√(2+x)/(2-x), then cot⁻¹u is being considered
Use chain rule for composite function
Simplify to get 1/4
Not using chain rule properly
😢 Uh oh! Incorrect answer, Try again
Break down into simpler parts using chain rule
Chain Rule, d/dx(cot⁻¹x)=-1/(1+x²)
Let u=√(2+x)/(2-x), then cot⁻¹u is being considered
Use chain rule for composite function
Simplify to get 1/4
Not using chain rule properly
Value of ⁴⁹C₃+⁴⁸C₃+⁴⁷C₃+⁴⁶C₃+⁴⁵C₃+⁴⁵C₄ is
🥳 Wohoo! Correct answer
ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁
Use combination formula
Apply Pascal's identity
Get ⁵⁰C₄
Not seeing pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern
ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁
Use combination formula
Apply Pascal's identity
Get ⁵⁰C₄
Not seeing pattern
If A={1,2,3,4,5,6}, then number of subsets of A which contain at least two elements is
🥳 Wohoo! Correct answer
• Total subsets = 2ⁿ • nCr formula
Find total subsets
Subtract empty set and single element sets
Simplify
Not subtracting correctly
😢 Uh oh! Incorrect answer, Try again
Use combination formula
• Total subsets = 2ⁿ • nCr formula
Find total subsets
Subtract empty set and single element sets
Simplify
Not subtracting correctly
Value of ¹⁶C₉ + ¹⁶C₁₀ - ¹⁶C₆ - ¹⁶C₇ is
🥳 Wohoo! Correct answer
• Pascal's identity • Combination properties
Use Pascal's identity
Combine terms
Simplify
Not using combination identities
😢 Uh oh! Incorrect answer, Try again
Look for patterns in combinations
• Pascal's identity • Combination properties
Use Pascal's identity
Combine terms
Simplify
Not using combination identities
Number of terms in expansion of (x+y+z)¹⁰ is
🥳 Wohoo! Correct answer
• Terms = (n+r-1)C(r-1) where r is number of variables
Use formula for terms in multinomial
Calculate (n+r-1)C(r-1)
Simplify
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Consider r=3 variables
• Terms = (n+r-1)C(r-1) where r is number of variables
Use formula for terms in multinomial
Calculate (n+r-1)C(r-1)
Simplify
Not using correct formula
If P(n): 2ⁿ < n!, Then the smallest positive integer for which P(n) is true is
🥳 Wohoo! Correct answer
• n! = n×(n-1)×...×1 • 2ⁿ grows slower than n! eventually
Compare 2ⁿ and n! for n=1,2,3,4
For n=4: 2⁴=16, 4!=24
Verify 4 is smallest such n
Not checking small values first
😢 Uh oh! Incorrect answer, Try again
Check values systematically
• n! = n×(n-1)×...×1 • 2ⁿ grows slower than n! eventually
Compare 2ⁿ and n! for n=1,2,3,4
For n=4: 2⁴=16, 4!=24
Verify 4 is smallest such n
Not checking small values first
In expansion of (1+x)ⁿ, C₁/C₀+2C₂/C₁+3C₃/C₂+...+nCₙ/Cₙ₋₁ equals
🥳 Wohoo! Correct answer
nCr=n!/r!(n-r)!
Write each term using nCr formula
Simplify ratio pattern
Sum to get n(n+1)/2
Not simplifying properly
😢 Uh oh! Incorrect answer, Try again
Look at coefficient pattern
nCr=n!/r!(n-r)!
Write each term using nCr formula
Simplify ratio pattern
Sum to get n(n+1)/2
Not simplifying properly
If sum of n terms of A.P is given by Sₙ=n²+n, then common difference of A.P is
🥳 Wohoo! Correct answer
• Sₙ=n/2[2a+(n-1)d] • d=aₙ-aₙ₋₁
Find aₙ by Sₙ-Sₙ₋₁
Compare consecutive terms
Calculate d
Not using correct method to find d
😢 Uh oh! Incorrect answer, Try again
Use Sₙ formula to find d
• Sₙ=n/2[2a+(n-1)d] • d=aₙ-aₙ₋₁
Find aₙ by Sₙ-Sₙ₋₁
Compare consecutive terms
Calculate d
Not using correct method to find d
If Sₙ stands for sum to n-terms of GP with 'a' as first term and 'r' as common ratio then Sₙ:S₂ₙ is
🥳 Wohoo! Correct answer
Sₙ = a(rⁿ-1)/(r-1)
Write Sₙ = a(rⁿ-1)/(r-1)
Write S₂ₙ = a(r²ⁿ-1)/(r-1)
Form ratio Sₙ/S₂ₙ = 1/(rⁿ+1)
Not using correct sum formula
😢 Uh oh! Incorrect answer, Try again
Use GP sum formula
Sₙ = a(rⁿ-1)/(r-1)
Write Sₙ = a(rⁿ-1)/(r-1)
Write S₂ₙ = a(r²ⁿ-1)/(r-1)
Form ratio Sₙ/S₂ₙ = 1/(rⁿ+1)
Not using correct sum formula
If A={a,b,c}, then number of binary operations on A is
🥳 Wohoo! Correct answer
• Binary operation maps A×A→A • Total possibilities = 3⁹
Understand binary operation definition
Count possible outputs for each input pair
Use multiplication principle
Not counting all possible combinations
😢 Uh oh! Incorrect answer, Try again
Remember total possible mappings
• Binary operation maps A×A→A • Total possibilities = 3⁹
Understand binary operation definition
Count possible outputs for each input pair
Use multiplication principle
Not counting all possible combinations
If f(x)= Determinant of |[x³,x+a,b+x],[x-a,x²-x,c+x],[x-b,x-c,0]|, then
🥳 Wohoo! Correct answer
• 3×3 determinant formula • Factor common terms
Set up determinant
Calculate determinant
Set equal to zero
Not evaluating determinant correctly
😢 Uh oh! Incorrect answer, Try again
Consider determinant properties
• 3×3 determinant formula • Factor common terms
Set up determinant
Calculate determinant
Set equal to zero
Not evaluating determinant correctly
If f(x)=[[1-cosKx]/xsinx if x≠0; AND 1/2 if x=0] is continuous at x=0, then value of K is
🥳 Wohoo! Correct answer
• Continuity requires limit=function value • L'Hôpital's rule
Find limit as x→0
Use L'Hôpital's rule
Equate to f(0)
Not applying L'Hôpital's rule correctly
😢 Uh oh! Incorrect answer, Try again
Consider continuity condition
• Continuity requires limit=function value • L'Hôpital's rule
Find limit as x→0
Use L'Hôpital's rule
Equate to f(0)
Not applying L'Hôpital's rule correctly
If a₁a₂a₃...a₉ are in A.P. then value of determinant |[a₁,a₂,a₃],[a₄,a₅,a₆],[a₇,a₈,a₉]| is
🥳 Wohoo! Correct answer
• A.P. formula: aₙ=a₁+(n-1)d • 3×3 determinant formula
Find pattern in A.P.
Set up determinant
Evaluate using properties
Not recognizing AP pattern in determinant
😢 Uh oh! Incorrect answer, Try again
Consider arithmetic sequence properties
• A.P. formula: aₙ=a₁+(n-1)d • 3×3 determinant formula
Find pattern in A.P.
Set up determinant
Evaluate using properties
Not recognizing AP pattern in determinant
If α and β are roots of x²+x+1=0, then α²+β² is
🥳 Wohoo! Correct answer
Vieta's formulas: α+β=-b/a, αβ=c/a
Apply Vieta's formulas: α+β=-1, αβ=1
Use (α+β)²=α²+2αβ+β²
Substitute values: α²+β²=(-1)²-2(1)=-1
Confusing sum of squares with square of sum
😢 Uh oh! Incorrect answer, Try again
Use Vieta's formulas for quadratic equations
Vieta's formulas: α+β=-b/a, αβ=c/a
Apply Vieta's formulas: α+β=-1, αβ=1
Use (α+β)²=α²+2αβ+β²
Substitute values: α²+β²=(-1)²-2(1)=-1
Confusing sum of squares with square of sum
Number of 4-digit numbers without repetition using 1,2,3,4,5,6,7 with two odd and two even digits
🥳 Wohoo! Correct answer
Combinations formula: nCr=n!/(r!(n-r)!)
Count odd (4) and even (3) digits
Calculate combinations possible
Multiply arrangements: C(4,2)×C(3,2)×4!
Not considering digit position arrangements
😢 Uh oh! Incorrect answer, Try again
Break into odd-even selection first
Combinations formula: nCr=n!/(r!(n-r)!)
Count odd (4) and even (3) digits
Calculate combinations possible
Multiply arrangements: C(4,2)×C(3,2)×4!
Not considering digit position arrangements
Number of terms in (x²+y²)²⁵-(x²-y²)²⁵ after simplification
🥳 Wohoo! Correct answer
Binomial expansion: (a+b)ⁿ
Expand using binomial theorem
Note terms cancel due to subtraction
Count remaining terms after cancellation
Missing term cancellations
😢 Uh oh! Incorrect answer, Try again
Look for pattern in term cancellation
Binomial expansion: (a+b)ⁿ
Expand using binomial theorem
Note terms cancel due to subtraction
Count remaining terms after cancellation
Missing term cancellations
Third term of GP is 9. Product of first five terms is
🥳 Wohoo! Correct answer
GP formula: Tₙ=ar^(n-1)
Let a be first term, r be ratio
Use a₃=ar²=9
Multiply a×ar×ar²×ar³×ar⁴
Not using GP properties correctly
😢 Uh oh! Incorrect answer, Try again
Use GP formula to find first term
GP formula: Tₙ=ar^(n-1)
Let a be first term, r be ratio
Use a₃=ar²=9
Multiply a×ar×ar²×ar³×ar⁴
Not using GP properties correctly
If AM and GM of roots of quadratic equation are 5 and 4 respectively, then quadratic equation is
🥳 Wohoo! Correct answer
AM=(α+β)/2, GM=√(αβ)
Use AM=(α+β)/2=5, GM=√(αβ)=4
Product of roots=16, sum=10
Form equation x²-10x+16=0
Confusing signs
😢 Uh oh! Incorrect answer, Try again
Relate AM-GM to roots
AM=(α+β)/2, GM=√(αβ)
Use AM=(α+β)/2=5, GM=√(αβ)=4
Product of roots=16, sum=10
Form equation x²-10x+16=0
Confusing signs
Value of √24.99 is
🥳 Wohoo! Correct answer
(1+x)^n ≈ 1+nx for small x
Use binomial expansion
Keep first two terms
Compare with options
Not using proper approximation
😢 Uh oh! Incorrect answer, Try again
Think about square root approximation
(1+x)^n ≈ 1+nx for small x
Use binomial expansion
Keep first two terms
Compare with options
Not using proper approximation
If U is the universal set with 10 elements; A and B are two sets such that n(A)=50, n(B)=60, n(A∩B)=20 then n(A'∩B') =
🥳 Wohoo! Correct answer
n(A∪B) = n(A) + n(B) - n(A∩B)
Use the formula for complement of intersection
Apply n(A'∩B') = n(U) - n(A∪B)
Calculate n(A∪B) using addition principle
Not using correct set operation laws
😢 Uh oh! Incorrect answer, Try again
Consider universal set size carefully
n(A∪B) = n(A) + n(B) - n(A∩B)
Use the formula for complement of intersection
Apply n(A'∩B') = n(U) - n(A∪B)
Calculate n(A∪B) using addition principle
Not using correct set operation laws
If P(n):2ⁿ
🥳 Wohoo! Correct answer
Exponential vs factorial growth
Compare 2ⁿ and n! for small n
Check values systematically
Find first n satisfying condition
Not checking systematically
😢 Uh oh! Incorrect answer, Try again
List out values for comparison
Exponential vs factorial growth
Compare 2ⁿ and n! for small n
Check values systematically
Find first n satisfying condition
Not checking systematically
Everybody in a room shakes hands with everybody else. Total handshakes is 45. Total persons in room is
🥳 Wohoo! Correct answer
nC₂ = n(n-1)/2
Use handshake formula nC₂ = 45
Solve n(n-1)/2 = 45
Get n² - n - 90 = 0, n = 10
Not considering both roots
😢 Uh oh! Incorrect answer, Try again
Total handshakes is combinations of 2
nC₂ = n(n-1)/2
Use handshake formula nC₂ = 45
Solve n(n-1)/2 = 45
Get n² - n - 90 = 0, n = 10
Not considering both roots
The constant term in expansion of (x²-1/x²)¹⁶ is
🥳 Wohoo! Correct answer
Binomial expansion
Find terms where powers sum to 0
Use binomial for (x²)ʳ(-1/x²)¹⁶⁻ʳ
r = 8 gives constant term
Missing negative signs
😢 Uh oh! Incorrect answer, Try again
Look for x⁰ term
Binomial expansion
Find terms where powers sum to 0
Use binomial for (x²)ʳ(-1/x²)¹⁶⁻ʳ
r = 8 gives constant term
Missing negative signs
Line parallel to 3x-4y+2=0 passing through (-2,3) is
🥳 Wohoo! Correct answer
y=mx+c format
Use parallel line form 3x-4y+k=0
Substitute (-2,3)
Solve for k=18
Sign errors in substitution
😢 Uh oh! Incorrect answer, Try again
Parallel lines have same slope
y=mx+c format
Use parallel line form 3x-4y+k=0
Substitute (-2,3)
Solve for k=18
Sign errors in substitution
If [(1-i)/(1+i)]⁹⁶ = a+ib then (a,b) is
🥳 Wohoo! Correct answer
Complex conjugates
Simplify (1-i)/(1+i)
Find 96th power
Get 1+0i
Power calculation errors
😢 Uh oh! Incorrect answer, Try again
Complex division trick
Complex conjugates
Simplify (1-i)/(1+i)
Find 96th power
Get 1+0i
Power calculation errors
The number of ways in which 5 girls and 3 boys can be seated in a row so that no two boys are together is
🥳 Wohoo! Correct answer
Permutation formula
Use gap method
Calculate 5!×⁶C₃×3!
Get 120×20×6=14400
Missing multiplication steps
😢 Uh oh! Incorrect answer, Try again
Consider gaps between girls
Permutation formula
Use gap method
Calculate 5!×⁶C₃×3!
Get 120×20×6=14400
Missing multiplication steps
If a,b,c are AP terms and x,y,z are GP terms, then x^(b-c)·y^(c-a)·z^(a-b) =
🥳 Wohoo! Correct answer
AP:d=common difference
Use AP: 2b=a+c
Use GP: y²=xz
Substitute and simplify
Not seeing pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern in powers
AP:d=common difference
Use AP: 2b=a+c
Use GP: y²=xz
Substitute and simplify
Not seeing pattern
If f(x) = [(√1+kx-√1-kx)/x] for x≠0, (2x+1)/(x-1) for 0≤x≤1 is continuous at x=0, then k=
🥳 Wohoo! Correct answer
Continuity conditions
Find limit as x→0 using L'Hôpital's rule
Equate with function value
Solve for k to get k=-1
L'Hôpital application errors
😢 Uh oh! Incorrect answer, Try again
Continuity means limits match
Continuity conditions
Find limit as x→0 using L'Hôpital's rule
Equate with function value
Solve for k to get k=-1
L'Hôpital application errors
If f(x) = [logₑx/(x-1)] for x≠1, k for x=1 is continuous at x=1, then k =
🥳 Wohoo! Correct answer
L'Hôpital's rule
Apply L'Hôpital's rule at x→1
Evaluate limit of logₑx/(x-1)
Get limit = 1, so k must equal 1
L'Hôpital application errors
😢 Uh oh! Incorrect answer, Try again
Use L'Hôpital for 0/0 form
L'Hôpital's rule
Apply L'Hôpital's rule at x→1
Evaluate limit of logₑx/(x-1)
Get limit = 1, so k must equal 1
L'Hôpital application errors
If A and B are finite sets and A ⊂ B, then
🥳 Wohoo! Correct answer
For A ⊂ B: n(A ∪ B) = n(B), n(A ∩ B) = n(A)
Since A ⊂ B, A is a proper subset of B, meaning all elements of A are in B
This means A ∩ B = A and A ∪ B = B
Therefore n(A ∪ B) = n(B)
Confusing union and intersection properties
😢 Uh oh! Incorrect answer, Try again
Draw a Venn diagram to visualize subset relationship
For A ⊂ B: n(A ∪ B) = n(B), n(A ∩ B) = n(A)
Since A ⊂ B, A is a proper subset of B, meaning all elements of A are in B
This means A ∩ B = A and A ∪ B = B
Therefore n(A ∪ B) = n(B)
Confusing union and intersection properties
3 + 5 + 7 + ...... to n term is
🥳 Wohoo! Correct answer
Sn = (n/2)[2a + (n-1)d]
Identify arithmetic sequence with a=3, d=2
Use arithmetic sequence sum formula Sn = (n/2)[2a + (n-1)d]
Simplify to get n(n+2)
Not identifying AP correctly
😢 Uh oh! Incorrect answer, Try again
Look for pattern in first few terms
Sn = (n/2)[2a + (n-1)d]
Identify arithmetic sequence with a=3, d=2
Use arithmetic sequence sum formula Sn = (n/2)[2a + (n-1)d]
Simplify to get n(n+2)
Not identifying AP correctly
If [(1+i)/(1-i)]^m = 1, then the least positive integral value of m is
🥳 Wohoo! Correct answer
i⁴ = 1, i² = -1
Convert complex fraction by rationalizing denominator
Find that [(1+i)/(1-i)] = i
Since i⁴ = 1, m = 4
Not using complex number properties
😢 Uh oh! Incorrect answer, Try again
Use properties of i
i⁴ = 1, i² = -1
Convert complex fraction by rationalizing denominator
Find that [(1+i)/(1-i)] = i
Since i⁴ = 1, m = 4
Not using complex number properties
If ⁿC₁₂ = ⁿC₈ then n is equal to
🥳 Wohoo! Correct answer
ⁿCᵣ = ⁿCₙ₋ᵣ
Use property ⁿCᵣ = ⁿCₙ₋ᵣ
Since ⁿC₁₂ = ⁿC₈, n-12 = 8
Solve to get n = 20
Not using combination properties
😢 Uh oh! Incorrect answer, Try again
Use combination formula symmetry
ⁿCᵣ = ⁿCₙ₋ᵣ
Use property ⁿCᵣ = ⁿCₙ₋ᵣ
Since ⁿC₁₂ = ⁿC₈, n-12 = 8
Solve to get n = 20
Not using combination properties
The total number of terms in expansion of (x+a)⁴⁷-(x-a)⁴⁷ after simplification is
🥳 Wohoo! Correct answer
(x+a)ⁿ-(x-a)ⁿ has (n+1)/2 terms when n is odd
Use odd power property difference of powers
Terms will be odd powers only up to 47
Count (47+1)/2 = 24 terms
Not recognizing pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern in expansion
(x+a)ⁿ-(x-a)ⁿ has (n+1)/2 terms when n is odd
Use odd power property difference of powers
Terms will be odd powers only up to 47
Count (47+1)/2 = 24 terms
Not recognizing pattern
Equation of line passing through (1,2) and perpendicular to y=3x-1 is
🥳 Wohoo! Correct answer
m₁m₂ = -1
Find slope of given line m₁=3
Perpendicular slope m₂=-1/3
Use point-slope form and simplify
Wrong perpendicular slope
😢 Uh oh! Incorrect answer, Try again
Remember perpendicular slopes product = -1
m₁m₂ = -1
Find slope of given line m₁=3
Perpendicular slope m₂=-1/3
Use point-slope form and simplify
Wrong perpendicular slope
The range of sec⁻¹x is
🥳 Wohoo! Correct answer
sec⁻¹x domain: (-∞,-1]∪[1,∞)
Consider domain of secx
Find principal values
Exclude π/2 as secπ/2 undefined
Including undefined points
😢 Uh oh! Incorrect answer, Try again
Draw graph
sec⁻¹x domain: (-∞,-1]∪[1,∞)
Consider domain of secx
Find principal values
Exclude π/2 as secπ/2 undefined
Including undefined points
Binary operation * on R-{-1} defined by a*b=a/(b+1) is
🥳 Wohoo! Correct answer
Commutative: ab = ba, Associative: (ab)c = a(bc)
Check commutative: ab = ba: a/(b+1) ≠ b/(a+1)
Check associative: (ab)c ≠ a(bc)
Neither property holds
Not checking both properties systematically
😢 Uh oh! Incorrect answer, Try again
Test with simple numbers
Commutative: ab = ba, Associative: (ab)c = a(bc)
Check commutative: ab = ba: a/(b+1) ≠ b/(a+1)
Check associative: (ab)c ≠ a(bc)
Neither property holds
Not checking both properties systematically
If f(x) is continuous at x=2, then value of K is
🥳 Wohoo! Correct answer
limₓ→c⁻f(x) = limₓ→c⁺f(x) = f(c)
Use continuity condition: limₓ→₂f(x)=f(2)
Evaluate limits from both sides
Solve for K to get 3/4
Not checking both sided limits
😢 Uh oh! Incorrect answer, Try again
Remember continuity definition
limₓ→c⁻f(x) = limₓ→c⁺f(x) = f(c)
Use continuity condition: limₓ→₂f(x)=f(2)
Evaluate limits from both sides
Solve for K to get 3/4
Not checking both sided limits
If y=tan⁻¹[(sinx+cosx)/(cosx-sinx)], then dy/dx equals
🥳 Wohoo! Correct answer
d/dx[tan⁻¹(x)]=1/(1+x²)
Simplify fraction inside tan⁻¹
Use identities to show it equals tan(π/4+x)
Differentiate to get dy/dx=1
Wrong chain rule application
😢 Uh oh! Incorrect answer, Try again
Look for standard angle forms
d/dx[tan⁻¹(x)]=1/(1+x²)
Simplify fraction inside tan⁻¹
Use identities to show it equals tan(π/4+x)
Differentiate to get dy/dx=1
Wrong chain rule application
If sinx=2t/(1+t²), tany=2t/(1-t²), then dy/dx equals
🥳 Wohoo! Correct answer
dy/dx=(dy/dt)÷(dx/dt)
Use parametric differentiation
Find dx/dt and dy/dt
Divide to get dy/dx=1
Wrong parametric differentiation
😢 Uh oh! Incorrect answer, Try again
Use chain rule
dy/dx=(dy/dt)÷(dx/dt)
Use parametric differentiation
Find dx/dt and dy/dt
Divide to get dy/dx=1
Wrong parametric differentiation
Derivative of cos⁻¹(2x²-1) w.r.t cos⁻¹x is
🥳 Wohoo! Correct answer
d/dx[cos⁻¹(x)]=-1/√(1-x²)
Let u=cos⁻¹(2x²-1) and v=1-cos⁻¹x
Find du/dx and dv/dx
Then du/dv = (du/dx)÷(dv/dx)=2
Wrong derivative formulas
😢 Uh oh! Incorrect answer, Try again
Use chain rule carefully
d/dx[cos⁻¹(x)]=-1/√(1-x²)
Let u=cos⁻¹(2x²-1) and v=1-cos⁻¹x
Find du/dx and dv/dx
Then du/dv = (du/dx)÷(dv/dx)=2
Wrong derivative formulas
If y=log(logx) then d²y/dx² equals
🥳 Wohoo! Correct answer
d²y/dx²=d/dx(dy/dx)
First find dy/dx=1/(xlogx)
Then differentiate again
Simplify to get given option
Wrong logarithm differentiation
😢 Uh oh! Incorrect answer, Try again
Use chain rule twice
d²y/dx²=d/dx(dy/dx)
First find dy/dx=1/(xlogx)
Then differentiate again
Simplify to get given option
Wrong logarithm differentiation
∫(cos2x-cos2θ)/(cosx-cosθ) dx equals
🥳 Wohoo! Correct answer
cos2A-cos2B=2(cosA+cosB)(cosA-cosB)
Use identity cos2A=2cos²A-1
Factor numerator
Integrate resulting form
Not factoring correctly
😢 Uh oh! Incorrect answer, Try again
Factor before integrating
cos2A-cos2B=2(cosA+cosB)(cosA-cosB)
Use identity cos2A=2cos²A-1
Factor numerator
Integrate resulting form
Not factoring correctly
Area of region bounded by y=x² and y=16 is
🥳 Wohoo! Correct answer
Area = ∫(upper-lower) dx
Find intersection points by solving x²=16
Area = ∫₍₋₄⁾⁴(16-x²) dx
Evaluate to get 256/3
Wrong limits
😢 Uh oh! Incorrect answer, Try again
Draw graph to visualize
Area = ∫(upper-lower) dx
Find intersection points by solving x²=16
Area = ∫₍₋₄⁾⁴(16-x²) dx
Evaluate to get 256/3
Wrong limits
The Set A has 4 elements and the set B has 5 elements then the number of injective mappings that can be defined from A to B is
🥳 Wohoo! Correct answer
P(n,r) = n!/(n-r)! where n=5, r=4
First identify that in injective mapping, each element of A must map to a unique element in B
Calculate number of choices: First element has 5 choices, second has 4, third has 3, and fourth has 2
Apply permutation formula: P(5,4) = 5!/(5-4)! = 5 × 4 × 3 × 2 = 120
Students might confuse with total possible mappings or combinations
😢 Uh oh! Incorrect answer, Try again
Think of arranging 4 distinct elements in 5 positions
P(n,r) = n!/(n-r)! where n=5, r=4
First identify that in injective mapping, each element of A must map to a unique element in B
Calculate number of choices: First element has 5 choices, second has 4, third has 3, and fourth has 2
Apply permutation formula: P(5,4) = 5!/(5-4)! = 5 × 4 × 3 × 2 = 120
Students might confuse with total possible mappings or combinations
Let * be binary operation defined on R by a * b = (a+b)/4 ∀ a,b∈R then the operation is *
🥳 Wohoo! Correct answer
For commutativity: ab = ba For associativity: (ab)c = a(bc)
Check commutative property: ab = (a+b)/4 = (b+a)/4 = ba
Check associative property: (ab)c = ((a+b)/4 + c)/4 ≠ (a + (b+c)/4)/4 = a(bc)
Conclude that operation is commutative but not associative
Not checking both properties independently
😢 Uh oh! Incorrect answer, Try again
Test with simple numbers to verify both properties
For commutativity: ab = ba For associativity: (ab)c = a(bc)
Check commutative property: ab = (a+b)/4 = (b+a)/4 = ba
Check associative property: (ab)c = ((a+b)/4 + c)/4 ≠ (a + (b+c)/4)/4 = a(bc)
Conclude that operation is commutative but not associative
Not checking both properties independently
The real part of (1-cosθ + i sinθ)⁻¹ is
🥳 Wohoo! Correct answer
(a+bi)⁻¹ = (a-bi)/(a²+b²)
Multiply numerator and denominator by conjugate
Simplify (1-cosθ - i sinθ)/[(1-cosθ)² + sin²θ]
Real part is 1/2
Not identifying conjugate method
😢 Uh oh! Incorrect answer, Try again
Use complex conjugate method
(a+bi)⁻¹ = (a-bi)/(a²+b²)
Multiply numerator and denominator by conjugate
Simplify (1-cosθ - i sinθ)/[(1-cosθ)² + sin²θ]
Real part is 1/2
Not identifying conjugate method
The simplified form of i^n + i^(n+1) + i^(n+2) + i^(n+3) is
🥳 Wohoo! Correct answer
i¹ = i, i² = -1, i³ = -i, i⁴ = 1
Write powers of i using cyclic pattern of 4
Group terms with same power
Sum equals 0
Not using cyclic property
😢 Uh oh! Incorrect answer, Try again
Use i⁴ = 1 pattern
i¹ = i, i² = -1, i³ = -i, i⁴ = 1
Write powers of i using cyclic pattern of 4
Group terms with same power
Sum equals 0
Not using cyclic property
The maximum value of (1/x)^x is
🥳 Wohoo! Correct answer
d/dx(ln(x^n)) = n/x
Take ln of expression: ln((1/x)^x) = -x ln(x)
Differentiate and set equal to zero: -ln(x) - 1 = 0
Solve to get x = 1/e, max value = e^(1/e)
Not using logarithmic differentiation
😢 Uh oh! Incorrect answer, Try again
Use logarithmic differentiation
d/dx(ln(x^n)) = n/x
Take ln of expression: ln((1/x)^x) = -x ln(x)
Differentiate and set equal to zero: -ln(x) - 1 = 0
Solve to get x = 1/e, max value = e^(1/e)
Not using logarithmic differentiation
The sum of 1st n terms of the series 1²/1 + 1²+2²/1+2 + 1²+2²+3²/1+2+3 +...... is
🥳 Wohoo! Correct answer
Sum of squares = n(n+1)(2n+1)/6
Find pattern in numerator and denominator
Use sum of squares formula for numerator
Get n(n-2)/3
Not identifying correct pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern in series
Sum of squares = n(n+1)(2n+1)/6
Find pattern in numerator and denominator
Use sum of squares formula for numerator
Get n(n-2)/3
Not identifying correct pattern
The 11th term in the expansion of [x + 1/√x]¹⁴ is
🥳 Wohoo! Correct answer
nCr = n!/(r!(n-r)!)
Use binomial expansion
Find general term: ¹⁴Cᵣ x^(r-r/2)
Put r=10 for 11th term
Error in power calculation
😢 Uh oh! Incorrect answer, Try again
Use binomial theorem
nCr = n!/(r!(n-r)!)
Use binomial expansion
Find general term: ¹⁴Cᵣ x^(r-r/2)
Put r=10 for 11th term
Error in power calculation
If the straight lines 2x + 3y - 3 = 0 and x + ky + 7 = 0 are perpendicular, then the value of k is
🥳 Wohoo! Correct answer
m₁m₂ = -1 for perpendicular lines
Find slopes of both lines: m₁ = -2/3 and m₂ = -1/k
Use perpendicular lines condition: m₁m₂ = -1
Solve to get k = -2/3
Not using negative reciprocal condition
😢 Uh oh! Incorrect answer, Try again
Perpendicular lines have negative reciprocal slopes
m₁m₂ = -1 for perpendicular lines
Find slopes of both lines: m₁ = -2/3 and m₂ = -1/k
Use perpendicular lines condition: m₁m₂ = -1
Solve to get k = -2/3
Not using negative reciprocal condition
Write the set builder form A = {-1, 1}
🥳 Wohoo! Correct answer
• Set builder notation rules • x² = 1 ⟺ x = ±1
Identify that -1 and 1 are roots of x² = 1
Check no other values satisfy
Verify set builder notation
Choosing too broad conditions
😢 Uh oh! Incorrect answer, Try again
Look for equation that gives exactly these values
• Set builder notation rules • x² = 1 ⟺ x = ±1
Identify that -1 and 1 are roots of x² = 1
Check no other values satisfy
Verify set builder notation
Choosing too broad conditions
If α and β are roots of x² - ax + b = 0, then a² + b² = 0, then a² + b² equals
🥳 Wohoo! Correct answer
• α + β = a • αβ = b
Use Vieta's formulas: α + β = a, αβ = b
Square both relations
Substitute in expression
Not using Vieta's formulas correctly
😢 Uh oh! Incorrect answer, Try again
Use relationships between roots and coefficients
• α + β = a • αβ = b
Use Vieta's formulas: α + β = a, αβ = b
Square both relations
Substitute in expression
Not using Vieta's formulas correctly
If 2nd and 5th terms of G.P. are 24 and 3 respectively, find sum of first 6 terms
🥳 Wohoo! Correct answer
• ar = a₁r^(n-1) • Sn = a(1-r^n)/(1-r)
Use ratio formula: ar = 24, ar⁴ = 3
Find r = 1/2, a = 48
Sum formula: a(1-r⁶)/(1-r)
Wrong ratio calculation
😢 Uh oh! Incorrect answer, Try again
Use geometric sequence properties
• ar = a₁r^(n-1) • Sn = a(1-r^n)/(1-r)
Use ratio formula: ar = 24, ar⁴ = 3
Find r = 1/2, a = 48
Sum formula: a(1-r⁶)/(1-r)
Wrong ratio calculation
Middle term of expansion of (10/x + x/10)¹⁰ is
🥳 Wohoo! Correct answer
• Middle term: r = n/2 • Binomial theorem
For middle term, r = n/2 = 5
Use binomial expansion
Find coefficient of x⁰
Wrong power calculation
😢 Uh oh! Incorrect answer, Try again
Consider even power expansion
• Middle term: r = n/2 • Binomial theorem
For middle term, r = n/2 = 5
Use binomial expansion
Find coefficient of x⁰
Wrong power calculation
The value of sin⁻¹(2√2/3) + sin⁻¹(1/3)
🥳 Wohoo! Correct answer
• sin⁻¹x + cos⁻¹x = π/2 • sin⁻¹(2√2/3)² + sin⁻¹(1/3)² = 1
Consider triangle with angles sin⁻¹(2√2/3) and sin⁻¹(1/3)
Use trigonometric identity sin⁻¹x + cos⁻¹x = π/2
Verify sum equals π/2
Not recognizing complementary angles
😢 Uh oh! Incorrect answer, Try again
Draw triangle to visualize angles
• sin⁻¹x + cos⁻¹x = π/2 • sin⁻¹(2√2/3)² + sin⁻¹(1/3)² = 1
Consider triangle with angles sin⁻¹(2√2/3) and sin⁻¹(1/3)
Use trigonometric identity sin⁻¹x + cos⁻¹x = π/2
Verify sum equals π/2
Not recognizing complementary angles
Function x^x; x>0 is strictly increasing at
🥳 Wohoo! Correct answer
d/dx(x^x)=x^x(1+lnx)
Take ln, differentiate
Find where derivative positive
Get x>1/e condition
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Use logarithmic differentiation
d/dx(x^x)=x^x(1+lnx)
Take ln, differentiate
Find where derivative positive
Get x>1/e condition
Wrong differentiation
If f(x) continuous, f(x) = 3x-8 if x≤5, f(x) = 2k if x>5. Find k
🥳 Wohoo! Correct answer
• Continuity: limx→a⁻ f(x) = limx→a⁺ f(x)
Use continuity at x=5
3(5)-8 = 2k
Solve k = 7/2
Wrong continuity application
😢 Uh oh! Incorrect answer, Try again
Consider continuity condition
• Continuity: limx→a⁻ f(x) = limx→a⁺ f(x)
Use continuity at x=5
3(5)-8 = 2k
Solve k = 7/2
Wrong continuity application
If 1,w,w² are cube roots of unity, (1-w+w²)(1+w-w²) equals
🥳 Wohoo! Correct answer
• w³ = 1 • 1+w+w² = 0
Use w³ = 1
Use 1+w+w² = 0
Multiply and simplify to 4
Wrong roots of unity
😢 Uh oh! Incorrect answer, Try again
Consider properties of cube roots
• w³ = 1 • 1+w+w² = 0
Use w³ = 1
Use 1+w+w² = 0
Multiply and simplify to 4
Wrong roots of unity
Solve for x: tan⁻¹((1-x)/(1+x)) = (1/2)tan⁻¹x, x > 0
🥳 Wohoo! Correct answer
• tan⁻¹((1-x)/(1+x)) formula • tan(A/2) formula
Use tan(A+B) formula
Let left side = θ, right side = φ
Solve to get x = 1/√3
Wrong inverse trig formulas
😢 Uh oh! Incorrect answer, Try again
Consider inverse tan properties
• tan⁻¹((1-x)/(1+x)) formula • tan(A/2) formula
Use tan(A+B) formula
Let left side = θ, right side = φ
Solve to get x = 1/√3
Wrong inverse trig formulas
Line through (2,2) perpendicular to 3x + y = 3, y-intercept
🥳 Wohoo! Correct answer
• m₁m₂ = -1 • Point-slope form
Find perpendicular slope
Write line equation
Find y-intercept
Wrong perpendicular slope
😢 Uh oh! Incorrect answer, Try again
Consider perpendicular lines
• m₁m₂ = -1 • Point-slope form
Find perpendicular slope
Write line equation
Find y-intercept
Wrong perpendicular slope
If A = [2-k 2; 1 3-k] is singular 2x2 matrix, then the value of 5k - k² is equal to
🥳 Wohoo! Correct answer
Det[2×2] = ad - bc
For singular matrix, determinant = 0. Calculate det(A) = (2-k)(3-k) - 2(1) = 0
Expand: (2-k)(3-k) - 2 = 6-5k+k² - 2 = 0
Solve k² - 5k + 4 = 0, therefore 5k - k² = 4
Not recognizing singular matrix condition
😢 Uh oh! Incorrect answer, Try again
For singular matrix, focus on determinant = 0
Det[2×2] = ad - bc
For singular matrix, determinant = 0. Calculate det(A) = (2-k)(3-k) - 2(1) = 0
Expand: (2-k)(3-k) - 2 = 6-5k+k² - 2 = 0
Solve k² - 5k + 4 = 0, therefore 5k - k² = 4
Not recognizing singular matrix condition
If Δ = [1 a a²; 1 b b²; 1 c c²] and Δ₁ = [1 1 1; bc ca ab; a b c], both are 3x3 matrix, then
🥳 Wohoo! Correct answer
Determinant properties
Evaluate determinant of given matrices
Compare corresponding elements and properties
Show that Δ₁ = -Δ using determinant properties
Not recognizing pattern in matrices
😢 Uh oh! Incorrect answer, Try again
Use determinant properties and compare matrices
Determinant properties
Evaluate determinant of given matrices
Compare corresponding elements and properties
Show that Δ₁ = -Δ using determinant properties
Not recognizing pattern in matrices
The modulus of the complex number ((1+i²)(1+3i))/((2-6i)(2-2i)) is
🥳 Wohoo! Correct answer
Complex number modulus formula
Simplify i² = -1 in numerator
Multiply numerator and denominator by conjugates
Calculate modulus
Wrong conjugate multiplication
😢 Uh oh! Incorrect answer, Try again
Use conjugate method
Complex number modulus formula
Simplify i² = -1 in numerator
Multiply numerator and denominator by conjugates
Calculate modulus
Wrong conjugate multiplication
Which of the following is an empty set?
🥳 Wohoo! Correct answer
Quadratic equation properties
Check if equation x²+1=0 has real solutions
Since x²+1>0 for all real x
No real solutions exist, set is empty
Not checking domain
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions
Quadratic equation properties
Check if equation x²+1=0 has real solutions
Since x²+1>0 for all real x
No real solutions exist, set is empty
Not checking domain
A line passes through (2,2) and is perpendicular to the line 3x+y=3. Its y-intercept is
🥳 Wohoo! Correct answer
Perpendicular slopes product=-1
Use perpendicular slopes: m₁m₂=-1
Form equation using point-slope
Find y-intercept=4/3
Wrong slope calculation
😢 Uh oh! Incorrect answer, Try again
Use perpendicular lines property
Perpendicular slopes product=-1
Use perpendicular slopes: m₁m₂=-1
Form equation using point-slope
Find y-intercept=4/3
Wrong slope calculation
If n is even and the middle term in the expansion of (x²+1/x)ⁿ is 924x⁶, then n is equal to
🥳 Wohoo! Correct answer
Binomial expansion formula
In middle term, powers of x should balance
Use binomial theorem: ⁿCᵣ coefficient = 924
Solve to get n=12
Wrong coefficient calculation
😢 Uh oh! Incorrect answer, Try again
Consider power balance in middle term
Binomial expansion formula
In middle term, powers of x should balance
Use binomial theorem: ⁿCᵣ coefficient = 924
Solve to get n=12
Wrong coefficient calculation
nᵗʰ term of the series 1+3/7+5/7²+1/7²+... is
🥳 Wohoo! Correct answer
Sequence general term formula
Find pattern in numerator: 2n-1
Find pattern in denominator: 7ⁿ⁻¹
General term = (2n-1)/7ⁿ⁻¹
Wrong pattern recognition
😢 Uh oh! Incorrect answer, Try again
Look for arithmetic-geometric pattern
Sequence general term formula
Find pattern in numerator: 2n-1
Find pattern in denominator: 7ⁿ⁻¹
General term = (2n-1)/7ⁿ⁻¹
Wrong pattern recognition
If p(1/q+1/r), q(1/r+1/p), r(1/p+1/q) are in A.P., then p,q,r
🥳 Wohoo! Correct answer
A.P. properties
Let terms be a,b,c in A.P.
Express in terms of p,q,r
Show p,q,r form A.P.
Wrong term arrangement
😢 Uh oh! Incorrect answer, Try again
Convert to common denominator
A.P. properties
Let terms be a,b,c in A.P.
Express in terms of p,q,r
Show p,q,r form A.P.
Wrong term arrangement
If 3x + i(4x - y) = 6 - i where x and y are real numbers, then the values of x and y respectively
🥳 Wohoo! Correct answer
z₁ = z₂ ⟺ Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)
Separate real and imaginary parts: 3x = 6 and 4x - y = -1
From real part, solve for x: x = 2
Substitute x in imaginary part to find y: y = 9
Not separating real and imaginary parts properly
😢 Uh oh! Incorrect answer, Try again
Always separate complex equation into real and imaginary parts
z₁ = z₂ ⟺ Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)
Separate real and imaginary parts: 3x = 6 and 4x - y = -1
From real part, solve for x: x = 2
Substitute x in imaginary part to find y: y = 9
Not separating real and imaginary parts properly
If a₁,a₂,a₃,...a₁₀ is a geometric progression and a₃/a₁ = 25, then a₉/a₅ equals
🥳 Wohoo! Correct answer
ar = a₁r^(r-1) for GP
Find r² = a₃/a₁ = 25, so r = 5
Note that a₉/a₅ = r⁴
Calculate r⁴ = 5⁴
Not identifying power of common ratio
😢 Uh oh! Incorrect answer, Try again
Use GP property: ar/ar-n = r^n
ar = a₁r^(r-1) for GP
Find r² = a₃/a₁ = 25, so r = 5
Note that a₉/a₅ = r⁴
Calculate r⁴ = 5⁴
Not identifying power of common ratio