Find value of limit: lim(y→0) ((3+y³-3)/y)
🥳 Wohoo! Correct answer
L'Hôpital's rule for 0/0 form
Apply L'Hôpital's rule as it's 0/0 form
Differentiate numerator and denominator
Simplify to get final answer
Not recognizing indeterminate form
😢 Uh oh! Incorrect answer, Try again
Identify indeterminate form first
L'Hôpital's rule for 0/0 form
Apply L'Hôpital's rule as it's 0/0 form
Differentiate numerator and denominator
Simplify to get final answer
Not recognizing indeterminate form
If f:R→R be defined by f(x)={2x : x>3; x : 1
🥳 Wohoo! Correct answer
Piecewise function evaluation
Calculate f(-1) = 3-(-1) = 4
Calculate f(2) = 2 (since 1<2≤3)
Calculate f(4) = 2(4) = 8, Add all: 4+2+8=14
Not checking intervals properly
😢 Uh oh! Incorrect answer, Try again
Check domain conditions carefully
Piecewise function evaluation
Calculate f(-1) = 3-(-1) = 4
Calculate f(2) = 2 (since 1<2≤3)
Calculate f(4) = 2(4) = 8, Add all: 4+2+8=14
Not checking intervals properly
Domain of cos⁻¹[x] is, where [.] denotes greatest integer function
🥳 Wohoo! Correct answer
Domain of cos⁻¹x is [-1,1]
Find range of greatest integer function
Intersect with domain of cos⁻¹
Get final domain
Not considering both functions
😢 Uh oh! Incorrect answer, Try again
Consider both functions' domains
Domain of cos⁻¹x is [-1,1]
Find range of greatest integer function
Intersect with domain of cos⁻¹
Get final domain
Not considering both functions
If f(1)=1, f'(1)=3 then derivative of f(f(f(x)))+(f(x))² at x=1 is
🥳 Wohoo! Correct answer
Chain rule: d/dx(f(g(x)))=f'(g(x))·g'(x)
Use chain rule for f(f(f(x)))
Add derivative of (f(x))²
Substitute given values
Not applying chain rule correctly
😢 Uh oh! Incorrect answer, Try again
Use chain rule carefully
Chain rule: d/dx(f(g(x)))=f'(g(x))·g'(x)
Use chain rule for f(f(f(x)))
Add derivative of (f(x))²
Substitute given values
Not applying chain rule correctly
If y = (1+x²)tan⁻¹x - x then dy/dx is
🥳 Wohoo! Correct answer
d/dx(tan⁻¹x) = 1/(1+x²)
Use product rule for first term
Subtract derivative of x
Simplify terms
Not applying product rule properly
😢 Uh oh! Incorrect answer, Try again
Break into parts using sum rule
d/dx(tan⁻¹x) = 1/(1+x²)
Use product rule for first term
Subtract derivative of x
Simplify terms
Not applying product rule properly
If x = eᶿsin θ, y = eᶿcos θ where θ is parameter, then dy/dx at (1,1) equals
🥳 Wohoo! Correct answer
dy/dx = (dy/dθ)/(dx/dθ)
Use parametric differentiation
Find dy/dθ and dx/dθ
Divide and substitute θ = π/4
Not handling parametric properly
😢 Uh oh! Incorrect answer, Try again
Use parametric derivative formula
dy/dx = (dy/dθ)/(dx/dθ)
Use parametric differentiation
Find dy/dθ and dx/dθ
Divide and substitute θ = π/4
Not handling parametric properly
The function f(x) = 4sin³x - 6sin²x + 12sinx + 100 is strictly
🥳 Wohoo! Correct answer
f'(x) < 0 for decreasing
Find f'(x) = 12(sin²x - sinx + 1)cosx
Analyze where f'(x) < 0
Verify in interval [π/2,π]
Not considering trigonometric period
😢 Uh oh! Incorrect answer, Try again
Check derivative sign for monotonicity
f'(x) < 0 for decreasing
Find f'(x) = 12(sin²x - sinx + 1)cosx
Analyze where f'(x) < 0
Verify in interval [π/2,π]
Not considering trigonometric period
Corner points of feasible region for LPP are (0,2),(3,0),(6,0),(6,8),(0,5). Let z=4x+6y be objective function. Minimum value occurs at
🥳 Wohoo! Correct answer
Linear programming principles
Plot points
Test objective function values
Compare values along line segment
Not checking all points
😢 Uh oh! Incorrect answer, Try again
Check all corner points
Linear programming principles
Plot points
Test objective function values
Compare values along line segment
Not checking all points
A dietician has to develop special diet using two foods X and Y. Each packet (30g) of X contains 12 units calcium, 4 units iron, 6 units cholesterol, 6 units vitamin A. Each packet of Y contains 3 units calcium, 20 units iron, 4 units cholesterol, 3 units vitamin A. Diet requires atleast 240 units calcium, atleast 460 units iron and atmost 300 units cholesterol. Corner points of feasible region are
🥳 Wohoo! Correct answer
Corner points found by intersection
Form constraints from given conditions
Graph inequalities
Find intersection points
Not graphing constraints properly
😢 Uh oh! Incorrect answer, Try again
Use linear programming method
Corner points found by intersection
Form constraints from given conditions
Graph inequalities
Find intersection points
Not graphing constraints properly
Corner points of feasible region of LPP are (0,2),(3,0),(6,0),(6,8),(0,5) then minimum value of z=4x+6y occurs at
🥳 Wohoo! Correct answer
Compare z values at corners
Plot points and form region
Calculate z at each point
Compare values
Not checking all corner points
😢 Uh oh! Incorrect answer, Try again
Check objective function values
Compare z values at corners
Plot points and form region
Calculate z at each point
Compare values
Not checking all corner points
Domain of function f(x)=1/log₁₀(1-x)+√(x+2) is
🥳 Wohoo! Correct answer
Domain includes all valid inputs
Check log denominator: 1-x>0
Check square root: x+2≥0
Combine conditions
Not considering all restrictions
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions
Domain includes all valid inputs
Check log denominator: 1-x>0
Check square root: x+2≥0
Combine conditions
Not considering all restrictions
Consider the following statements: Statement 1: lim(x→1)(ax²+bx+c)/(cx²+bx+a) is 1 where a+b+c≠0. Statement 2: lim(x→2)(1/x+1/2)/(x+2) is 1/4
🥳 Wohoo! Correct answer
L'Hôpital's rule for 0/0 form
For Statement 1: Put x=2 in numerator and denominator
For Statement 2: Use factorization method
Compare results with given values
Not checking all conditions
😢 Uh oh! Incorrect answer, Try again
Verify by direct substitution
L'Hôpital's rule for 0/0 form
For Statement 1: Put x=2 in numerator and denominator
For Statement 2: Use factorization method
Compare results with given values
Not checking all conditions
If a and b are fixed non-zero constants, then the derivative of (a/x⁴)-(b/x²)+Cosx is ma+nb-p where
🥳 Wohoo! Correct answer
d/dx(1/x^n) = -n/x^(n+1)
Find derivative of each term separately
Combine terms with constants
Identify coefficients
Wrong power rule application
😢 Uh oh! Incorrect answer, Try again
Use power rule for each term
d/dx(1/x^n) = -n/x^(n+1)
Find derivative of each term separately
Combine terms with constants
Identify coefficients
Wrong power rule application
f:R→R defined by f(x)={2x; when x>3, x²; when 1
🥳 Wohoo! Correct answer
Piecewise function notation
Identify function values for each x
Add the values correctly
Simplify final answer
Not checking domains properly
😢 Uh oh! Incorrect answer, Try again
Check domain conditions carefully
Piecewise function notation
Identify function values for each x
Add the values correctly
Simplify final answer
Not checking domains properly
Let A={x:x∈R; x is not a positive integer) Define f:A→R as f(x)=2x/(x-1), then f is
🥳 Wohoo! Correct answer
f'(x)=2/(x-1)²
Check one-one by derivative test
Check onto by range analysis
Combine results
Not checking all conditions
😢 Uh oh! Incorrect answer, Try again
Use derivative test for injection
f'(x)=2/(x-1)²
Check one-one by derivative test
Check onto by range analysis
Combine results
Not checking all conditions
The function f(x)=√3 sin2x - cos2x + 4 is one-one in the interval
🥳 Wohoo! Correct answer
f'(x)=2√3cos2x+2sin2x
Find derivative of function
Set f'(x)>0 or f'(x)<0
Find interval where strictly monotonic
Not checking monotonicity
😢 Uh oh! Incorrect answer, Try again
Check derivative sign
f'(x)=2√3cos2x+2sin2x
Find derivative of function
Set f'(x)>0 or f'(x)<0
Find interval where strictly monotonic
Not checking monotonicity
Domain of the function f(x)=1/√([x]²-[x]-6) where [x] is greatest integer ≤ x is
🥳 Wohoo! Correct answer
Domain: x where x²-[x]-6>0
Find where denominator is real
Check where denominator≠0
Combine intervals
Not handling floor function correctly
😢 Uh oh! Incorrect answer, Try again
Consider floor function carefully
Domain: x where x²-[x]-6>0
Find where denominator is real
Check where denominator≠0
Combine intervals
Not handling floor function correctly
If f(x)={Cosx 1 0; 0 2cosx 3; 0 1 2cosx} then lim(x→π)f(x)=
🥳 Wohoo! Correct answer
cos(π)=-1
Evaluate determinant as x→π
Substitute x=π in the matrix
Simplify the result
Not evaluating determinant correctly
😢 Uh oh! Incorrect answer, Try again
Consider trigonometric limits
cos(π)=-1
Evaluate determinant as x→π
Substitute x=π in the matrix
Simplify the result
Not evaluating determinant correctly
If y=(cosx²)², then dy/dx is equal to
🥳 Wohoo! Correct answer
d/dx(cos²x)=-2cosx·sinx
Use chain rule twice
Combine terms properly
Simplify the result
Not applying chain rule correctly
😢 Uh oh! Incorrect answer, Try again
Use chain rule carefully
d/dx(cos²x)=-2cosx·sinx
Use chain rule twice
Combine terms properly
Simplify the result
Not applying chain rule correctly
For constant a, d/dx(x^x + x^a + a^x + a^a) is
🥳 Wohoo! Correct answer
d/dx(x^x)=x^x(1+lnx)
Differentiate each term separately
Use logarithmic differentiation where needed
Combine all terms
Not using logarithmic differentiation
😢 Uh oh! Incorrect answer, Try again
Consider different types of exponents
d/dx(x^x)=x^x(1+lnx)
Differentiate each term separately
Use logarithmic differentiation where needed
Combine all terms
Not using logarithmic differentiation
Consider the following statements: Statement 1: If y=log₁₀x + logₑx then dy/dx=(log₁₀e)/x + 1/x Statement 2: If d/dx(log₁₀x)=logx/log10 and d/dx(logₑx)=logx/loge
🥳 Wohoo! Correct answer
d/dx(logₐx)=1/(x·lna)
Check derivative formulas for logarithms
Verify statement 1 using chain rule
Compare with standard results
Not using correct log differentiation rules
😢 Uh oh! Incorrect answer, Try again
Remember log base change formula
d/dx(logₐx)=1/(x·lna)
Check derivative formulas for logarithms
Verify statement 1 using chain rule
Compare with standard results
Not using correct log differentiation rules
If [x]²-5[x]+6=0, where [x] is greatest integer function, then
🥳 Wohoo! Correct answer
[x] ≤ x < [x]+1
Let [x]=k, solve k²-5k+6=0
Get k=2 or k=3
Find x range for these values
Not understanding GIF
😢 Uh oh! Incorrect answer, Try again
Consider GIF properties
[x] ≤ x < [x]+1
Let [x]=k, solve k²-5k+6=0
Get k=2 or k=3
Find x range for these values
Not understanding GIF
If y=(x-1)²(x-2)³(x-3)⁵ then dy/dx at x=4 is equal to
🥳 Wohoo! Correct answer
d/dx(u^n)=nu^(n-1)·du/dx
Use product rule and chain rule
Find derivative of each factor
Substitute x=4
Not using product rule correctly
😢 Uh oh! Incorrect answer, Try again
Break into simpler parts
d/dx(u^n)=nu^(n-1)·du/dx
Use product rule and chain rule
Find derivative of each factor
Substitute x=4
Not using product rule correctly
The function f(x)=x²-2x is strictly decreasing in the interval
🥳 Wohoo! Correct answer
f'(x)=2x-2
Find f'(x)=2x-2
Set f'(x)<0
Solve inequality
Not checking derivative sign correctly
😢 Uh oh! Incorrect answer, Try again
Decreasing when f'(x)<0
f'(x)=2x-2
Find f'(x)=2x-2
Set f'(x)<0
Solve inequality
Not checking derivative sign correctly
The maximum slope of the curve y=-x³+3x²+2x-27 is
🥳 Wohoo! Correct answer
d²y/dx²=-6x+6
Find dy/dx=(-3x²+6x+2)
Find critical points using d²y/dx²=0
Evaluate slope at critical points
Not finding all critical points
😢 Uh oh! Incorrect answer, Try again
Maximum slope at critical point
d²y/dx²=-6x+6
Find dy/dx=(-3x²+6x+2)
Find critical points using d²y/dx²=0
Evaluate slope at critical points
Not finding all critical points
If 2ˣ+2ʸ=2ˣ⁺ʸ, then dy/dx is
🥳 Wohoo! Correct answer
• d/dx(2ˣ) = 2ˣln(2) • Chain Rule
Differentiate both sides with respect to x: d/dx(2ˣ+2ʸ) = d/dx(2ˣ⁺ʸ)
Apply chain rule: 2ˣln(2) + 2ʸln(2)(dy/dx) = 2ˣ⁺ʸln(2)(1 + dy/dx)
Solve for dy/dx to get -2ʸ⁻ˣ
Forgetting chain rule application
😢 Uh oh! Incorrect answer, Try again
Remember that d/dx(2ˣ) = 2ˣln(2)
• d/dx(2ˣ) = 2ˣln(2) • Chain Rule
Differentiate both sides with respect to x: d/dx(2ˣ+2ʸ) = d/dx(2ˣ⁺ʸ)
Apply chain rule: 2ˣln(2) + 2ʸln(2)(dy/dx) = 2ˣ⁺ʸln(2)(1 + dy/dx)
Solve for dy/dx to get -2ʸ⁻ˣ
Forgetting chain rule application
The right hand and left hand limit of f(x)={(e¹/ˣ-1)/(e¹/ˣ+1), if x≠0; OR 0, if x=0} are respectively
🥳 Wohoo! Correct answer
• One-sided limits • Exponential properties
Evaluate right hand limit as x→0⁺
Evaluate left hand limit as x→0⁻
Compare both limits with given function value at x=0
Not considering direction of approach
😢 Uh oh! Incorrect answer, Try again
Consider behavior of exponential function near zero
• One-sided limits • Exponential properties
Evaluate right hand limit as x→0⁺
Evaluate left hand limit as x→0⁻
Compare both limits with given function value at x=0
Not considering direction of approach
Corner points of feasible region determined by linear constraints are (0,3), (1,1) and (3,0). Let z=px+qy, where p,q>0. Condition on p and q so minimum of z occurs at (3,0) and (1,1) is
🥳 Wohoo! Correct answer
• Linear programming optimality conditions
Compare z values at given points
Use minimum condition
Solve inequalities
Not comparing correct points
😢 Uh oh! Incorrect answer, Try again
Consider optimal point conditions
• Linear programming optimality conditions
Compare z values at given points
Use minimum condition
Solve inequalities
Not comparing correct points
If n(A)=2 and total number of possible relations from Set A to set B is 1024, then n(B) is
🥳 Wohoo! Correct answer
• Number of relations = 2ⁿ⁽ᴬ⁾ˣⁿ⁽ᴮ⁾
Use formula for total possible relations
Express 1024 as 2ᵃ
Solve for n(B)
Not using correct relation formula
😢 Uh oh! Incorrect answer, Try again
Remember total relations = 2ⁿ⁽ᴬ⁾ˣⁿ⁽ᴮ⁾
• Number of relations = 2ⁿ⁽ᴬ⁾ˣⁿ⁽ᴮ⁾
Use formula for total possible relations
Express 1024 as 2ᵃ
Solve for n(B)
Not using correct relation formula
Negation of statement "For all real numbers x and y, x+y=y+x" is
🥳 Wohoo! Correct answer
• Negation of ∀: ∃ • Negate statement too
Identify universal statement
Negate quantifier and statement
Express negation correctly
Not negating both quantifier and statement
😢 Uh oh! Incorrect answer, Try again
Remember negation rules
• Negation of ∀: ∃ • Negate statement too
Identify universal statement
Negate quantifier and statement
Express negation correctly
Not negating both quantifier and statement
lim(x→0) [tanx / √(2x+4) -2)] is equal to
🥳 Wohoo! Correct answer
• L'Hôpital's rule for 0/0 form
Use L'Hôpital's rule
Simplify numerator and denominator
Take limit
Not recognizing indeterminate form
😢 Uh oh! Incorrect answer, Try again
Look for indeterminate form
• L'Hôpital's rule for 0/0 form
Use L'Hôpital's rule
Simplify numerator and denominator
Take limit
Not recognizing indeterminate form
If relation R on set {1,2,3} be defined by R={(1,1)}, then R is
🥳 Wohoo! Correct answer
• Reflexive: (a,a) ∈ R • Symmetric: (a,b) ∈ R ⟹ (b,a) ∈ R
Check reflexive property
Check symmetric property
Check transitive property
Not checking all properties
😢 Uh oh! Incorrect answer, Try again
Test each property separately
• Reflexive: (a,a) ∈ R • Symmetric: (a,b) ∈ R ⟹ (b,a) ∈ R
Check reflexive property
Check symmetric property
Check transitive property
Not checking all properties
Let f:[2,∞)→R be function defined by f(x)=x²-4x+5, then range of f is
🥳 Wohoo! Correct answer
• Range for quadratic function • Minimum point
Find derivative
Find minimum point
Determine range
Not considering domain restrictions
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions
• Range for quadratic function • Minimum point
Find derivative
Find minimum point
Determine range
Not considering domain restrictions
The domain of function defined by f(x)=cos⁻¹√(x-1) is
🥳 Wohoo! Correct answer
• Domain of cos⁻¹ is [-1,1] • Domain of √x is [0,∞)
Consider domain of square root
Consider domain of cos⁻¹
Combine restrictions
Not considering both function restrictions
😢 Uh oh! Incorrect answer, Try again
Remember cos⁻¹ has range [0,π]
• Domain of cos⁻¹ is [-1,1] • Domain of √x is [0,∞)
Consider domain of square root
Consider domain of cos⁻¹
Combine restrictions
Not considering both function restrictions
If ∛y√x = ∜(x + y)⁵, then dy/dx =
🥳 Wohoo! Correct answer
Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
Differentiate both sides with respect to x
Apply chain rule to right side
Solve for dy/dx
Not applying chain rule correctly
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation
Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
Differentiate both sides with respect to x
Apply chain rule to right side
Solve for dy/dx
Not applying chain rule correctly
Interval where f(x)=x³-6x²+9x-10 is increasing
🥳 Wohoo! Correct answer
f'(x)>0 for increasing function
Find f'(x)=3x²-12x+9
Solve f'(x)=0
Check sign changes of f'
Not checking interval endpoints
😢 Uh oh! Incorrect answer, Try again
Use first derivative test
f'(x)>0 for increasing function
Find f'(x)=3x²-12x+9
Solve f'(x)=0
Check sign changes of f'
Not checking interval endpoints
If f:IR→IR and g:[0,∞)→IR are defined by f(x)=x² and g(x)=√x. Which is not true?
🥳 Wohoo! Correct answer
Function composition rules
Find gof(x) and fog(x)
Evaluate at given points
Compare with given options
Confusing order of composition
😢 Uh oh! Incorrect answer, Try again
Check composition order carefully
Function composition rules
Find gof(x) and fog(x)
Evaluate at given points
Compare with given options
Confusing order of composition
If A = {x|x ∈ N, x ≤ 5}B = {x|x ∈ Z, x² - 5x + 6 = 0}then the number of onto functions from A to B
🥳 Wohoo! Correct answer
Onto function conditions
Find elements of sets A and B
Count possible functions
Check onto condition
Not checking onto condition properly
😢 Uh oh! Incorrect answer, Try again
Consider range and domain carefully
Onto function conditions
Find elements of sets A and B
Count possible functions
Check onto condition
Not checking onto condition properly
The domain of the function f:ℝ→ℝ defined by f(x)=√(x²-7x+12) is
🥳 Wohoo! Correct answer
Domain requires expression ≥0 under √
Find where expression under root is ≥0
Solve quadratic inequality x²-7x+12≥0
Express answer in interval notation
Not checking endpoint conditions
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions for square root
Domain requires expression ≥0 under √
Find where expression under root is ≥0
Solve quadratic inequality x²-7x+12≥0
Express answer in interval notation
Not checking endpoint conditions
limx→π/4 (√2 cos x - 1) / (cot x - 1) equals
🥳 Wohoo! Correct answer
L'Hospital's Rule for 0/0
Apply L'Hospital's Rule
Differentiate num & denom
Get limit = 1/2
Not recognizing 0/0
😢 Uh oh! Incorrect answer, Try again
Use L'Hospital when 0/0
L'Hospital's Rule for 0/0
Apply L'Hospital's Rule
Differentiate num & denom
Get limit = 1/2
Not recognizing 0/0
Negation of "All continuous functions are differentiable" is
🥳 Wohoo! Correct answer
Negation of universal statement
Understand logical negation
Apply negation rules
Choose correct logical form
Confusing negation rules
😢 Uh oh! Incorrect answer, Try again
Consider logical equivalence
Negation of universal statement
Understand logical negation
Apply negation rules
Choose correct logical form
Confusing negation rules
Negation of "For every real number x; x²+5 is positive" is
🥳 Wohoo! Correct answer
Negation of ∀ is ∃
Understand logical negation
Negate universal quantifier
Get existential statement
Wrong quantifier negation
😢 Uh oh! Incorrect answer, Try again
Logical negation rules
Negation of ∀ is ∃
Understand logical negation
Negate universal quantifier
Get existential statement
Wrong quantifier negation
If [x] represents greatest integer function and f(x)=x-[x]-cos x then f'(π/2) =
🥳 Wohoo! Correct answer
Greatest Integer Function properties
Find derivative parts separately
Consider discontinuities
Evaluate at π/2
Not handling GIF derivative
😢 Uh oh! Incorrect answer, Try again
Consider GIF properties
Greatest Integer Function properties
Find derivative parts separately
Consider discontinuities
Evaluate at π/2
Not handling GIF derivative
If f(x) = sin⁻¹[2^(x+1)/(1 + 4^x)], then f'(0) =
🥳 Wohoo! Correct answer
d/dx(sin⁻¹x) = 1/√(1-x²)
Use chain rule
Simplify derivative
Evaluate at x=0
Not applying chain rule correctly
😢 Uh oh! Incorrect answer, Try again
Use inverse sine derivative
d/dx(sin⁻¹x) = 1/√(1-x²)
Use chain rule
Simplify derivative
Evaluate at x=0
Not applying chain rule correctly
If x=asec²θ, y=atanθ then d²y/dx² =
🥳 Wohoo! Correct answer
d²y/dx² = (d²y/dθ²)/(dx/dθ)² - (dy/dθ)(d²x/dθ²)/(dx/dθ)³
Use chain rule twice
Simplify expressions
Find second derivative
Not using correct parametric formula
😢 Uh oh! Incorrect answer, Try again
Consider parametric derivatives
d²y/dx² = (d²y/dθ²)/(dx/dθ)² - (dy/dθ)(d²x/dθ²)/(dx/dθ)³
Use chain rule twice
Simplify expressions
Find second derivative
Not using correct parametric formula
If P(n): "2²ⁿ-1 is divisible by k for all n∈N" is true, then k =
🥳 Wohoo! Correct answer
Divisibility rules
Check n=1: 2²⁻¹=3
Check n=2: 2⁴-1=15
Find common factor: 3
Not checking enough cases
😢 Uh oh! Incorrect answer, Try again
Check first few terms
Divisibility rules
Check n=1: 2²⁻¹=3
Check n=2: 2⁴-1=15
Find common factor: 3
Not checking enough cases
The value of lim(x→0) |x|/x is
🥳 Wohoo! Correct answer
One-sided limits
Check left limit
Check right limit
Limits not equal
Forgetting to check both sides
😢 Uh oh! Incorrect answer, Try again
Check both sides
One-sided limits
Check left limit
Check right limit
Limits not equal
Forgetting to check both sides
Let f(x) = x - (1/x) then f'(-1) is
🥳 Wohoo! Correct answer
d/dx(1/x) = -1/x²
Take f(x) = x - (1/x) and apply differentiation rule: f'(x) = 1 + (1/x²)
Substitute x = -1 into the derivative f'(x) = 1 + (1/x²)
f'(-1) = 1 + (1/1) = 2
Forgetting positive nature of x²
😢 Uh oh! Incorrect answer, Try again
Use chain rule and product rule for differentiation
d/dx(1/x) = -1/x²
Take f(x) = x - (1/x) and apply differentiation rule: f'(x) = 1 + (1/x²)
Substitute x = -1 into the derivative f'(x) = 1 + (1/x²)
f'(-1) = 1 + (1/1) = 2
Forgetting positive nature of x²
The negation of "72 is divisible by 2 and 3" is
🥳 Wohoo! Correct answer
¬(p ∧ q) ≡ ¬p ∨ ¬q
Let p: "72 is divisible by 2" and q: "72 is divisible by 3"
Negation of (p ∧ q) is (¬p ∨ ¬q)
Therefore: "not divisible by 2 OR not divisible by 3"
Confusing AND/OR in negation
😢 Uh oh! Incorrect answer, Try again
De Morgan's law for negation
¬(p ∧ q) ≡ ¬p ∨ ¬q
Let p: "72 is divisible by 2" and q: "72 is divisible by 3"
Negation of (p ∧ q) is (¬p ∨ ¬q)
Therefore: "not divisible by 2 OR not divisible by 3"
Confusing AND/OR in negation
If f,g:R→R, f(x)=|x|+x, g(x)=|x|-x, then (fog)(x) for x<0 =
🥳 Wohoo! Correct answer
Composite function rules
First find g(x) for x<0: g(x)=-x-x=-2x
Then find |g(x)| + g(x) = |-2x| + (-2x)
For x<0, result is 2x + (-2x) = -4x
Sign errors in composition
😢 Uh oh! Incorrect answer, Try again
Compose functions carefully
Composite function rules
First find g(x) for x<0: g(x)=-x-x=-2x
Then find |g(x)| + g(x) = |-2x| + (-2x)
For x<0, result is 2x + (-2x) = -4x
Sign errors in composition
Let f:R→R be given by f(x)=tanx. Then f⁻¹(1) is
🥳 Wohoo! Correct answer
tan(π/4)=1
Find x where tanx=1
Solve to get x=π/4
Check uniqueness
Not considering periodicity
😢 Uh oh! Incorrect answer, Try again
Inverse function basics
tan(π/4)=1
Find x where tanx=1
Solve to get x=π/4
Check uniqueness
Not considering periodicity
A has 6 distinct elements. Number of non-bijective functions A→A =
🥳 Wohoo! Correct answer
Functions = n^n
Total functions = 6⁶
Number of bijective functions = 6!
Non-bijective = Total - Bijective = 6⁶ - 6!
Confusing permutations
😢 Uh oh! Incorrect answer, Try again
Bijective needs 1-1 and onto
Functions = n^n
Total functions = 6⁶
Number of bijective functions = 6!
Non-bijective = Total - Bijective = 6⁶ - 6!
Confusing permutations
Let f : R → R be defined by f(x)={2x; when x>3, x²; when 1
🥳 Wohoo! Correct answer
Piecewise function rules
f(-1)=3(-1)=-3, f(2)=2²=4, f(4)=2(4)=8
Add all values: -3+4+8
Final sum = 9
Domain confusion
😢 Uh oh! Incorrect answer, Try again
Check domain for each input
Piecewise function rules
f(-1)=3(-1)=-3, f(2)=2²=4, f(4)=2(4)=8
Add all values: -3+4+8
Final sum = 9
Domain confusion
If cosy = xcos(a+y) with cosa ≠ ±1, then dy/dx =
🥳 Wohoo! Correct answer
Chain rule
Implicitly differentiate both sides
Collect dy/dx terms
Solve to get dy/dx=cos(a+y)/sinα
Chain rule errors
😢 Uh oh! Incorrect answer, Try again
Implicit differentiation
Chain rule
Implicitly differentiate both sides
Collect dy/dx terms
Solve to get dy/dx=cos(a+y)/sinα
Chain rule errors
For the function f(x)=x³-6x²+12x-3; x=2 is
🥳 Wohoo! Correct answer
Second derivative test
Find f'(x)=3x²-12x+12
Find f''(x)=6x-12
At x=2: f''(2)=0, f'''(2)≠0
Confusing inflection with extrema
😢 Uh oh! Incorrect answer, Try again
Check signs of derivatives
Second derivative test
Find f'(x)=3x²-12x+12
Find f''(x)=6x-12
At x=2: f''(2)=0, f'''(2)≠0
Confusing inflection with extrema
Let f:R→R be defined by f(x)=x²+1. Then pre-images of 17 and -3 are
🥳 Wohoo! Correct answer
Quadratic equation
Solve x²+1=17
Solve x²+1=-3 (impossible)
Get {4,-4,ϕ}
Not checking impossibility
😢 Uh oh! Incorrect answer, Try again
Check if equations possible
Quadratic equation
Solve x²+1=17
Solve x²+1=-3 (impossible)
Get {4,-4,ϕ}
Not checking impossibility
Let g∘f(x)=sinx and f∘g(x)=(sinx)². Then
🥳 Wohoo! Correct answer
(g∘f)(x)=g(f(x))
Use composition definition
Verify both compositions
Check f and g satisfy both
Wrong composition order
😢 Uh oh! Incorrect answer, Try again
Function composition rules
(g∘f)(x)=g(f(x))
Use composition definition
Verify both compositions
Check f and g satisfy both
Wrong composition order
Let A={2,3,4,5,...,16,17,18}. Let R be relation on set A of ordered pairs defined by (a,b)R(c,d) if and only if ad=bc. Then number of ordered pairs in equivalence class of (8,2) is
🥳 Wohoo! Correct answer
ad=bc means a/b=c/d
Identify ratio 8:2=4:1
Find all equivalent ratios
Count pairs: (3,2),(6,4),(9,6),(12,8),(18,12),(15,10)
Missing some equivalent pairs
😢 Uh oh! Incorrect answer, Try again
Look for equal ratios
ad=bc means a/b=c/d
Identify ratio 8:2=4:1
Find all equivalent ratios
Count pairs: (3,2),(6,4),(9,6),(12,8),(18,12),(15,10)
Missing some equivalent pairs
For LPP maximize z=x+4y subject to the constraints x+2y≤2, x+2y≥8, x,y≥0
🥳 Wohoo! Correct answer
LP constraints
Graph constraints
Check feasible region
Find no intersection of feasible regions
Constraint plotting errors
😢 Uh oh! Incorrect answer, Try again
Feasibility check
LP constraints
Graph constraints
Check feasible region
Find no intersection of feasible regions
Constraint plotting errors
The value of lim[θ→0] (1-cos4θ)/(1-cos6θ) is
🥳 Wohoo! Correct answer
limₓ→₀ sinx/x = 1
Apply L'Hôpital's rule as it's 0/0 form
Use cos2A = 1-2sin²A to rewrite
Simplify to get 4/9
Not identifying indeterminate form
😢 Uh oh! Incorrect answer, Try again
Convert to sin² form
limₓ→₀ sinx/x = 1
Apply L'Hôpital's rule as it's 0/0 form
Use cos2A = 1-2sin²A to rewrite
Simplify to get 4/9
Not identifying indeterminate form
The contrapositive of "If x is a prime number, then x is odd" is
🥳 Wohoo! Correct answer
Contrapositive: q→p
Convert to p→q form
Contrapositive is q→p
Therefore: If x is not odd, then x is not prime
Confusing converse with contrapositive
😢 Uh oh! Incorrect answer, Try again
Use logical equivalence
Contrapositive: q→p
Convert to p→q form
Contrapositive is q→p
Therefore: If x is not odd, then x is not prime
Confusing converse with contrapositive
The range of function f(x) = √(9-x²) is
🥳 Wohoo! Correct answer
Range of √(a²-x²) is [0,a]
Domain: x² ≤ 9
Find y = √(9-x²) limits
y ∈ [0,3]
Not considering endpoints
😢 Uh oh! Incorrect answer, Try again
Draw graph for visualization
Range of √(a²-x²) is [0,a]
Domain: x² ≤ 9
Find y = √(9-x²) limits
y ∈ [0,3]
Not considering endpoints
Let f:R→R be defined by f(x)=x⁴, then
🥳 Wohoo! Correct answer
For onto: range=codomain
Check if f(x₁)=f(x₂)⟹x₁=x₂
Check if range=codomain
Function fails both tests
Not checking both conditions
😢 Uh oh! Incorrect answer, Try again
Draw graph
For onto: range=codomain
Check if f(x₁)=f(x₂)⟹x₁=x₂
Check if range=codomain
Function fails both tests
Not checking both conditions
If f(x)=8x³, g(x)=x^(1/3), then fog(x) is
🥳 Wohoo! Correct answer
fog(x) = f(g(x))
Find g(x) first
Apply f to g(x)
Simplify to get 8x
Wrong order of composition
😢 Uh oh! Incorrect answer, Try again
Understand composition
fog(x) = f(g(x))
Find g(x) first
Apply f to g(x)
Simplify to get 8x
Wrong order of composition
Function f(x)=x²+2x-5 is strictly increasing in interval
🥳 Wohoo! Correct answer
Increasing when f'(x)>0
Find f'(x)=2x+2
Solve f'(x)>0
Get x>-1, so interval is (-1,∞)
Wrong interval notation
😢 Uh oh! Incorrect answer, Try again
Check derivative sign
Increasing when f'(x)>0
Find f'(x)=2x+2
Solve f'(x)>0
Get x>-1, so interval is (-1,∞)
Wrong interval notation
If LPP admits optimal solution at two consecutive vertices, then
🥳 Wohoo! Correct answer
Linear objective function
Optimal value is constant along line
All points on line give same value
Therefore all points are optimal
Not understanding optimality
😢 Uh oh! Incorrect answer, Try again
Understand convexity
Linear objective function
Optimal value is constant along line
All points on line give same value
Therefore all points are optimal
Not understanding optimality
Let f: R -> R be defined by f(x) =2x+6 which is a bijective mapping then f⁻¹(x) is given by
🥳 Wohoo! Correct answer
y = f⁻¹(x) where f(f⁻¹(x)) = x
Identify that for inverse function: y = 2x + 6
Replace y with f⁻¹(x) and solve for x: x = 2y + 6
Solve for y: y = (x-6)/2 = x/2 - 3
Forgetting to swap x and y, Error in algebraic manipulation
😢 Uh oh! Incorrect answer, Try again
To find inverse, swap x and y then solve for y
y = f⁻¹(x) where f(f⁻¹(x)) = x
Identify that for inverse function: y = 2x + 6
Replace y with f⁻¹(x) and solve for x: x = 2y + 6
Solve for y: y = (x-6)/2 = x/2 - 3
Forgetting to swap x and y, Error in algebraic manipulation
Let f(x) be the determinant of the 3×3 matrix:|cos x x 1 ||2sin x x 2x||sin x x x |Find the limit of f(x)/x² as x approaches 0
🥳 Wohoo! Correct answer
L'Hospital for 0/0 form
Find determinant
Apply L'Hospital's Rule twice
Get limit = 0
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Expand determinant first
L'Hospital for 0/0 form
Find determinant
Apply L'Hospital's Rule twice
Get limit = 0
Wrong differentiation
The function f(x) = [x] where [x] the greatest integer function, is continuous at
🥳 Wohoo! Correct answer
[x] = greatest integer ≤ x
Evaluate left hand limit at 1.5
Evaluate right hand limit at 1.5
Check if function value equals both limits
Not understanding GIF discontinuities
😢 Uh oh! Incorrect answer, Try again
Check continuity at non-integer points
[x] = greatest integer ≤ x
Evaluate left hand limit at 1.5
Evaluate right hand limit at 1.5
Check if function value equals both limits
Not understanding GIF discontinuities
Which one of the following observations is correct for the features of logarithm function to and base b>1?
🥳 Wohoo! Correct answer
logₐ1=0 for all bases
Check domain (x>0)
Check range (all reals)
Verify logₐ1=0
Wrong domain/range
😢 Uh oh! Incorrect answer, Try again
Think about properties
logₐ1=0 for all bases
Check domain (x>0)
Check range (all reals)
Verify logₐ1=0
Wrong domain/range
lim(x→0) (xe^x - sinx)/x is equal to
🥳 Wohoo! Correct answer
L'Hospital's rule for 0/0
Apply L'Hospital's rule as 0/0 form
Differentiate numerator and denominator
Get limit as 0
Not recognizing indeterminate form
😢 Uh oh! Incorrect answer, Try again
Identify indeterminate form
L'Hospital's rule for 0/0
Apply L'Hospital's rule as 0/0 form
Differentiate numerator and denominator
Get limit as 0
Not recognizing indeterminate form
If y=2x^3x, then dy/dx at x=1 is
🥳 Wohoo! Correct answer
d/dx(a^x) = a^x ln(a)
Take ln of both sides: ln y = ln(2x^3x)
Differentiate using ln rule: 1/y * dy/dx = 3ln(2) + 3
Substitute x=1 to get dy/dx = 6
Not using log differentiation
😢 Uh oh! Incorrect answer, Try again
Use logarithmic differentiation
d/dx(a^x) = a^x ln(a)
Take ln of both sides: ln y = ln(2x^3x)
Differentiate using ln rule: 1/y * dy/dx = 3ln(2) + 3
Substitute x=1 to get dy/dx = 6
Not using log differentiation
If f(x)=xe^x(1-x) then f'(x) is
🥳 Wohoo! Correct answer
f''(x) determines increasing/decreasing
Find f'(x)=e^x(1-x²)
Find f''(x)=e^x(x-2x)
Determine where f''(x)>0
Wrong interval analysis
😢 Uh oh! Incorrect answer, Try again
Check second derivative
f''(x) determines increasing/decreasing
Find f'(x)=e^x(1-x²)
Find f''(x)=e^x(x-2x)
Determine where f''(x)>0
Wrong interval analysis
The contra positive of the converse of the statement "If x is a prime number then x is odd" is
🥳 Wohoo! Correct answer
Contrapositive: ~q → ~p
Write converse: If x is odd then x is prime
Take contrapositive: If x is not prime then x is not odd
Verify answer D
Not understanding logical equivalence
😢 Uh oh! Incorrect answer, Try again
Understand logical implication rules
Contrapositive: ~q → ~p
Write converse: If x is odd then x is prime
Take contrapositive: If x is not prime then x is not odd
Verify answer D
Not understanding logical equivalence
f(x) = 1/2 - tan(πx/2), -1 < x < 1 and g(x) = √(3 + 4x - 4x²), find domain of (f+x)
🥳 Wohoo! Correct answer
• Domain of √x: x ≥ 0 • Domain intersection rules
For g(x): Under square root ≥ 0: 3 + 4x - 4x² ≥ 0, 4x² - 4x - 3 ≤ 0, (2x - 1)(2x - 3) ≤ 0
Solving quadratic: x ∈ [1/2, 3/2]
Domain intersection: [-1/2, 1)
Forgetting to intersect domains
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions from both functions
• Domain of √x: x ≥ 0 • Domain intersection rules
For g(x): Under square root ≥ 0: 3 + 4x - 4x² ≥ 0, 4x² - 4x - 3 ≤ 0, (2x - 1)(2x - 3) ≤ 0
Solving quadratic: x ∈ [1/2, 3/2]
Domain intersection: [-1/2, 1)
Forgetting to intersect domains
If operation ⊕ defined by a⊕b = a² + b² for all real numbers, find (2⊕3)⊕4
🥳 Wohoo! Correct answer
• a⊕b = a² + b² • Associative property
Calculate 2⊕3 = 2² + 3² = 13
Find (13)⊕4 = 13² + 4²
Simplify: 169 + 16 = 185
Not following order of operations
😢 Uh oh! Incorrect answer, Try again
Follow operation definition step by step
• a⊕b = a² + b² • Associative property
Calculate 2⊕3 = 2² + 3² = 13
Find (13)⊕4 = 13² + 4²
Simplify: 169 + 16 = 185
Not following order of operations
Slope of Normal to curve y = x² - 1/x² at (-1,0)
🥳 Wohoo! Correct answer
• Normal slope = -1/(dy/dx) • Product rule
Find dy/dx = 2x + 2/x³
At x = -1, dy/dx = -4
Normal slope = -1/(dy/dx) = 1/4
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Normal is perpendicular to tangent
• Normal slope = -1/(dy/dx) • Product rule
Find dy/dx = 2x + 2/x³
At x = -1, dy/dx = -4
Normal slope = -1/(dy/dx) = 1/4
Wrong differentiation
If f:R→R defined by f(x)=x/(x+2), find f(f(2))
🥳 Wohoo! Correct answer
• Function composition • Fraction arithmetic
Calculate f(2) = 2/4 = 1/2
Find f(1/2) = (1/2)/(5/2) = 1/5
Simplify to 10/29
Wrong composition order
😢 Uh oh! Incorrect answer, Try again
Apply function composition
• Function composition • Fraction arithmetic
Calculate f(2) = 2/4 = 1/2
Find f(1/2) = (1/2)/(5/2) = 1/5
Simplify to 10/29
Wrong composition order
limx→0(1-cosx)/x² equals
🥳 Wohoo! Correct answer
• L'Hospital's rule • Derivatives of trig functions
Apply L'Hospital's rule twice
First derivative: sinx/2x
Second derivative gives 1/2
Wrong application of L'Hospital
😢 Uh oh! Incorrect answer, Try again
Use L'Hospital for 0/0 form
• L'Hospital's rule • Derivatives of trig functions
Apply L'Hospital's rule twice
First derivative: sinx/2x
Second derivative gives 1/2
Wrong application of L'Hospital
If f(x) = 2x², find (f(3.8)-f(4)) / (3.8-4)
🥳 Wohoo! Correct answer
• Difference quotient • f'(x) = limh→0[f(x+h)-f(x)]/h
Use difference quotient
Substitute values
Simplify to 15.6
Wrong arithmetic
😢 Uh oh! Incorrect answer, Try again
Consider derivative definition
• Difference quotient • f'(x) = limh→0[f(x+h)-f(x)]/h
Use difference quotient
Substitute values
Simplify to 15.6
Wrong arithmetic
Remainder when 1!+2!+3!+...+11! divided by 12
🥳 Wohoo! Correct answer
• Modular arithmetic • Factorial divisibility
Group terms by divisibility
Find remainder of each
Sum remainders mod 12
Wrong modular arithmetic
😢 Uh oh! Incorrect answer, Try again
Consider factorial properties
• Modular arithmetic • Factorial divisibility
Group terms by divisibility
Find remainder of each
Sum remainders mod 12
Wrong modular arithmetic
Function f(x)=[x] where [x] is greatest integer function is continuous at
🥳 Wohoo! Correct answer
• Left/right limit equality • Greatest integer function
Check left and right limits
Find function values
Verify continuity at 1.5
Wrong limit calculation
😢 Uh oh! Incorrect answer, Try again
Consider floor function properties
• Left/right limit equality • Greatest integer function
Check left and right limits
Find function values
Verify continuity at 1.5
Wrong limit calculation
y = log((1-x²)/(1+x²)), then dy/dx equals
🥳 Wohoo! Correct answer
• Log derivative rules • Chain rule
Use logarithm properties
Differentiate using chain rule
Simplify to -4x/(1-x⁴)
Wrong chain rule
😢 Uh oh! Incorrect answer, Try again
Consider log derivative
• Log derivative rules • Chain rule
Use logarithm properties
Differentiate using chain rule
Simplify to -4x/(1-x⁴)
Wrong chain rule
Let f: R→R defined by f(x) = 1/x, then f is
🥳 Wohoo! Correct answer
• Function properties • Domain/Range
Check domain includes 0
Verify function properties
Conclude not defined
Wrong function analysis
😢 Uh oh! Incorrect answer, Try again
Consider function domain
• Function properties • Domain/Range
Check domain includes 0
Verify function properties
Conclude not defined
Wrong function analysis
Let the function satisfy f(x+y)=f(x)f(y) for all x,y∈R, where f(0)=0. If f(5)=3 and f'(0)=2, then f'(5) is
🥳 Wohoo! Correct answer
f(x+y)=f(x)f(y) is key functional equation
Use functional equation to get derivative
f'(x)=f'(0)f(x)
Substitute values to get f'(5)=6
Not recognizing functional equation pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern in functional equation
f(x+y)=f(x)f(y) is key functional equation
Use functional equation to get derivative
f'(x)=f'(0)f(x)
Substitute values to get f'(5)=6
Not recognizing functional equation pattern
If the function is f(x) = 1/(x+2), then the point of discontinuity of the composite function y = f(f(x)) is
🥳 Wohoo! Correct answer
Composite functions
Find f(f(x)) = 1/(1/(x+2) + 2)
Simplify to (x+2)/(x+5)
Point of discontinuity when x+5 = 0, x = -5/2
Wrong composition order
😢 Uh oh! Incorrect answer, Try again
Look for denominator = 0
Composite functions
Find f(f(x)) = 1/(1/(x+2) + 2)
Simplify to (x+2)/(x+5)
Point of discontinuity when x+5 = 0, x = -5/2
Wrong composition order
The function f(x) = cotx is discontinuous on every point of the set
🥳 Wohoo! Correct answer
Domain of trigonometric functions
Analyze where cotx is undefined
Identify points where denominator = 0
Conclude at x = nπ, n∈Z
Missing periodicity
😢 Uh oh! Incorrect answer, Try again
Check domain of cotangent
Domain of trigonometric functions
Analyze where cotx is undefined
Identify points where denominator = 0
Conclude at x = nπ, n∈Z
Missing periodicity
If f(x) and g(x) are two functions with g(x) = x - 1/x and fog(x) = x³ - 1/x³ then f(x) =
🥳 Wohoo! Correct answer
Function composition
Use definition of composite function
Find f(g(x)) = x³ - 1/x³
Deduce f(x) = 2x³ - 3
Wrong composition order
😢 Uh oh! Incorrect answer, Try again
Compare composite function results
Function composition
Use definition of composite function
Find f(g(x)) = x³ - 1/x³
Deduce f(x) = 2x³ - 3
Wrong composition order
lim[n→∞]Σ(n/(n²+r²)) from r=1 to n is
🥳 Wohoo! Correct answer
lim Σ = ∫ for Riemann sums
Convert to Riemann sum
Recognize as integral form
Evaluate definite integral to get tan⁻¹2
Not recognizing Riemann sum
😢 Uh oh! Incorrect answer, Try again
Look for standard limit pattern
lim Σ = ∫ for Riemann sums
Convert to Riemann sum
Recognize as integral form
Evaluate definite integral to get tan⁻¹2
Not recognizing Riemann sum
Let A={x,y,z,u} and B={a,b}. A function f:A→B is selected randomly. The probability that the function is an onto function is
🥳 Wohoo! Correct answer
Probability = favorable/total
Total functions = 2⁴
Count onto functions = 2⁴-2
Probability = 14/16 = 7/8
Missing cases in counting
😢 Uh oh! Incorrect answer, Try again
Consider range coverage
Probability = favorable/total
Total functions = 2⁴
Count onto functions = 2⁴-2
Probability = 14/16 = 7/8
Missing cases in counting
If f(x)=ax+b, where a and b are integers, f(-1)=-5 and f(3)=3 then a and b are respectively
🥳 Wohoo! Correct answer
Linear function formula
Use f(-1)=-5: -a+b=-5
Use f(3)=3: 3a+b=3
Solve simultaneously to get a=2,b=-3
Wrong equation formation
😢 Uh oh! Incorrect answer, Try again
Form system of linear equations
Linear function formula
Use f(-1)=-5: -a+b=-5
Use f(3)=3: 3a+b=3
Solve simultaneously to get a=2,b=-3
Wrong equation formation
If limₓ→₀[(sin(2+x)-sin(2-x))/x] = AcosB, then the values of A and B respectively are
🥳 Wohoo! Correct answer
lim(sinx/x)=1 as x→0
Write sin(2+x)-sin(2-x) = 2cos2·sinx
Apply limit x→0: sinx/x = 1
Therefore A=2, B=2
Wrong trigonometric formula
😢 Uh oh! Incorrect answer, Try again
Use trigonometric difference formula
lim(sinx/x)=1 as x→0
Write sin(2+x)-sin(2-x) = 2cos2·sinx
Apply limit x→0: sinx/x = 1
Therefore A=2, B=2
Wrong trigonometric formula
Let f:R→R be defined by f(x)=3x²-5 and g:R→R by g(x)=x/(x²+1) then gof is
🥳 Wohoo! Correct answer
Composite function formula
Find f(x)=3x²-5
Substitute in g(x): g(f(x))
Simplify to get final form
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Use function composition rules
Composite function formula
Find f(x)=3x²-5
Substitute in g(x): g(f(x))
Simplify to get final form
Wrong substitution
Let the relation R be defined in N by aRb if 3a+2b=27 then R is
🥳 Wohoo! Correct answer
Relation definition
List all pairs (a,b) satisfying 3a+2b=27
Check if b is natural number
Get R={(1,12),(3,9),(5,6),(7,3)}
Not checking natural numbers
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions
Relation definition
List all pairs (a,b) satisfying 3a+2b=27
Check if b is natural number
Get R={(1,12),(3,9),(5,6),(7,3)}
Not checking natural numbers
Let f(x)=sin2x+cos2x and f²(x)=x-1, then g(f(x)) is invertible in the domain
🥳 Wohoo! Correct answer
Invertible function conditions
Find g(f(x))=sin4x
Check when function is one-to-one
Domain is [-π/8,π/8]
Wrong domain calculation
😢 Uh oh! Incorrect answer, Try again
Consider function composition
Invertible function conditions
Find g(f(x))=sin4x
Check when function is one-to-one
Domain is [-π/8,π/8]
Wrong domain calculation
The contrapositive of the statement "If two lines do not intersect in the same plane then they are parallel" is
🥳 Wohoo! Correct answer
Contrapositive: q→p ≡ p→q
Write original statement: p→q
Form contrapositive: q→p
Get "not parallel → intersect"
Wrong negation
😢 Uh oh! Incorrect answer, Try again
Use logical equivalence rules
Contrapositive: q→p ≡ p→q
Write original statement: p→q
Form contrapositive: q→p
Get "not parallel → intersect"
Wrong negation
f:R→R and g:[0,∞)→R are defined by f(x)=x² and g(x)=√x. Which one of the following is not true?
🥳 Wohoo! Correct answer
Function composition domain
Check each composition
Note g(x) only defined for x≥0
fog(-4) undefined
Missing domain check
😢 Uh oh! Incorrect answer, Try again
Consider domain restrictions
Function composition domain
Check each composition
Note g(x) only defined for x≥0
fog(-4) undefined
Missing domain check
If f(x) = {x² - 1, when 0 < x < 2 and it is 2x + 3, when 2 ≤ x < 3}, find quadratic equation with roots lim(x→2⁻) f(x) and lim(x→2⁺) f(x)
🥳 Wohoo! Correct answer
(x - α)(x - β) = x² - (α+β)x + αβ where α,β are roots
Calculate left-hand limit at x=2: lim(x→2⁻) (x² - 1) = 3
Calculate right-hand limit at x=2⁺: lim(x→2⁺) (2x + 3) = 7
Form quadratic equation with roots 3 and 7
Not checking both sides of limit
😢 Uh oh! Incorrect answer, Try again
Check both pieces of function at x=2
(x - α)(x - β) = x² - (α+β)x + αβ where α,β are roots
Calculate left-hand limit at x=2: lim(x→2⁻) (x² - 1) = 3
Calculate right-hand limit at x=2⁺: lim(x→2⁺) (2x + 3) = 7
Form quadratic equation with roots 3 and 7
Not checking both sides of limit
Let relation R be defined in N by aRb, if 3a+2b=27 then R is
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Relation as set of ordered pairs
Express b in terms of a: b = (27-3a)/2
Find valid ordered pairs with natural numbers
List all pairs satisfying condition
Not checking for natural numbers
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Check both components are natural numbers
Relation as set of ordered pairs
Express b in terms of a: b = (27-3a)/2
Find valid ordered pairs with natural numbers
List all pairs satisfying condition
Not checking for natural numbers