Family of curves whose x,y intercepts of tangent at any point are double x,y coordinates of that point is
Point-slope form of tangent
Use point-slope form
Apply intercept condition
Get xy=C family
Wrong differential equation
Think about ratio property
Point-slope form of tangent
Use point-slope form
Apply intercept condition
Get xy=C family
Wrong differential equation
Solution of dy/dx=e^x+1,y(0)=3 is
Integration of e^x
Separate variables
Integrate both sides
Use initial condition
Wrong integration
Use separation of variables
Integration of e^x
Separate variables
Integrate both sides
Use initial condition
Wrong integration
Volume of parallelopiped whose coterminous edges are (ĵ + k̂), (î + k̂) and (î + ĵ) is
Volume= a·(b×c)
Set up determinant
Calculate 3×3 determinant
Get 2 cubic units
Wrong determinant calculation
Use scalar triple product and convert to determinant
Volume= a·(b×c)
Set up determinant
Calculate 3×3 determinant
Get 2 cubic units
Wrong determinant calculation
Let â and b̂ be two unit vectors and θ is the angle between them. Then â + b̂ is a unit vector if
a+b²=2(1+cosθ)
For â + b̂ to be a unit vector, its magnitude must be 1:
Use a+b²=2(1+cosθ)
Set equal to 1. Solve for θ=2π/3
Wrong vector addition
Unit vector has magnitude 1
a+b²=2(1+cosθ)
For â + b̂ to be a unit vector, its magnitude must be 1:
Use a+b²=2(1+cosθ)
Set equal to 1. Solve for θ=2π/3
Wrong vector addition
Maximum volume of right circular cone with slant height 6 units is
V=πr²h/3, l²=r²+h²
Express volume in terms of radius
Use constraint l²=r²+h²
Maximize V=πr²h/3
Not using constraints correctly
Use calculus to maximize
V=πr²h/3, l²=r²+h²
Express volume in terms of radius
Use constraint l²=r²+h²
Maximize V=πr²h/3
Not using constraints correctly
Vectors AB=3i+4k and AC=5i-2j+4k are sides of ΔABC. Length of median through A is
Median length formula
Find median vector=(AB+AC)/2
Calculate magnitude
Get √33
Wrong median formula
Use median formula
Median length formula
Find median vector=(AB+AC)/2
Calculate magnitude
Get √33
Wrong median formula
Function x^x;x>0 is strictly increasing at
d/dx(x^x)=x^x(1+lnx)
Take ln, differentiate
Find where derivative positive
Get x>1/e condition
Wrong differentiation
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
∫(sinx/(3+4cos²x))dx =
Integration of rational trig functions
Use substitution u=cosx
Rearrange to standard form
Integrate to get arctan form
Wrong substitution
Look for rational function in cosx
Integration of rational trig functions
Use substitution u=cosx
Rearrange to standard form
Integrate to get arctan form
Wrong substitution
∫(-π to π)(1-x²)sinx·cos²xdx =
Odd function over symmetric interval
Use odd/even function properties
Observe integrand is odd
Integral over [-π,π] is 0
Not recognizing odd function
Check symmetry of function
Odd function over symmetric interval
Use odd/even function properties
Observe integrand is odd
Integral over [-π,π] is 0
Not recognizing odd function
If f(x)=xe^x(1-x) then f'(x) is
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
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
If x-1 1 1;x-1 dB/dA and B=1 x 1;1 1 x, then is
d(XAX)/dA formula
Calculate dB/dA using derivative
Apply chain rule
Get 3A
Wrong differentiation rules
Use matrix differentiation
d(XAX)/dA formula
Calculate dB/dA using derivative
Apply chain rule
Get 3A
Wrong differentiation rules
If P=[1 α 3; 1 3 3; 2 4 4] is adjoint of 3×3 matrix A and |A| = 4, then α is equal to
Calculate determinant of P
|P| = |A - A| = |A|² =16
2 α - 6 = 16, α = 11
• Wrong adjoint determinant property • Errors in determinant calculation • Sign errors in solving equation
Use property of adjoint matrix determinant and equate
Calculate determinant of P
|P| = |A - A| = |A|² =16
2 α - 6 = 16, α = 11
• Wrong adjoint determinant property • Errors in determinant calculation • Sign errors in solving equation
Which observation is correct for logarithm function to base b>1?
logₐ1=0 for all bases
Check domain (x>0)
Check range (all reals)
Verify logₐ1=0
Wrong domain/range
Think about properties
logₐ1=0 for all bases
Check domain (x>0)
Check range (all reals)
Verify logₐ1=0
Wrong domain/range
Let f(x)=[cosx x 1;2sinx x 2x;sinx x x]. Then lim[x→0]f(x)/x² equals
L'Hospital for 0/0 form
Find determinant
Apply L'Hospital's Rule twice
Get limit = 0
Wrong differentiation
Expand determinant first
L'Hospital for 0/0 form
Find determinant
Apply L'Hospital's Rule twice
Get limit = 0
Wrong differentiation
If f(x)=determinant[(x-3) (2x²-18) (2x³-81); (x-5) (2x²-50) (4x³-500); (1) (2) (3)], then f(1)·f(3)+f(3)·f(5)+f(5)·f(1) is
3×3 determinant formula
Calculate determinant at x=1,3,5
Multiply pairs of values
Sum products
Calculation errors
Use determinant properties
3×3 determinant formula
Calculate determinant at x=1,3,5
Multiply pairs of values
Sum products
Calculation errors
If A=[1 1;1 1], then A¹⁰ equals
A^n = k^(n-1)A pattern
Find A² = 2A
Use pattern A^n = 2^(n-1)A
Get A¹⁰ = 2⁹A
Not recognizing pattern
Look for pattern in powers
A^n = k^(n-1)A pattern
Find A² = 2A
Use pattern A^n = 2^(n-1)A
Get A¹⁰ = 2⁹A
Not recognizing pattern
If A is square matrix such that A²=A, then (I+A)³ equals
(I+A)³ = I + 3A + 3A² + A³
Expand (I+A)³
Use A²=A
Simplify to get 7A+I
Not using A²=A properly
Use binomial expansion
(I+A)³ = I + 3A + 3A² + A³
Expand (I+A)³
Use A²=A
Simplify to get 7A+I
Not using A²=A properly
If 2sin⁻¹x-3cos⁻¹x=4,x∈[-1,1] then 2sin⁻¹x+3cos⁻¹x equals
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
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 cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π, then x(y+z) + y(z+x) + z(x+y) equals
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
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
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
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
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
Let g∘f(x)=sinx and f∘g(x)=(sinx)². Then
(g∘f)(x)=g(f(x))
Use composition definition
Verify both compositions
Check f and g satisfy both
Wrong composition order
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 f:R→R be given by f(x)=tanx. Then f⁻¹(1) is
tan(π/4)=1
Find x where tanx=1
Solve to get x=π/4
Check uniqueness
Not considering periodicity
Inverse function basics
tan(π/4)=1
Find x where tanx=1
Solve to get x=π/4
Check uniqueness
Not considering periodicity
Let f:R→R be defined by f(x)=x²+1. Then pre-images of 17 and -3 are
Quadratic equation
Solve x²+1=17
Solve x²+1=-3 (impossible)
Get {4,-4,ϕ}
Not checking impossibility
Check if equations possible
Quadratic equation
Solve x²+1=17
Solve x²+1=-3 (impossible)
Get {4,-4,ϕ}
Not checking impossibility
If a,b,c,d,e are observations with mean m and SD S, then SD of a+k,b+k,c+k,d+k,e+k is
SD(X+k) = SD(X)
Understand effect of adding constant
Adding k doesn't change spread
SD remains S
Think k affects SD
Constant addition property
SD(X+k) = SD(X)
Understand effect of adding constant
Adding k doesn't change spread
SD remains S
Think k affects SD
Negation of "For every real number x; x²+5 is positive" is
Negation of ∀ is ∃
Understand logical negation
Negate universal quantifier
Get existential statement
Wrong quantifier negation
Logical negation rules
Negation of ∀ is ∃
Understand logical negation
Negate universal quantifier
Get existential statement
Wrong quantifier negation
limx→π/4/(cotx-1) equals
L'Hospital's Rule for 0/0
Apply L'Hospital's Rule
Differentiate num & denom
Get limit = 1/2
Not recognizing 0/0
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
The distance between two planes 2x+3y+4z=4 and 4x+6y+8z=12 is
Distance = c₁-c₂ /√(a²+b²+c²)
Identify parallel planes- The planes are parallel as coefficients are proportional (4x + 6y + 8z = 12 is 2 times of 2x + 3y + 4z = 4)
Use distance formula, For first plane: a = 2, b = 3, c = 4, N₁ = 4. For second plane: a = 4, b = 6, c = 8, N₂ = 12
Substituting in formula: Simplify to get 2/√29
• Not checking for parallel planes first • Using wrong coefficients in formula • Forgetting to normalize direction cosines
For parallel planes, use perpendicular distance formula and identify coefficients
Distance = c₁-c₂ /√(a²+b²+c²)
Identify parallel planes- The planes are parallel as coefficients are proportional (4x + 6y + 8z = 12 is 2 times of 2x + 3y + 4z = 4)
Use distance formula, For first plane: a = 2, b = 3, c = 4, N₁ = 4. For second plane: a = 4, b = 6, c = 8, N₂ = 12
Substituting in formula: Simplify to get 2/√29
• Not checking for parallel planes first • Using wrong coefficients in formula • Forgetting to normalize direction cosines
If random variable X follows binomial distribution with n=5, p and P(X=2)=9P(X=3), then p equals
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
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
Equation of parabola whose focus is (6,0) and directrix is x=-6 is
y²=4ax for vertical axis
Use standard form (y²=4ax)
Identify a=6 from focus
Write equation y²=24x
Wrong standard form
Focus-directrix definition
y²=4ax for vertical axis
Use standard form (y²=4ax)
Identify a=6 from focus
Write equation y²=24x
Wrong standard form
Angle between line x+y=3 and line joining points (1,1) and (-3,4) is
tan θ = (m₁-m₂)/(1+m₁m₂)
Find slope of x+y=3 (m₁=-1)
Find slope of second line m₂=3/4
Use tan θ = (m₁-m₂)/(1+m₁m₂)
• Wrong slope calculation • Missing absolute value signs • Wrong angle formula
Use slope formula and angle between lines formula
tan θ = (m₁-m₂)/(1+m₁m₂)
Find slope of x+y=3 (m₁=-1)
Find slope of second line m₂=3/4
Use tan θ = (m₁-m₂)/(1+m₁m₂)
• Wrong slope calculation • Missing absolute value signs • Wrong angle formula
If AM and GM of roots of quadratic equation are 5 and 4 respectively, then quadratic equation is
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
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
If Sₙ stands for sum to n-terms of GP with 'a' as first term and 'r' as common ratio then Sₙ:S₂ₙ is
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
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
In expansion of (1+x)ⁿ, C₁/C₀+2C₂/C₁+3C₃/C₂+...+nCₙ/Cₙ₋₁ equals
nCr=n!/r!(n-r)!
Write each term using nCr formula
Simplify ratio pattern
Sum to get n(n+1)/2
Not simplifying properly
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
Value of ⁴⁹C₃+⁴⁸C₃+⁴⁷C₃+⁴⁶C₃+⁴⁵C₃+⁴⁵C₄ is
ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁
Use combination formula
Apply Pascal's identity
Get ⁵⁰C₄
Not seeing pattern
Look for pattern
ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁
Use combination formula
Apply Pascal's identity
Get ⁵⁰C₄
Not seeing pattern
Length of rectangle is five times breadth. If minimum perimeter is 180cm, then
P=2(l+b)
Let l=5b
Use P=2(l+b)=180
Solve for b≥15
Not using inequality
Use perimeter formula
P=2(l+b)
Let l=5b
Use P=2(l+b)=180
Solve for b≥15
Not using inequality
Real value of α for which (1-isinα)/(1+2isinα) is purely real is
z is real if Im(z)=0
Rationalize denominator
Set imaginary part to zero
Solve for α=nπ
Not rationalizing properly
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
If ABC is right angled at C, then tanA·tanB is
tanθ = opposite/adjacent
Use right triangle ratios
Express tanA and tanB
Multiply and simplify
Not using right angle property
Use Pythagorean theorem
tanθ = opposite/adjacent
Use right triangle ratios
Express tanA and tanB
Multiply and simplify
Not using right angle property
If in two circles, arcs of same length subtend angles 30° and 78° at centre, ratio of radii is
l=rθ where θ in radians
Use arc length formula l=rθ
Set up ratio equation
Solve for r₁/r₂ = 13/5
Not converting to radians
Arc lengths are equal
l=rθ where θ in radians
Use arc length formula l=rθ
Set up ratio equation
Solve for r₁/r₂ = 13/5
Not converting to radians
If [x]²-5[x]+6=0, where [x] is greatest integer function, then
[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
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
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
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
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
Two finite sets have m,n elements. Total subsets of first set is 56 more than second set. Values of m,n are
Number of subsets = 2ⁿ
Use 2ᵐ-2ⁿ=56
Try values of m,n
Verify m=6,n=3 satisfies
Not considering all possibilities
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
A die is thrown 10 times. Probability that odd number will come up at least once is
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
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
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
Linear programming principles
Plot points
Test objective function values
Compare values along line segment
Not checking all points
Check all corner points
Linear programming principles
Plot points
Test objective function values
Compare values along line segment
Not checking all points
The sine of angle between line (x-2) / 3= (y-3) / 4= (4-z) /-5 and plane 2x-2y+z=5 is
sin θ = ax₁+by₁+cz₁ / √[(a²+b²+c²)(x₁²+y₁²+z₁²)]
Find direction cosines of line. Direction ratios of line are a = 3, b = 4, c = -5. Normal to plane is (2,-2,1).
Use angle formula with plane
Compute sine, Therefore, sin θ = 7/√450 = 7/(15√2) = 1/ (5√2)
• Wrong direction ratios • Wrong normal vector • Error in absolute value
Use direction ratios
sin θ = ax₁+by₁+cz₁ / √[(a²+b²+c²)(x₁²+y₁²+z₁²)]
Find direction cosines of line. Direction ratios of line are a = 3, b = 4, c = -5. Normal to plane is (2,-2,1).
Use angle formula with plane
Compute sine, Therefore, sin θ = 7/√450 = 7/(15√2) = 1/ (5√2)
• Wrong direction ratios • Wrong normal vector • Error in absolute value
The equation xy=0 in three-dimensional space represents
Equation of planes
Consider what xy=0 means
Factorize equation
Recognize as x=0 or y=0
Thinking in 2D only
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
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
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
Refer image
Vector triple product formula
Calculate each cross product
Find dot products
Sum to get 3
Not using vector identities correctly
Use vector triple product
Vector triple product formula
Calculate each cross product
Find dot products
Sum to get 3
Not using vector identities correctly
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
Perpendicular vectors dot product = 0
Use direction vectors
Apply perpendicularity condition
Solve for k
Wrong direction vectors
Direction vectors must be perpendicular
Perpendicular vectors dot product = 0
Use direction vectors
Apply perpendicularity condition
Solve for k
Wrong direction vectors
The area of the region bounded by the line y=x and the curve y=x³ is
Area between curves formula
Find intersection points: x=0,1
Area = ∫₀¹(x-x³)dx
Evaluate to get 0.5
Wrong integration limits
Consider region between curves
Area between curves formula
Find intersection points: x=0,1
Area = ∫₀¹(x-x³)dx
Evaluate to get 0.5
Wrong integration limits
lim(n→∞)[n/(n²+1²) + n/(n²+2²) + n/(n²+3²) + ....... + 1/5n] =
lim Σ = ∫ for Riemann sums
The given sum can be written as lim(n→∞)Σᵣ₌₁²ⁿ [n/(n²+r²)]
lim(n→∞)Σᵣ₌₁²ⁿ [1/(n/r²+1)] = ∫₀²[1/(1+x²)]dx
[tan⁻¹x]₀² = tan⁻¹2 - 0 = tan⁻¹2
Not recognizing Riemann sum
Look for standard limit pattern
lim Σ = ∫ for Riemann sums
The given sum can be written as lim(n→∞)Σᵣ₌₁²ⁿ [n/(n²+r²)]
lim(n→∞)Σᵣ₌₁²ⁿ [1/(n/r²+1)] = ∫₀²[1/(1+x²)]dx
[tan⁻¹x]₀² = tan⁻¹2 - 0 = tan⁻¹2
Not recognizing Riemann sum
The area of the region bounded by the line y=3x and the curve y=x² in sq. units is
Area = ∫(upper - lower)dx
Find intersection points by solving 3x=x²
Area = ∫(3x-x²)dx from 0 to 3
Evaluate to get 9/2
Not finding correct limits
Look for points where curves meet
Area = ∫(upper - lower)dx
Find intersection points by solving 3x=x²
Area = ∫(3x-x²)dx from 0 to 3
Evaluate to get 9/2
Not finding correct limits
∫₁⁵ (|x-3| + |1-x|) dx =
Split integral at x=3 and x=1
∫₁⁵(|x-3| + |x+1|)dx = ∫₁³(-(x-3))dx + ∫₃⁵(x-3)dx + ∫₋₁¹(-(x+1))dx + ∫₁⁵(x+1)dx
∫₁⁵2dx + ∫₃⁵(2x-4)dx = 2(5-1) + [x² - 4x]₃⁵
8 + [(25 - 20) - (9 - 12)] = 8 + 4 = 12
Mathematics
Split integral at x=3 and x=1
Split integral at x=3 and x=1
∫₁⁵(|x-3| + |x+1|)dx = ∫₁³(-(x-3))dx + ∫₃⁵(x-3)dx + ∫₋₁¹(-(x+1))dx + ∫₁⁵(x+1)dx
∫₁⁵2dx + ∫₃⁵(2x-4)dx = 2(5-1) + [x² - 4x]₃⁵
8 + [(25 - 20) - (9 - 12)] = 8 + 4 = 12
Mathematics
∫[( sin(5x/2))/( sin(x/2))]dx =
Integration of trigonometric functions
Simplify fraction first
Break into partial fractions
Integrate term by term
Not simplifying before integration
Look for cancellation in fraction
Integration of trigonometric functions
Simplify fraction first
Break into partial fractions
Integrate term by term
Not simplifying before integration
Let the function satisfy f(x+y)=f(x)f(y) for all x,y∈R, where f(0)not equal to 0. If f(5)=3 and f'(0)=2, then f'(5) is
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
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
∫ 1/ {x [6(log x)² + 7log x + 2] } dx =
• dx = xd(log x) • Partial fraction decomposition • Integration of rational functions
Let t = log x ⟹ dx = xdt. This transforms integral to ∫dt/[6t² + 7t + 2]
Using partial fractions on 1/[6t² + 7t + 2] = 1/[(2t + 1)(3t + 2)] = A/(2t + 1) + B/(3t + 2), where A and B are constants
After integration and substituting back t = log x: log
• Wrong substitution • Errors in partial fractions • Missing absolute value signs
Use substitution followed by partial fractions
• dx = xd(log x) • Partial fraction decomposition • Integration of rational functions
Let t = log x ⟹ dx = xdt. This transforms integral to ∫dt/[6t² + 7t + 2]
Using partial fractions on 1/[6t² + 7t + 2] = 1/[(2t + 1)(3t + 2)] = A/(2t + 1) + B/(3t + 2), where A and B are constants
After integration and substituting back t = log x: log
• Wrong substitution • Errors in partial fractions • Missing absolute value signs
If y=2x^3x, then dy/dx at x=1 is
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
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
For the function f(x)=x³-6x²+12x-3; x=2 is
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
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
The function f(x)= |Cos x| is
Check continuity
At x = π/2, 3π/2, etc. (odd multiples of π/2), derivative from left = -sin x and from right = sin x
Since left hand and right hand derivatives are not equal at these points, function is not differentiable at odd multiples of π/2, but remains continuous
• Confusing continuity with differentiability • Missing critical points
Examine behavior at points where cos x changes from positive to negative
Check continuity
At x = π/2, 3π/2, etc. (odd multiples of π/2), derivative from left = -sin x and from right = sin x
Since left hand and right hand derivatives are not equal at these points, function is not differentiable at odd multiples of π/2, but remains continuous
• Confusing continuity with differentiability • Missing critical points
d/dx[cos²{cot⁻¹(√(2+x)/(2-x))}] is
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
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
The value of C in (0, 2) satisfying the mean value theorem for the function f(x) = x(x-1)², x∈[0,2] is equal to
MVT: f'(c)=[f(b)-f(a)]/(b-a)
Calculate f(2)=2 and f(0)=0
Apply MVT: [f(2)-f(0)]/2 = f'(c)
Find f'(x)=2x(x-1)+(x-1)² and solve f'(c)=1
Forgetting to verify if c lies in (0,2)
Use MVT formula and verify c lies in interval
MVT: f'(c)=[f(b)-f(a)]/(b-a)
Calculate f(2)=2 and f(0)=0
Apply MVT: [f(2)-f(0)]/2 = f'(c)
Find f'(x)=2x(x-1)+(x-1)² and solve f'(c)=1
Forgetting to verify if c lies in (0,2)
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