🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
🥳 Wohoo! Correct answer
😢 Uh oh! Incorrect answer, Try again
If standard deviation of numbers -1,0,1,k is √5 where k>0, then k equals
🥳 Wohoo! Correct answer
σ² = Σ(x-x̄)²/n
Write variance formula with given numbers
Substitute σ = √5 and solve for k
Simplify to get k = 2√6
Not using correct variance formula
😢 Uh oh! Incorrect answer, Try again
Use variance formula σ² = Σ(x-x̄)²/n
σ² = Σ(x-x̄)²/n
Write variance formula with given numbers
Substitute σ = √5 and solve for k
Simplify to get k = 2√6
Not using correct variance formula
If A=[0 1; 0 0], then (aI+bA)ⁿ is (where I is identity matrix of order 2)
🥳 Wohoo! Correct answer
Matrix powers: (aI+bA)ⁿ
Write first few powers of matrix
Identify pattern in expansion
Use binomial expansion pattern
Not seeing binomial pattern
😢 Uh oh! Incorrect answer, Try again
Look for pattern in powers
Matrix powers: (aI+bA)ⁿ
Write first few powers of matrix
Identify pattern in expansion
Use binomial expansion pattern
Not seeing binomial pattern
If A is matrix of order 3×3, then (A²)⁻¹ equals
🥳 Wohoo! Correct answer
(Aⁿ)⁻¹=(A⁻¹)ⁿ
Use properties of inverse and powers
Apply (A²)⁻¹=(A⁻¹)²
Simplify
Not applying inverse rules correctly
😢 Uh oh! Incorrect answer, Try again
Remember inverse of power rule
(Aⁿ)⁻¹=(A⁻¹)ⁿ
Use properties of inverse and powers
Apply (A²)⁻¹=(A⁻¹)²
Simplify
Not applying inverse rules correctly
If 2x2 matrix A= [2 -1; 3 -2], then inverse of matrix A³ is
🥳 Wohoo! Correct answer
For inverse: AA⁻¹=A⁻¹A=I
Calculate A³
Find it equals I
Therefore A³A=I
Not verifying matrix multiplication
😢 Uh oh! Incorrect answer, Try again
Check if A³=I
For inverse: AA⁻¹=A⁻¹A=I
Calculate A³
Find it equals I
Therefore A³A=I
Not verifying matrix multiplication
If A is skew symmetric matrix, then A²⁰²¹ is
🥳 Wohoo! Correct answer
For skew symmetric: A^T=-A
Use property A^T=-A
For odd powers, matrix remains skew symmetric
2021 is odd
Not considering power parity
😢 Uh oh! Incorrect answer, Try again
Check matrix power properties
For skew symmetric: A^T=-A
Use property A^T=-A
For odd powers, matrix remains skew symmetric
2021 is odd
Not considering power parity
Coordinates of point on x+y=6 at which tangent is equally inclined to axes is
🥳 Wohoo! Correct answer
dy/dx = -1 for x+y=c
Use condition dy/dx = ±1
Substitute in equation x+y=6
Solve for x and y
Not using slope condition
😢 Uh oh! Incorrect answer, Try again
Equal inclination means slope = ±1
dy/dx = -1 for x+y=c
Use condition dy/dx = ±1
Substitute in equation x+y=6
Solve for x and y
Not using slope condition
The Standard Deviation of the numbers 31, 32, 33......46, 47 is
🥳 Wohoo! Correct answer
SD = √[(Σx²/n)-(x̄)²]
Find number of terms (n=17)
Use formula for sum of squares
Substitute in SD formula
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Use AP series formulas
SD = √[(Σx²/n)-(x̄)²]
Find number of terms (n=17)
Use formula for sum of squares
Substitute in SD formula
Not using correct formula
Given two matrices:A = [1 -2 1] B = [2 1][2 1 3] [3 2][1 1]Find the transpose of their product (AB)'
🥳 Wohoo! Correct answer
(AB)' = B'A'
Multiply matrices A and B first
Find transpose of resulting matrix
Verify dimensions
Not following matrix multiplication rules correctly
😢 Uh oh! Incorrect answer, Try again
Check matrix multiplication compatibility
(AB)' = B'A'
Multiply matrices A and B first
Find transpose of resulting matrix
Verify dimensions
Not following matrix multiplication rules correctly
Let M be 2 x 2 symmetric matrix with integer entries, then M is invertible if
🥳 Wohoo! Correct answer
det(M)≠0 for invertibility
Check properties of symmetric matrix
Examine determinant conditions
Verify diagonal dominance
Not understanding symmetric matrix properties
😢 Uh oh! Incorrect answer, Try again
Consider determinant for invertibility
det(M)≠0 for invertibility
Check properties of symmetric matrix
Examine determinant conditions
Verify diagonal dominance
Not understanding symmetric matrix properties
If the parabola y=αx²-6x+β passes through the point (0,2) and has its tangent at x=3/2 parallel to x axis, then
🥳 Wohoo! Correct answer
dy/dx=2αx-6
Use point condition y(0)=2
Use derivative condition dy/dx=0 at x=3/2
Solve system of equations
Not using both conditions together
😢 Uh oh! Incorrect answer, Try again
Parallel to x-axis means slope=0
dy/dx=2αx-6
Use point condition y(0)=2
Use derivative condition dy/dx=0 at x=3/2
Solve system of equations
Not using both conditions together
If in two circles, arcs of same length subtend angles 30° and 78° at centre, ratio of radii is
🥳 Wohoo! Correct answer
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
😢 Uh oh! Incorrect answer, Try again
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 the area of the Ellipse x²/25 + y²/λ² = 1 is 20π square units, then λ is
🥳 Wohoo! Correct answer
A = π√(ab) where a,b are semi-axes
Use area formula for ellipse
Substitute given area value
Solve for λ
Not using correct area formula
😢 Uh oh! Incorrect answer, Try again
Area of ellipse = πab
A = π√(ab) where a,b are semi-axes
Use area formula for ellipse
Substitute given area value
Solve for λ
Not using correct area formula
If y = 2xⁿ⁺¹ + 3/xⁿ, then x² d²y/dx² is
🥳 Wohoo! Correct answer
• Power Rule: d/dx(xⁿ) = nxⁿ⁻¹ • Chain Rule
First find dy/dx using power rule and chain rule
Find d²y/dx² by differentiating dy/dx
Multiply result by x² and simplify
Not multiplying by x² at final step
😢 Uh oh! Incorrect answer, Try again
Remember to apply chain rule twice
• Power Rule: d/dx(xⁿ) = nxⁿ⁻¹ • Chain Rule
First find dy/dx using power rule and chain rule
Find d²y/dx² by differentiating dy/dx
Multiply result by x² and simplify
Not multiplying by x² at final step
If curves 2x=y² and 2xy=K intersect perpendicularly, then K² is
🥳 Wohoo! Correct answer
• Perpendicular slopes: m₁m₂ = -1
Find dy/dx for both curves
Apply perpendicular condition: m₁m₂ = -1
Solve for K²
Forgetting to square K
😢 Uh oh! Incorrect answer, Try again
Use condition for perpendicular lines
• Perpendicular slopes: m₁m₂ = -1
Find dy/dx for both curves
Apply perpendicular condition: m₁m₂ = -1
Solve for K²
Forgetting to square K
If (xe)ʸ=eʸ, then dy/dx is
🥳 Wohoo! Correct answer
• d/dx(ln x) = 1/x • Implicit differentiation
Take natural log of both sides
Differentiate implicitly with respect to x
Solve for dy/dx
Not using log properties correctly
😢 Uh oh! Incorrect answer, Try again
Use logarithmic differentiation
• d/dx(ln x) = 1/x • Implicit differentiation
Take natural log of both sides
Differentiate implicitly with respect to x
Solve for dy/dx
Not using log properties correctly
If side of cube is increased by 5%, surface area increases by
🥳 Wohoo! Correct answer
• % change formula • Surface area of cube
Express new side in terms of original
Calculate % change in surface area
Simplify to get final percentage
Not using correct percentage formula
😢 Uh oh! Incorrect answer, Try again
Remember surface area = 6x²
• % change formula • Surface area of cube
Express new side in terms of original
Calculate % change in surface area
Simplify to get final percentage
Not using correct percentage formula
Maximum value of (logₑx)/x, if x>0 is
🥳 Wohoo! Correct answer
• d/dx(log x) = 1/x • Product rule
Find derivative and set to zero
Solve for critical points
Check second derivative
Not checking domain restrictions
😢 Uh oh! Incorrect answer, Try again
Use derivative test for max/min
• d/dx(log x) = 1/x • Product rule
Find derivative and set to zero
Solve for critical points
Check second derivative
Not checking domain restrictions
Area of region bounded by curve y²=8x and line y=2x is
🥳 Wohoo! Correct answer
• Area = ∫(upper - lower) dx
Find points of intersection
Set up integral for area
Evaluate definite integral
Not identifying correct bounds
😢 Uh oh! Incorrect answer, Try again
Area between curves formula
• Area = ∫(upper - lower) dx
Find points of intersection
Set up integral for area
Evaluate definite integral
Not identifying correct bounds
Length of rectangle is five times breadth. If minimum perimeter is 180cm, then
🥳 Wohoo! Correct answer
P=2(l+b)
Let l=5b
Use P=2(l+b)=180
Solve for b≥15
Not using inequality
😢 Uh oh! Incorrect answer, Try again
Use perimeter formula
P=2(l+b)
Let l=5b
Use P=2(l+b)=180
Solve for b≥15
Not using inequality
Two vectors î+ĵ+k̂ and î+3ĵ+5k̂ represent sides AB and AC of triangle ABC. Length of median through A is
🥳 Wohoo! Correct answer
• Median formula • Vector operations
Find coordinates of centroid
Calculate vector AD
Find magnitude
Not using correct median formula
😢 Uh oh! Incorrect answer, Try again
Use median formula
• Median formula • Vector operations
Find coordinates of centroid
Calculate vector AD
Find magnitude
Not using correct median formula
The curve passing through point (1, 2) given that slope of tangent at any point (x, y) is 3x/y represents
🥳 Wohoo! Correct answer
• dy/dx = 3x/y • Separable differential equations
Write differential equation dy/dx = 3x/y
Separate variables and integrate
Use point (1,2) to find constant
Not recognizing hyperbola form
😢 Uh oh! Incorrect answer, Try again
Consider standard forms of conics
• dy/dx = 3x/y • Separable differential equations
Write differential equation dy/dx = 3x/y
Separate variables and integrate
Use point (1,2) to find constant
Not recognizing hyperbola form
If vectors 2î-3ĵ+4k̂, 2î+ĵ-k̂ and λî-ĵ+2k̂ are coplanar, then value of λ is
🥳 Wohoo! Correct answer
• Triple product = 0 for coplanar vectors
Form triple product equation
Expand determinant
Solve for λ
Not setting up triple product correctly
😢 Uh oh! Incorrect answer, Try again
Coplanar vectors have zero triple product
• Triple product = 0 for coplanar vectors
Form triple product equation
Expand determinant
Solve for λ
Not setting up triple product correctly
The two lines lx+my=n and l'x+m'y=n' are perpendicular if
🥳 Wohoo! Correct answer
• m₁m₂=-1 for perpendicular lines
Find slopes of lines
Apply perpendicular condition
Express in terms of coefficients
Not relating coefficients to slopes
😢 Uh oh! Incorrect answer, Try again
Remember perpendicular slopes multiply to -1
• m₁m₂=-1 for perpendicular lines
Find slopes of lines
Apply perpendicular condition
Express in terms of coefficients
Not relating coefficients to slopes
If parabola x²=4ay passes through point (2,1), then length of latus rectum is
🥳 Wohoo! Correct answer
• Latus rectum = 4a for x²=4ay
Substitute point in equation
Solve for parameter a
Calculate latus rectum = 4a
Not using correct latus rectum formula
😢 Uh oh! Incorrect answer, Try again
Remember latus rectum formula
• Latus rectum = 4a for x²=4ay
Substitute point in equation
Solve for parameter a
Calculate latus rectum = 4a
Not using correct latus rectum formula
Standard deviation of data 6,7,8,9,10 is
🥳 Wohoo! Correct answer
• σ = √(Σ(x-x̄)²/n)
Find mean
Calculate sum of squares of deviations
Take square root
Not dividing by correct n
😢 Uh oh! Incorrect answer, Try again
Remember to divide by n
• σ = √(Σ(x-x̄)²/n)
Find mean
Calculate sum of squares of deviations
Take square root
Not dividing by correct n
If 3x3 matrix A=[[0,0,1],[0,1,0],[1,0,0]], then A⁴ is equal to
🥳 Wohoo! Correct answer
• Matrix multiplication • Identity matrix properties
Calculate A² first
Calculate A⁴=A²×A²
Compare with identity matrix
Not seeing pattern in matrix powers
😢 Uh oh! Incorrect answer, Try again
Notice pattern in powers
• Matrix multiplication • Identity matrix properties
Calculate A² first
Calculate A⁴=A²×A²
Compare with identity matrix
Not seeing pattern in matrix powers
If [[2,1],[3,2]]A=[[1,0],[0,1]], then matrix A is
🥳 Wohoo! Correct answer
• (AB)=(I) implies B=A⁻¹ • Matrix inverse formulas
Use matrix multiplication
Solve matrix equation
Verify solution
Not solving matrix equation correctly
😢 Uh oh! Incorrect answer, Try again
Consider inverse matrix properties
• (AB)=(I) implies B=A⁻¹ • Matrix inverse formulas
Use matrix multiplication
Solve matrix equation
Verify solution
Not solving matrix equation correctly
If A and B are square matrices of same order and B is skew symmetric matrix, then A'BA is
🥳 Wohoo! Correct answer
• (AB)'=B'A' • Skew symmetric: B'=-B
Use properties of transpose
Apply skew symmetric property B'=-B
Verify skew symmetric property
Not applying transpose properties correctly
😢 Uh oh! Incorrect answer, Try again
Remember matrix transpose rules
• (AB)'=B'A' • Skew symmetric: B'=-B
Use properties of transpose
Apply skew symmetric property B'=-B
Verify skew symmetric property
Not applying transpose properties correctly
Line cuts equal intercepts on axes. Angle with positive X-axis is
🥳 Wohoo! Correct answer
Angle formula: θ=tan⁻¹(m)
Use intercept form equation
Equal intercepts means a=b in ax+by=ab
Calculate angle using tan⁻¹(-a/b)
Confusing slope with angle
😢 Uh oh! Incorrect answer, Try again
Remember equal intercepts property
Angle formula: θ=tan⁻¹(m)
Use intercept form equation
Equal intercepts means a=b in ax+by=ab
Calculate angle using tan⁻¹(-a/b)
Confusing slope with angle
Unit vector perpendicular to plane containing î+2ĵ+k̂ and -2î+ĵ+3k̂ is
🥳 Wohoo! Correct answer
Cross product formula
Find cross product of vectors
Normalize resulting vector
Take negative for given direction
Not normalizing final vector
😢 Uh oh! Incorrect answer, Try again
Use right-hand rule
Cross product formula
Find cross product of vectors
Normalize resulting vector
Take negative for given direction
Not normalizing final vector
Equation of parabola whose focus is (6,0) and directrix is x=-6 is
🥳 Wohoo! Correct answer
y²=4ax for vertical axis
Use standard form (y²=4ax)
Identify a=6 from focus
Write equation y²=24x
Wrong standard form
😢 Uh oh! Incorrect answer, Try again
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
If P and Q are symmetric matrices of the same order then PQ-QP is
🥳 Wohoo! Correct answer
For skew symmetric, A^T = -A
Apply matrix multiplication
Compare with transpose
Check skew symmetric property
Not understanding matrix properties
😢 Uh oh! Incorrect answer, Try again
Consider properties of symmetric matrices
For skew symmetric, A^T = -A
Apply matrix multiplication
Compare with transpose
Check skew symmetric property
Not understanding matrix properties
If 3A+4B' = matrix [7 -10 17; 0 6 31] and 2B-3A' = matrix [-1 18; 4 0; -5 -7] then B=
🥳 Wohoo! Correct answer
Matrix transpose properties
Form system of equations
Solve for A and B
Verify solution
Not handling transpose correctly
😢 Uh oh! Incorrect answer, Try again
Use matrix properties systematically
Matrix transpose properties
Form system of equations
Solve for A and B
Verify solution
Not handling transpose correctly
The eccentricity of ellipse 9x²+25y²=225 is
🥳 Wohoo! Correct answer
e = √(1-b²/a²) for ellipse
Convert to standard form
Identify a and b
Calculate eccentricity
Not identifying semi-axes correctly
😢 Uh oh! Incorrect answer, Try again
Compare with standard form
e = √(1-b²/a²) for ellipse
Convert to standard form
Identify a and b
Calculate eccentricity
Not identifying semi-axes correctly
Mean and standard deviation of 100 items are 50 and 4 respectively. Sum of squares of items is
🥳 Wohoo! Correct answer
σ² = (Σx²/n) - (x̄)²
Use formula for standard deviation
Substitute given values
Solve for sum of squares
Not using correct formula
😢 Uh oh! Incorrect answer, Try again
Remember variance formula
σ² = (Σx²/n) - (x̄)²
Use formula for standard deviation
Substitute given values
Solve for sum of squares
Not using correct formula
Area of region above X-axis included between parabola y²=x and circle x²+y²=2x in square units is
🥳 Wohoo! Correct answer
Area = ∫(upper curve - lower curve) dx
Find intersection points
Set up definite integral
Calculate area difference
Not identifying correct region
😢 Uh oh! Incorrect answer, Try again
Consider region boundaries carefully
Area = ∫(upper curve - lower curve) dx
Find intersection points
Set up definite integral
Calculate area difference
Not identifying correct region
Area of region bounded by Y-axis, y=cos x and y=sin x; 0≤x≤π/2 is
🥳 Wohoo! Correct answer
Area between curves formula
Find intersection points
Set up integral
Calculate area
Not setting correct limits
😢 Uh oh! Incorrect answer, Try again
Consider curve orientations
Area between curves formula
Find intersection points
Set up integral
Calculate area
Not setting correct limits
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
🥳 Wohoo! Correct answer
SD(X+k) = SD(X)
Understand effect of adding constant
Adding k doesn't change spread
SD remains S
Think k affects SD
😢 Uh oh! Incorrect answer, Try again
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
The distance between the foci of a hyperbola is 16 and its eccentricity is √2. Its equation is
🥳 Wohoo! Correct answer
e²=1+b²/a²
Use 2ae=16, e=√2
Find a=4√2, b=4√2
Get x²-y²=32
Confusion in standard forms
😢 Uh oh! Incorrect answer, Try again
Use standard form
e²=1+b²/a²
Use 2ae=16, e=√2
Find a=4√2, b=4√2
Get x²-y²=32
Confusion in standard forms
If 2x2 matrix A = [2 -2; -2 2], then Aⁿ = 2ᵏA, Where k =
🥳 Wohoo! Correct answer
Matrix multiplication
Calculate A² = 2²[2 -2; -2 2] = 2²A
Calculate A³ = 2⁴A
Observe pattern: each power increases exponent by 2
Missing pattern in powers
😢 Uh oh! Incorrect answer, Try again
Look for pattern in consecutive powers
Matrix multiplication
Calculate A² = 2²[2 -2; -2 2] = 2²A
Calculate A³ = 2⁴A
Observe pattern: each power increases exponent by 2
Missing pattern in powers
If matrix [1 1; -1 1][x; y] = [2; 4], then x,y =
🥳 Wohoo! Correct answer
System solution methods
Form equations: x + y = 2 and -x + y = 4
Solve system using elimination: 2y = 6
Get y = 3, x = -1
Sign errors in equations
😢 Uh oh! Incorrect answer, Try again
Matrix multiplication gives system of equations
System solution methods
Form equations: x + y = 2 and -x + y = 4
Solve system using elimination: 2y = 6
Get y = 3, x = -1
Sign errors in equations
If 2x2 matrix A = [cosα sinα; -sinα cosα], then AA' =
🥳 Wohoo! Correct answer
AA' = I for orthogonal matrix
Multiply A with A' elementwise
Simplify using trig identities cos²α + sin²α = 1
Get identity matrix I
Forgetting trig identities
😢 Uh oh! Incorrect answer, Try again
Orthogonal matrix property
AA' = I for orthogonal matrix
Multiply A with A' elementwise
Simplify using trig identities cos²α + sin²α = 1
Get identity matrix I
Forgetting trig identities
Approximate change in volume V of cube side x metres when increased by 3% is
🥳 Wohoo! Correct answer
ΔV = dV/dx × Δx
Use ΔV = dV/dx × Δx
Δx = 0.03x (3% of x)
ΔV = 3x²(0.03x) = 0.09x³
Unit conversion errors
😢 Uh oh! Incorrect answer, Try again
Small change approximation
ΔV = dV/dx × Δx
Use ΔV = dV/dx × Δx
Δx = 0.03x (3% of x)
ΔV = 3x²(0.03x) = 0.09x³
Unit conversion errors
Maximum value of (1/x)ˣ is
🥳 Wohoo! Correct answer
Derivative tests
Let y = (1/x)ˣ, take ln of both sides
Find dy/dx using implicit differentiation
Maximum occurs at x = 1/e
Logarithm errors
😢 Uh oh! Incorrect answer, Try again
Use calculus for optimization
Derivative tests
Let y = (1/x)ˣ, take ln of both sides
Find dy/dx using implicit differentiation
Maximum occurs at x = 1/e
Logarithm errors
f(x) = xˣ has stationary point at
🥳 Wohoo! Correct answer
Stationary points: dy/dx = 0
Let y = xˣ, take ln of both sides: lny = xlnx
Differentiate implicitly to get dy/dx
Set dy/dx = 0 to get x = 1/e
Chain rule errors
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation
Stationary points: dy/dx = 0
Let y = xˣ, take ln of both sides: lny = xlnx
Differentiate implicitly to get dy/dx
Set dy/dx = 0 to get x = 1/e
Chain rule errors
Maximum area of rectangle inscribed in circle (x+1)² + (y-3)² = 64 is
🥳 Wohoo! Correct answer
Area = length × width
Transform to standard form (x+1)² + (y-3)² = 64
Express area as function of one variable
Maximize using calculus to get 128
Transformation errors
😢 Uh oh! Incorrect answer, Try again
Use optimization techniques
Area = length × width
Transform to standard form (x+1)² + (y-3)² = 64
Express area as function of one variable
Maximize using calculus to get 128
Transformation errors
If a,b are mutually perpendicular unit vectors, then (3a+2b)·(5a-6b) =
🥳 Wohoo! Correct answer
Unit vector properties
Use a·a=1, b·b=1, a·b=0
Expand expression
Get 15-12=3
Dot product errors
😢 Uh oh! Incorrect answer, Try again
Perpendicular property
Unit vector properties
Use a·a=1, b·b=1, a·b=0
Expand expression
Get 15-12=3
Dot product errors
If the vectors ai+j+k, i+bj+k, i+j+ck are coplanar, a≠b≠c≠1, then abc-(a+b+c) =
🥳 Wohoo! Correct answer
Determinant for coplanarity
Write determinant condition
Expand and simplify
Get abc-(a+b+c)=-2
Determinant errors
😢 Uh oh! Incorrect answer, Try again
Coplanar vectors condition
Determinant for coplanarity
Write determinant condition
Expand and simplify
Get abc-(a+b+c)=-2
Determinant errors
If a = i + λj + 2k, b = μi + j - k are orthogonal and |a| = |b| then (λ, μ) =
🥳 Wohoo! Correct answer
a·b = 0 for orthogonal vectors
Use orthogonality: a·b = 0 to get μ + λ = 2
Use equal magnitudes: 1 + λ² + 4 = μ² + 1 + 1
Solve to get λ = 1/4, μ = 7/4
Missing equality conditions
😢 Uh oh! Incorrect answer, Try again
Orthogonal vectors are perpendicular
a·b = 0 for orthogonal vectors
Use orthogonality: a·b = 0 to get μ + λ = 2
Use equal magnitudes: 1 + λ² + 4 = μ² + 1 + 1
Solve to get λ = 1/4, μ = 7/4
Missing equality conditions
The eccentricity of ellipse x²/36 + y²/16 = 1 is
🥳 Wohoo! Correct answer
e = √(1-b²/a²)
Identify a²=36, b²=16
Use e = √(1-b²/a²)
Simplify to get 2√5/6
Wrong substitution in formula
😢 Uh oh! Incorrect answer, Try again
Compare with standard form
e = √(1-b²/a²)
Identify a²=36, b²=16
Use e = √(1-b²/a²)
Simplify to get 2√5/6
Wrong substitution in formula
If coefficient of variation is 60 and standard deviation is 24, then Arithmetic mean is
🥳 Wohoo! Correct answer
CV = (σ/μ)×100
Use CV = (σ/μ)×100
Substitute 60 = (24/μ)×100
Solve to get μ = 40
Wrong formula application
😢 Uh oh! Incorrect answer, Try again
Remember CV formula
CV = (σ/μ)×100
Use CV = (σ/μ)×100
Substitute 60 = (24/μ)×100
Solve to get μ = 40
Wrong formula application
If matrix A is both symmetric and skew-symmetric, then
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Symmetric: A=Aᵀ, Skew: A=-Aᵀ
Apply A=Aᵀ (symmetric)
Apply A=-Aᵀ (skew-symmetric)
Therefore A=0
Not combining conditions
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Compare conditions
Symmetric: A=Aᵀ, Skew: A=-Aᵀ
Apply A=Aᵀ (symmetric)
Apply A=-Aᵀ (skew-symmetric)
Therefore A=0
Not combining conditions
If 2[1 3; 0 x] + [y 0; 1 2] = [5 6; 1 8], then values of x and y are
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Matrix equation: AX=B means comparing corresponding elements
Equate corresponding elements: 2(1)+y=5, 2(3)=6, 2(0)+1=1, 2x+2=8
From second equation: 6=6 (verified), From third equation: 1=1 (verified)
From remaining equations: 2+y=5, so y=3 and 2x+2=8, so x=3
Not equating corresponding elements correctly
😢 Uh oh! Incorrect answer, Try again
Matrix addition is element-wise
Matrix equation: AX=B means comparing corresponding elements
Equate corresponding elements: 2(1)+y=5, 2(3)=6, 2(0)+1=1, 2x+2=8
From second equation: 6=6 (verified), From third equation: 1=1 (verified)
From remaining equations: 2+y=5, so y=3 and 2x+2=8, so x=3
Not equating corresponding elements correctly
If A is square matrix such that A²=A, then (I+A)³ equals
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(I+A)³ = I + 3A + 3A² + A³
Expand (I+A)³
Use A²=A
Simplify to get 7A+I
Not using A²=A properly
😢 Uh oh! Incorrect answer, Try again
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 A is a 2×2 matrix where A = [[1,1],[1,1]], find A¹⁰
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A^n = k^(n-1)A pattern
Find A² = 2A
Use pattern A^n = 2^(n-1)A
Get A¹⁰ = 2⁹A
Not recognizing pattern
😢 Uh oh! Incorrect answer, Try again
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=2i+λj+k and b=i+2j+3k are orthogonal, then λ=
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a·b=0 for orthogonal vectors
Use a·b=0 for orthogonal vectors
Expand 2·1+λ·2+1·3=0
Solve to get λ=-5/2
Wrong dot product
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Remember orthogonal condition
a·b=0 for orthogonal vectors
Use a·b=0 for orthogonal vectors
Expand 2·1+λ·2+1·3=0
Solve to get λ=-5/2
Wrong dot product
If A = [cos2θ -sin2θ; sin2θ cos2θ] and B = [1/π(-cos⁻¹(πx) tan⁻¹(πx); sin⁻¹(x/π) -tan⁻¹(πx)] then A-B is equal to
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Matrix subtraction: [a b; c d] - [p q; r s] = [a-p b-q; c-r d-s]
Convert matrix B to standard form
Subtract matrices element by element
Simplify to get (1/2)I
Errors in matrix subtraction
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Compare corresponding elements
Matrix subtraction: [a b; c d] - [p q; r s] = [a-p b-q; c-r d-s]
Convert matrix B to standard form
Subtract matrices element by element
Simplify to get (1/2)I
Errors in matrix subtraction
If A is a matrix of order m × n and B is a matrix such that AB' and B'A are both defined, the order of the matrix B is
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For AB: columns of A = rows of B
Consider dimensions for matrix multiplication AB'
Consider dimensions for matrix multiplication B'A
Conclude B must be m × n for both products to be defined
Not checking both multiplication conditions
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Remember matrix multiplication compatibility rules
For AB: columns of A = rows of B
Consider dimensions for matrix multiplication AB'
Consider dimensions for matrix multiplication B'A
Conclude B must be m × n for both products to be defined
Not checking both multiplication conditions
The length of latus rectum of the parabola 4y² + 3x + 3y + 1 = 0 is
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For y² = 4ax, latus rectum = 4a
Rearrange to standard form: 4(y² + 3y/4) = -3x - 1
Complete square: 4[(y + 3/8)² - 9/64] = -3x - 1
Length of latus rectum = 4a where a = coefficient of x = 3/4
Not identifying standard form
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Convert to vertex form first
For y² = 4ax, latus rectum = 4a
Rearrange to standard form: 4(y² + 3y/4) = -3x - 1
Complete square: 4[(y + 3/8)² - 9/64] = -3x - 1
Length of latus rectum = 4a where a = coefficient of x = 3/4
Not identifying standard form
If A is the determinant of the 2×2 matrix A = |x 1||1 x|and B is the determinant of the 3×3 matrix B = |x 1 1||1 x 1||1 1 x|find dB/dx
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d(XAX)/dA formula
Calculate dB/dA using derivative
Apply chain rule
Get 3A
Wrong differentiation rules
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Use matrix differentiation
d(XAX)/dA formula
Calculate dB/dA using derivative
Apply chain rule
Get 3A
Wrong differentiation rules
The equation of the normal to the curve y(1 + x²) = 2 - x where the tangent crosses x-axis is
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For normal: y-y₁ = -1/m(x-x₁)
Find dy/dx = -((1+x²) + 2x(2-x))/y(1+x²)²
Find point where tangent crosses x-axis (y=0)
Write equation of normal at this point
Confusion between tangent and normal
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Find point of intersection with x-axis first
For normal: y-y₁ = -1/m(x-x₁)
Find dy/dx = -((1+x²) + 2x(2-x))/y(1+x²)²
Find point where tangent crosses x-axis (y=0)
Write equation of normal at this point
Confusion between tangent and normal
The coefficient of variation of two distributions are 60 and 70. The standard deviations are 21 and 16 respectively, then their mean is
🥳 Wohoo! Correct answer
CV = (σ/μ)×100
Use CV = (σ/μ)×100 for both distributions
Substitute given values: 60 = 21/μ₁×100 and 70 = 16/μ₂×100
Solve to get μ₁ = 35
Not understanding CV formula
😢 Uh oh! Incorrect answer, Try again
Use coefficient of variation formula
CV = (σ/μ)×100
Use CV = (σ/μ)×100 for both distributions
Substitute given values: 60 = 21/μ₁×100 and 70 = 16/μ₂×100
Solve to get μ₁ = 35
Not understanding CV formula
If 2x2 matrix A = [cos2θ -sin2θ; sin2θ cos2θ] and A+A^T =I, Where I is the unit matrix of 2×2 & A^T is the transpose of A, then the value of θ is equal to
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Matrix addition is element-wise
Write A^T = [cos2θ sin2θ; -sin2θ cos2θ]
Add A and A^T to get [2cos2θ 0; 0 2cos2θ] = [1 0; 0 1]
Solve 2cos2θ = 1 to get θ = π/6
Not using matrix transpose correctly
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Use matrix addition properties
Matrix addition is element-wise
Write A^T = [cos2θ sin2θ; -sin2θ cos2θ]
Add A and A^T to get [2cos2θ 0; 0 2cos2θ] = [1 0; 0 1]
Solve 2cos2θ = 1 to get θ = π/6
Not using matrix transpose correctly
If 2x2 matrix A = [3 1; -1 2] then A²-5A is equal to
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(A²-5A) = A(A-5I)
Calculate A² by matrix multiplication
Multiply 5 and A
Subtract to get -7I
Errors in matrix multiplication
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Use matrix multiplication
(A²-5A) = A(A-5I)
Calculate A² by matrix multiplication
Multiply 5 and A
Subtract to get -7I
Errors in matrix multiplication
If x = 2 + 3cosθ and y = 1 - 3sinθ represent a circle then the Centre and radius is
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(x-h)² + (y-k)² = r²
Write x-2 = 3cosθ and y-1 = -3sinθ
Square and add equations
Get (x-2)²/9 + (y-1)²/9 = 1
Not identifying standard form of circle
😢 Uh oh! Incorrect answer, Try again
Recognize circle equation form
(x-h)² + (y-k)² = r²
Write x-2 = 3cosθ and y-1 = -3sinθ
Square and add equations
Get (x-2)²/9 + (y-1)²/9 = 1
Not identifying standard form of circle
If cosα, cosβ, cosγ are the direction cosines of a vector a then cos²α + cos²β + cos²γ is equal to
🥳 Wohoo! Correct answer
cos²α + cos²β + cos²γ = 1
Use direction cosines property
Sum of squares of direction cosines = 1
Therefore cos²α + cos²β + cos²γ = -1
Not knowing direction cosines property
😢 Uh oh! Incorrect answer, Try again
Direction cosines property
cos²α + cos²β + cos²γ = 1
Use direction cosines property
Sum of squares of direction cosines = 1
Therefore cos²α + cos²β + cos²γ = -1
Not knowing direction cosines property
If eccentricity of hyperbola x²/a² - y²/b² = 1 is 5/4 and 2x + 3y - 6 = 0 is focal chord, length of transverse axis
🥳 Wohoo! Correct answer
• e² = 1 + b²/a² • Transverse axis = 2a
Use focal chord equation in hyperbola
Apply e = √(1 + b²/a²) = 5/4
Calculate 2a = 24/5
Wrong focal chord application
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Relate eccentricity to axes
• e² = 1 + b²/a² • Transverse axis = 2a
Use focal chord equation in hyperbola
Apply e = √(1 + b²/a²) = 5/4
Calculate 2a = 24/5
Wrong focal chord application
If a→=i+2j+k, b→=i-j+k, c→=i+j-k, a vector in the plane a→ and whose projection on it is
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• Projection formula • Vector operations
Find direction of plane
Use projection formula
Verify conditions
Wrong projection
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Consider plane equation
• Projection formula • Vector operations
Find direction of plane
Use projection formula
Verify conditions
Wrong projection
Mean deviation from data 3,10,10,4,7,10,5
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• Mean deviation formula • Absolute values
Find mean = 7
Calculate deviations
Average absolute deviations
Wrong mean calculation
😢 Uh oh! Incorrect answer, Try again
Consider ordered data
• Mean deviation formula • Absolute values
Find mean = 7
Calculate deviations
Average absolute deviations
Wrong mean calculation
Maximum volume of right circular cone with slant height 6 units is
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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
😢 Uh oh! Incorrect answer, Try again
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
System x+y+z=6, x+2y+3z=10, x+2y+az=6 has no solutions when
🥳 Wohoo! Correct answer
• Determinant = 0 • Matrix rank
Form augmented matrix
Apply consistency conditions
Find a=3, b≠10
Wrong consistency check
😢 Uh oh! Incorrect answer, Try again
Consider system consistency
• Determinant = 0 • Matrix rank
Form augmented matrix
Apply consistency conditions
Find a=3, b≠10
Wrong consistency check
Vectors AB=3i+4k and AC=5i-2j+4k are sides of ΔABC. Length of median through A is
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Median length formula
Find median vector=(AB+AC)/2
Calculate magnitude
Get √33
Wrong median formula
😢 Uh oh! Incorrect answer, Try again
Use median formula
Median length formula
Find median vector=(AB+AC)/2
Calculate magnitude
Get √33
Wrong median formula
If 2x2 matrix A=[0 1; 1 0], then A² equals
🥳 Wohoo! Correct answer
• Matrix multiplication • Square matrix properties
Multiply matrix by itself
Simplify elements
Get identity matrix
Wrong multiplication
😢 Uh oh! Incorrect answer, Try again
Consider matrix multiplication
• Matrix multiplication • Square matrix properties
Multiply matrix by itself
Simplify elements
Get identity matrix
Wrong multiplication
If A and B are two matrices such that AB = B and BA = A then A² + B² = ?
🥳 Wohoo! Correct answer
Matrix multiplication: (PQ)R = P(QR)
Using BA = A, multiply both sides by A: A² = (BA)A = B(AA) = BA = A
Using AB = B, multiply both sides by B: B² = B(B) = (AB)B = A(BB) = AB = B
Therefore A² + B² = A + B
1. Confusing AB = B with AB = BA\n2. Forgetting associative property
😢 Uh oh! Incorrect answer, Try again
Use given conditions AB = B and BA = A systematically
Matrix multiplication: (PQ)R = P(QR)
Using BA = A, multiply both sides by A: A² = (BA)A = B(AA) = BA = A
Using AB = B, multiply both sides by B: B² = B(B) = (AB)B = A(BB) = AB = B
Therefore A² + B² = A + B
1. Confusing AB = B with AB = BA\n2. Forgetting associative property
The distance between the foci of a hyperbola is 16 and its eccentricity is √2. Its equation is
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Hyperbola equation: x²/a²-y²/b²=1
Use 2ae=16 and e=√2
Calculate a=4√2 and b=4
Get standard form x²-y²=32
Wrong standard form
😢 Uh oh! Incorrect answer, Try again
Use hyperbola standard form
Hyperbola equation: x²/a²-y²/b²=1
Use 2ae=16 and e=√2
Calculate a=4√2 and b=4
Get standard form x²-y²=32
Wrong standard form
The mean of 100 observations is 50 and their standard deviation is 5. Then the sum of squares of all observations is
🥳 Wohoo! Correct answer
σ² = Σ(x-μ)²/n
Use formula: σ² = Σ(x-μ)²/n
Expand: Σx²/n - μ² = σ²
Calculate Σx² = 252500
Wrong formula application
😢 Uh oh! Incorrect answer, Try again
Use variance formula
σ² = Σ(x-μ)²/n
Use formula: σ² = Σ(x-μ)²/n
Expand: Σx²/n - μ² = σ²
Calculate Σx² = 252500
Wrong formula application
If the straight line 2x-3y+17=0 is perpendicular to the line passing through points (7,17) and (15,β), then β equals
🥳 Wohoo! Correct answer
m₁m₂ = -1 for perpendicular lines
Find slope m₁ = -2/3 of given line
Use perpendicular slopes property: m₁m₂ = -1
Calculate β using point-slope form
Not using perpendicular slopes property correctly
😢 Uh oh! Incorrect answer, Try again
Remember perpendicular lines have negative reciprocal slopes
m₁m₂ = -1 for perpendicular lines
Find slope m₁ = -2/3 of given line
Use perpendicular slopes property: m₁m₂ = -1
Calculate β using point-slope form
Not using perpendicular slopes property correctly