KCET PYQ MATHEMATICS Group-5-Questions

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

3×3 determinant expansion

📚 Problem Solving Strategy

Expand determinant using first row

Get quadratic in 'a'

Add roots of quadratic

⚠️ Common Mistakes

Not expanding determinant correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use determinant expansion

This is some text inside of a div block.

💡 Key Formula

3×3 determinant expansion

📚 Problem Solving Strategy

Expand determinant using first row

Get quadratic in 'a'

Add roots of quadratic

⚠️ Common Mistakes

Not expanding determinant correctly

The function f(x) = log(1+x) - 2x/(2+x) is increasing on

(-∞,∞)

(-∞,-1)

(-1,∞)

(-∞,0)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

f'(x) > 0 for increasing

📚 Problem Solving Strategy

Find f'(x)

Find where f'(x) > 0

Solve inequality

⚠️ Common Mistakes

Not checking domain

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Check derivative sign

This is some text inside of a div block.

💡 Key Formula

f'(x) > 0 for increasing

📚 Problem Solving Strategy

Find f'(x)

Find where f'(x) > 0

Solve inequality

⚠️ Common Mistakes

Not checking domain

If A={1,2,3,...,10} then number of subsets of A containing only odd numbers is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Number of subsets = 2ⁿ

📚 Problem Solving Strategy

Identify odd numbers in A: {1,3,5,7,9}

Count possible subsets: 2⁵

Subtract 1 for empty set: 32-1=31

⚠️ Common Mistakes

Not considering empty set

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use power set formula

This is some text inside of a div block.

💡 Key Formula

Number of subsets = 2ⁿ

📚 Problem Solving Strategy

Identify odd numbers in A: {1,3,5,7,9}

Count possible subsets: 2⁵

Subtract 1 for empty set: 32-1=31

⚠️ Common Mistakes

Not considering empty set

In a certain two 65% families own cell phones, 15000 families own scooter and 15% families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is

20000

30000

40000

50000

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

n(A∪B) = n(A) + n(B) - n(A∩B)

📚 Problem Solving Strategy

Use Venn diagram

Apply addition principle

Solve equation

⚠️ Common Mistakes

Not considering intersection properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Draw Venn diagram first

This is some text inside of a div block.

💡 Key Formula

n(A∪B) = n(A) + n(B) - n(A∩B)

📚 Problem Solving Strategy

Use Venn diagram

Apply addition principle

Solve equation

⚠️ Common Mistakes

Not considering intersection properly

If A and B are non-singleton sets where n(A×B) = 35 and B ⊂ A, then ⁿ⁽ᴬ⁾ C ₙ₍ʙ₎ =

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

n(A×B)=n(A)×n(B)

📚 Problem Solving Strategy

Since B⊂A, let n(B)=k and n(A)=k+r

Use n(A×B)=n(A)×n(B)=35

Solve equation k(k+r)=35

⚠️ Common Mistakes

Not using subset property correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Remember B is subset of A

This is some text inside of a div block.

💡 Key Formula

n(A×B)=n(A)×n(B)

📚 Problem Solving Strategy

Since B⊂A, let n(B)=k and n(A)=k+r

Use n(A×B)=n(A)×n(B)=35

Solve equation k(k+r)=35

⚠️ Common Mistakes

Not using subset property correctly

If ((1+i)/(1-i))^x = 1 then

x=4n+1; n∈N

x=2n+1; n∈N

x=2n; n∈N

x=4n; n∈N

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

(1+i)/(1-i) = i

📚 Problem Solving Strategy

Convert complex fraction to standard form

Take log of both sides

Solve for x

⚠️ Common Mistakes

Not simplifying complex fraction

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use properties of complex numbers

This is some text inside of a div block.

💡 Key Formula

(1+i)/(1-i) = i

📚 Problem Solving Strategy

Convert complex fraction to standard form

Take log of both sides

Solve for x

⚠️ Common Mistakes

Not simplifying complex fraction

A student has to answer 10 questions, choosing at least 4 from each of the parts A and B. If there are 6 questions in part A and 7 in part B, then the number of ways can the student choose 10 questions is

256

352

266

426

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

nCr formula

📚 Problem Solving Strategy

List possible combinations of questions from A and B

Use combination formula for each case

Sum all possibilities

⚠️ Common Mistakes

Not considering all cases

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Make table of possible selections

This is some text inside of a div block.

💡 Key Formula

nCr formula

📚 Problem Solving Strategy

List possible combinations of questions from A and B

Use combination formula for each case

Sum all possibilities

⚠️ Common Mistakes

Not considering all cases

If the middle term of the A.P is 300 then the sum of its first 51 terms is

15300

14800

16500

14300

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Sn = n/2[2a+(n-1)d]

📚 Problem Solving Strategy

Use middle term formula

Find first term and common difference

Apply sum formula

⚠️ Common Mistakes

Not using middle term properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Middle term = a + (n-1)d/2

This is some text inside of a div block.

💡 Key Formula

Sn = n/2[2a+(n-1)d]

📚 Problem Solving Strategy

Use middle term formula

Find first term and common difference

Apply sum formula

⚠️ Common Mistakes

Not using middle term properly

The equation of straight line which passes through the point (acos3,asin3) and perpendicular to xsec θ+ycosec θ=a is

x/a + y/a = cosθ

xcosθ-ysinθ=acos²θ

xcosθ+ysinθ=acos²θ

xcosθ-ysinθ=-acos²θ

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1

📚 Problem Solving Strategy

Find slope of given line

Use perpendicular slope condition

Form equation through point

⚠️ Common Mistakes

Not using perpendicular condition correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Perpendicular lines have negative reciprocal slopes

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1

📚 Problem Solving Strategy

Find slope of given line

Use perpendicular slope condition

Form equation through point

⚠️ Common Mistakes

Not using perpendicular condition correctly

The mid points of the sides of triangle are (1,5,-1) (0,4,-2) and (2,3,4) then centroid of the triangle

(1,4,3)

(1/3,4,1/3)

(-1,4,3)

(1/3,2,4/3)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Centroid = (x₁+x₂+x₃)/3

📚 Problem Solving Strategy

Use midpoint formula to find vertices

Average the vertices

Simplify coordinates

⚠️ Common Mistakes

Not using centroid formula correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Centroid divides median in ratio 2:1

This is some text inside of a div block.

💡 Key Formula

Centroid = (x₁+x₂+x₃)/3

📚 Problem Solving Strategy

Use midpoint formula to find vertices

Average the vertices

Simplify coordinates

⚠️ Common Mistakes

Not using centroid formula correctly

Two finite sets have m,n elements. Total subsets of first set is 56 more than second set. Values of m,n are

7,6

5,1

6,3

8,7

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Number of subsets = 2ⁿ

📚 Problem Solving Strategy

Use 2ᵐ-2ⁿ=56

Try values of m,n

Verify m=6,n=3 satisfies

⚠️ Common Mistakes

Not considering all possibilities

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Think about powers of 2

This is some text inside of a div block.

💡 Key Formula

Number of subsets = 2ⁿ

📚 Problem Solving Strategy

Use 2ᵐ-2ⁿ=56

Try values of m,n

Verify m=6,n=3 satisfies

⚠️ Common Mistakes

Not considering all possibilities

At x=1, the function f(x)={x³-1 when 1

continuous and differentiable

continuous and non-differentiable

discontinuous and differentiable

discontinuous and non-differentiable

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Left and right hand derivatives

📚 Problem Solving Strategy

Check left and right hand limits

Check derivatives from both sides

Compare results

⚠️ Common Mistakes

Not checking both conditions separately

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Check both continuity and differentiability

This is some text inside of a div block.

💡 Key Formula

Left and right hand derivatives

📚 Problem Solving Strategy

Check left and right hand limits

Check derivatives from both sides

Compare results

⚠️ Common Mistakes

Not checking both conditions separately

If f(x)=sin⁻¹(2x/(1+x²)), then f'(√3) is

-1/2

1/2

1/√3

-1/√3

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• d/dx(sin⁻¹x) = 1/√(1-x²)

📚 Problem Solving Strategy

Use formula for derivative of arcsin: f'(x) = 1/√(1-x²)

Substitute x=√3 into the derivative formula

Simplify to get 1/2

⚠️ Common Mistakes

Not properly applying inverse function derivative formula

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use derivative formula for inverse sine carefully

This is some text inside of a div block.

💡 Key Formula

• d/dx(sin⁻¹x) = 1/√(1-x²)

📚 Problem Solving Strategy

Use formula for derivative of arcsin: f'(x) = 1/√(1-x²)

Substitute x=√3 into the derivative formula

Simplify to get 1/2

⚠️ Common Mistakes

Not properly applying inverse function derivative formula

Real value of α for which (1-isinα) / (1+2isinα) is purely real is

(n+1)π/2,n∈N

(2n+1)π/2,n∈N

nπ,n∈N

(2n-1)π/2,n∈N

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

z is real if Im(z)=0

📚 Problem Solving Strategy

Rationalize denominator

Set imaginary part to zero

Solve for α=nπ

⚠️ Common Mistakes

Not rationalizing properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Complex number is real if Im=0

This is some text inside of a div block.

💡 Key Formula

z is real if Im(z)=0

📚 Problem Solving Strategy

Rationalize denominator

Set imaginary part to zero

Solve for α=nπ

⚠️ Common Mistakes

Not rationalizing properly

d/dx[cos²{cot⁻¹(√(2+x)/√(2-x))}] is

-3/4

-1/2

1/2

1/4

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Chain Rule, d/dx(cot⁻¹x)=-1/(1+x²)

📚 Problem Solving Strategy

Let u=√(2+x)/(2-x), then cot⁻¹u is being considered

Use chain rule for composite function

Simplify to get 1/4

⚠️ Common Mistakes

Not using chain rule properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Break down into simpler parts using chain rule

This is some text inside of a div block.

💡 Key Formula

Chain Rule, d/dx(cot⁻¹x)=-1/(1+x²)

📚 Problem Solving Strategy

Let u=√(2+x)/(2-x), then cot⁻¹u is being considered

Use chain rule for composite function

Simplify to get 1/4

⚠️ Common Mistakes

Not using chain rule properly

Value of ⁴⁹C₃+⁴⁸C₃+⁴⁷C₃+⁴⁶C₃+⁴⁵C₃+⁴⁵C₄ is

⁵⁰C₄

⁵⁰C₃

⁵⁰C₂

⁵⁰C₁

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁

📚 Problem Solving Strategy

Use combination formula

Apply Pascal's identity

Get ⁵⁰C₄

⚠️ Common Mistakes

Not seeing pattern

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern

This is some text inside of a div block.

💡 Key Formula

ⁿCᵣ+ⁿCᵣ₊₁=ⁿ⁺¹Cᵣ₊₁

📚 Problem Solving Strategy

Use combination formula

Apply Pascal's identity

Get ⁵⁰C₄

⚠️ Common Mistakes

Not seeing pattern

If A={1,2,3,4,5,6}, then number of subsets of A which contain at least two elements is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Total subsets = 2ⁿ • nCr formula

📚 Problem Solving Strategy

Find total subsets

Subtract empty set and single element sets

Simplify

⚠️ Common Mistakes

Not subtracting correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use combination formula

This is some text inside of a div block.

💡 Key Formula

• Total subsets = 2ⁿ • nCr formula

📚 Problem Solving Strategy

Find total subsets

Subtract empty set and single element sets

Simplify

⚠️ Common Mistakes

Not subtracting correctly

Value of ¹⁶C₉ + ¹⁶C₁₀ - ¹⁶C₆ - ¹⁶C₇ is

¹⁷C₁₀

¹⁷C₃

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Pascal's identity • Combination properties

📚 Problem Solving Strategy

Use Pascal's identity

Combine terms

Simplify

⚠️ Common Mistakes

Not using combination identities

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for patterns in combinations

This is some text inside of a div block.

💡 Key Formula

• Pascal's identity • Combination properties

📚 Problem Solving Strategy

Use Pascal's identity

Combine terms

Simplify

⚠️ Common Mistakes

Not using combination identities

Number of terms in expansion of (x+y+z)¹⁰ is

142

110

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Terms = (n+r-1)C(r-1) where r is number of variables

📚 Problem Solving Strategy

Use formula for terms in multinomial

Calculate (n+r-1)C(r-1)

Simplify

⚠️ Common Mistakes

Not using correct formula

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider r=3 variables

This is some text inside of a div block.

💡 Key Formula

• Terms = (n+r-1)C(r-1) where r is number of variables

📚 Problem Solving Strategy

Use formula for terms in multinomial

Calculate (n+r-1)C(r-1)

Simplify

⚠️ Common Mistakes

Not using correct formula

If P(n): 2ⁿ < n!, Then the smallest positive integer for which P(n) is true is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• n! = n×(n-1)×...×1 • 2ⁿ grows slower than n! eventually

📚 Problem Solving Strategy

Compare 2ⁿ and n! for n=1,2,3,4

For n=4: 2⁴=16, 4!=24

Verify 4 is smallest such n

⚠️ Common Mistakes

Not checking small values first

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Check values systematically

This is some text inside of a div block.

💡 Key Formula

• n! = n×(n-1)×...×1 • 2ⁿ grows slower than n! eventually

📚 Problem Solving Strategy

Compare 2ⁿ and n! for n=1,2,3,4

For n=4: 2⁴=16, 4!=24

Verify 4 is smallest such n

⚠️ Common Mistakes

Not checking small values first

In expansion of (1+x)ⁿ, C₁/C₀+2C₂/C₁+3C₃/C₂+...+nCₙ/Cₙ₋₁ equals

n(n+1)/2

n/2

(n+1)/2

3n(n+1)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

nCr=n!/r!(n-r)!

📚 Problem Solving Strategy

Write each term using nCr formula

Simplify ratio pattern

Sum to get n(n+1)/2

⚠️ Common Mistakes

Not simplifying properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look at coefficient pattern

This is some text inside of a div block.

💡 Key Formula

nCr=n!/r!(n-r)!

📚 Problem Solving Strategy

Write each term using nCr formula

Simplify ratio pattern

Sum to get n(n+1)/2

⚠️ Common Mistakes

Not simplifying properly

If sum of n terms of A.P is given by Sₙ=n²+n, then common difference of A.P is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Sₙ=n/2[2a+(n-1)d] • d=aₙ-aₙ₋₁

📚 Problem Solving Strategy

Find aₙ by Sₙ-Sₙ₋₁

Compare consecutive terms

Calculate d

⚠️ Common Mistakes

Not using correct method to find d

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use Sₙ formula to find d

This is some text inside of a div block.

💡 Key Formula

• Sₙ=n/2[2a+(n-1)d] • d=aₙ-aₙ₋₁

📚 Problem Solving Strategy

Find aₙ by Sₙ-Sₙ₋₁

Compare consecutive terms

Calculate d

⚠️ Common Mistakes

Not using correct method to find d

If Sₙ stands for sum to n-terms of GP with 'a' as first term and 'r' as common ratio then Sₙ:S₂ₙ is

rⁿ+1

1/(rⁿ+1)

rⁿ-1

1/(rⁿ-1)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Sₙ = a(rⁿ-1)/(r-1)

📚 Problem Solving Strategy

Write Sₙ = a(rⁿ-1)/(r-1)

Write S₂ₙ = a(r²ⁿ-1)/(r-1)

Form ratio Sₙ/S₂ₙ = 1/(rⁿ+1)

⚠️ Common Mistakes

Not using correct sum formula

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use GP sum formula

This is some text inside of a div block.

💡 Key Formula

Sₙ = a(rⁿ-1)/(r-1)

📚 Problem Solving Strategy

Write Sₙ = a(rⁿ-1)/(r-1)

Write S₂ₙ = a(r²ⁿ-1)/(r-1)

Form ratio Sₙ/S₂ₙ = 1/(rⁿ+1)

⚠️ Common Mistakes

Not using correct sum formula

If A={a,b,c}, then number of binary operations on A is

3⁶

3³

3⁹

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Binary operation maps A×A→A • Total possibilities = 3⁹

📚 Problem Solving Strategy

Understand binary operation definition

Count possible outputs for each input pair

Use multiplication principle

⚠️ Common Mistakes

Not counting all possible combinations

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Remember total possible mappings

This is some text inside of a div block.

💡 Key Formula

• Binary operation maps A×A→A • Total possibilities = 3⁹

📚 Problem Solving Strategy

Understand binary operation definition

Count possible outputs for each input pair

Use multiplication principle

⚠️ Common Mistakes

Not counting all possible combinations

If f(x)= Determinant of |[x³,x+a,b+x],[x-a,x²-x,c+x],[x-b,x-c,0]|, then

f(1)=0

f(2)=0

f(0)=0

f(-1)=0

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• 3×3 determinant formula • Factor common terms

📚 Problem Solving Strategy

Set up determinant

Calculate determinant

Set equal to zero

⚠️ Common Mistakes

Not evaluating determinant correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider determinant properties

This is some text inside of a div block.

💡 Key Formula

• 3×3 determinant formula • Factor common terms

📚 Problem Solving Strategy

Set up determinant

Calculate determinant

Set equal to zero

⚠️ Common Mistakes

Not evaluating determinant correctly

If f(x)=[[1-cosKx]/xsinx if x≠0; AND 1/2 if x=0] is continuous at x=0, then value of K is

±1/2

±2

±1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Continuity requires limit=function value • L'Hôpital's rule

📚 Problem Solving Strategy

Find limit as x→0

Use L'Hôpital's rule

Equate to f(0)

⚠️ Common Mistakes

Not applying L'Hôpital's rule correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider continuity condition

This is some text inside of a div block.

💡 Key Formula

• Continuity requires limit=function value • L'Hôpital's rule

📚 Problem Solving Strategy

Find limit as x→0

Use L'Hôpital's rule

Equate to f(0)

⚠️ Common Mistakes

Not applying L'Hôpital's rule correctly

If a₁a₂a₃...a₉ are in A.P. then value of determinant |[a₁,a₂,a₃],[a₄,a₅,a₆],[a₇,a₈,a₉]| is

(a₁+a₉)/2

a₁+a₉

loge·loge·e

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• A.P. formula: aₙ=a₁+(n-1)d • 3×3 determinant formula

📚 Problem Solving Strategy

Find pattern in A.P.

Set up determinant

Evaluate using properties

⚠️ Common Mistakes

Not recognizing AP pattern in determinant

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider arithmetic sequence properties

This is some text inside of a div block.

💡 Key Formula

• A.P. formula: aₙ=a₁+(n-1)d • 3×3 determinant formula

📚 Problem Solving Strategy

Find pattern in A.P.

Set up determinant

Evaluate using properties

⚠️ Common Mistakes

Not recognizing AP pattern in determinant

If α and β are roots of x²+x+1=0, then α²+β² is

(-1+i√3)/2

(-1-i√3)/2

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Vieta's formulas: α+β=-b/a, αβ=c/a

📚 Problem Solving Strategy

Apply Vieta's formulas: α+β=-1, αβ=1

Use (α+β)²=α²+2αβ+β²

Substitute values: α²+β²=(-1)²-2(1)=-1

⚠️ Common Mistakes

Confusing sum of squares with square of sum

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use Vieta's formulas for quadratic equations

This is some text inside of a div block.

💡 Key Formula

Vieta's formulas: α+β=-b/a, αβ=c/a

📚 Problem Solving Strategy

Apply Vieta's formulas: α+β=-1, αβ=1

Use (α+β)²=α²+2αβ+β²

Substitute values: α²+β²=(-1)²-2(1)=-1

⚠️ Common Mistakes

Confusing sum of squares with square of sum

Number of 4-digit numbers without repetition using 1,2,3,4,5,6,7 with two odd and two even digits

432

436

450

454

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Combinations formula: nCr=n!/(r!(n-r)!)

📚 Problem Solving Strategy

Count odd (4) and even (3) digits

Calculate combinations possible

Multiply arrangements: C(4,2)×C(3,2)×4!

⚠️ Common Mistakes

Not considering digit position arrangements

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Break into odd-even selection first

This is some text inside of a div block.

💡 Key Formula

Combinations formula: nCr=n!/(r!(n-r)!)

📚 Problem Solving Strategy

Count odd (4) and even (3) digits

Calculate combinations possible

Multiply arrangements: C(4,2)×C(3,2)×4!

⚠️ Common Mistakes

Not considering digit position arrangements

Number of terms in (x²+y²)²⁵-(x²-y²)²⁵ after simplification

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Binomial expansion: (a+b)ⁿ

📚 Problem Solving Strategy

Expand using binomial theorem

Note terms cancel due to subtraction

Count remaining terms after cancellation

⚠️ Common Mistakes

Missing term cancellations

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern in term cancellation

This is some text inside of a div block.

💡 Key Formula

Binomial expansion: (a+b)ⁿ

📚 Problem Solving Strategy

Expand using binomial theorem

Note terms cancel due to subtraction

Count remaining terms after cancellation

⚠️ Common Mistakes

Missing term cancellations

Third term of GP is 9. Product of first five terms is

3⁵

3⁹

3¹⁰

3¹²

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

GP formula: Tₙ=ar^(n-1)

📚 Problem Solving Strategy

Let a be first term, r be ratio

Use a₃=ar²=9

Multiply a×ar×ar²×ar³×ar⁴

⚠️ Common Mistakes

Not using GP properties correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use GP formula to find first term

This is some text inside of a div block.

💡 Key Formula

GP formula: Tₙ=ar^(n-1)

📚 Problem Solving Strategy

Let a be first term, r be ratio

Use a₃=ar²=9

Multiply a×ar×ar²×ar³×ar⁴

⚠️ Common Mistakes

Not using GP properties correctly

If AM and GM of roots of quadratic equation are 5 and 4 respectively, then quadratic equation is

x²-10x-16=0

x²+10x+16=0

x²-10x+16=0

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

AM=(α+β)/2, GM=√(αβ)

📚 Problem Solving Strategy

Use AM=(α+β)/2=5, GM=√(αβ)=4

Product of roots=16, sum=10

Form equation x²-10x+16=0

⚠️ Common Mistakes

Confusing signs

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Relate AM-GM to roots

This is some text inside of a div block.

💡 Key Formula

AM=(α+β)/2, GM=√(αβ)

📚 Problem Solving Strategy

Use AM=(α+β)/2=5, GM=√(αβ)=4

Product of roots=16, sum=10

Form equation x²-10x+16=0

⚠️ Common Mistakes

Confusing signs

Value of √24.99 is

4.999

4.899

5.001

4.897

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

(1+x)^n ≈ 1+nx for small x

📚 Problem Solving Strategy

Use binomial expansion

Keep first two terms

Compare with options

⚠️ Common Mistakes

Not using proper approximation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Think about square root approximation

This is some text inside of a div block.

💡 Key Formula

(1+x)^n ≈ 1+nx for small x

📚 Problem Solving Strategy

Use binomial expansion

Keep first two terms

Compare with options

⚠️ Common Mistakes

Not using proper approximation

If U is the universal set with 10 elements; A and B are two sets such that n(A)=50, n(B)=60, n(A∩B)=20 then n(A'∩B') =

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

n(A∪B) = n(A) + n(B) - n(A∩B)

📚 Problem Solving Strategy

Use the formula for complement of intersection

Apply n(A'∩B') = n(U) - n(A∪B)

Calculate n(A∪B) using addition principle

⚠️ Common Mistakes

Not using correct set operation laws

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider universal set size carefully

This is some text inside of a div block.

💡 Key Formula

n(A∪B) = n(A) + n(B) - n(A∩B)

📚 Problem Solving Strategy

Use the formula for complement of intersection

Apply n(A'∩B') = n(U) - n(A∪B)

Calculate n(A∪B) using addition principle

⚠️ Common Mistakes

Not using correct set operation laws

If P(n):2ⁿ

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Exponential vs factorial growth

📚 Problem Solving Strategy

Compare 2ⁿ and n! for small n

Check values systematically

Find first n satisfying condition

⚠️ Common Mistakes

Not checking systematically

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

List out values for comparison

This is some text inside of a div block.

💡 Key Formula

Exponential vs factorial growth

📚 Problem Solving Strategy

Compare 2ⁿ and n! for small n

Check values systematically

Find first n satisfying condition

⚠️ Common Mistakes

Not checking systematically

Everybody in a room shakes hands with everybody else. Total handshakes is 45. Total persons in room is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

nC₂ = n(n-1)/2

📚 Problem Solving Strategy

Use handshake formula nC₂ = 45

Solve n(n-1)/2 = 45

Get n² - n - 90 = 0, n = 10

⚠️ Common Mistakes

Not considering both roots

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Total handshakes is combinations of 2

This is some text inside of a div block.

💡 Key Formula

nC₂ = n(n-1)/2

📚 Problem Solving Strategy

Use handshake formula nC₂ = 45

Solve n(n-1)/2 = 45

Get n² - n - 90 = 0, n = 10

⚠️ Common Mistakes

Not considering both roots

The constant term in expansion of (x²-1/x²)¹⁶ is

¹⁶C₈

¹⁶C₇

¹⁶C₉

¹⁶C₁₀

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Binomial expansion

📚 Problem Solving Strategy

Find terms where powers sum to 0

Use binomial for (x²)ʳ(-1/x²)¹⁶⁻ʳ

r = 8 gives constant term

⚠️ Common Mistakes

Missing negative signs

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for x⁰ term

This is some text inside of a div block.

💡 Key Formula

Binomial expansion

📚 Problem Solving Strategy

Find terms where powers sum to 0

Use binomial for (x²)ʳ(-1/x²)¹⁶⁻ʳ

r = 8 gives constant term

⚠️ Common Mistakes

Missing negative signs

Line parallel to 3x-4y+2=0 passing through (-2,3) is

3x-4y+18=0

3x-4y-18=0

3x+4y+18=0

3x+4y-18=0

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

y=mx+c format

📚 Problem Solving Strategy

Use parallel line form 3x-4y+k=0

Substitute (-2,3)

Solve for k=18

⚠️ Common Mistakes

Sign errors in substitution

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Parallel lines have same slope

This is some text inside of a div block.

💡 Key Formula

y=mx+c format

📚 Problem Solving Strategy

Use parallel line form 3x-4y+k=0

Substitute (-2,3)

Solve for k=18

⚠️ Common Mistakes

Sign errors in substitution

If [(1-i)/(1+i)]⁹⁶ = a+ib then (a,b) is

(1,1)

(1,0)

(0,1)

(0,-1)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Complex conjugates

📚 Problem Solving Strategy

Simplify (1-i)/(1+i)

Find 96th power

Get 1+0i

⚠️ Common Mistakes

Power calculation errors

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Complex division trick

This is some text inside of a div block.

💡 Key Formula

Complex conjugates

📚 Problem Solving Strategy

Simplify (1-i)/(1+i)

Find 96th power

Get 1+0i

⚠️ Common Mistakes

Power calculation errors

The number of ways in which 5 girls and 3 boys can be seated in a row so that no two boys are together is

14040

14440

14000

14400

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Permutation formula

📚 Problem Solving Strategy

Use gap method

Calculate 5!×⁶C₃×3!

Get 120×20×6=14400

⚠️ Common Mistakes

Missing multiplication steps

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider gaps between girls

This is some text inside of a div block.

💡 Key Formula

Permutation formula

📚 Problem Solving Strategy

Use gap method

Calculate 5!×⁶C₃×3!

Get 120×20×6=14400

⚠️ Common Mistakes

Missing multiplication steps

If a,b,c are AP terms and x,y,z are GP terms, then x^(b-c)·y^(c-a)·z^(a-b) =

xyz

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

AP:d=common difference

📚 Problem Solving Strategy

Use AP: 2b=a+c

Use GP: y²=xz

Substitute and simplify

⚠️ Common Mistakes

Not seeing pattern

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern in powers

This is some text inside of a div block.

💡 Key Formula

AP:d=common difference

📚 Problem Solving Strategy

Use AP: 2b=a+c

Use GP: y²=xz

Substitute and simplify

⚠️ Common Mistakes

Not seeing pattern

If f(x) = [(√1+kx-√1-kx)/x] for x≠0, (2x+1)/(x-1) for 0≤x≤1 is continuous at x=0, then k=

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Continuity conditions

📚 Problem Solving Strategy

Find limit as x→0 using L'Hôpital's rule

Equate with function value

Solve for k to get k=-1

⚠️ Common Mistakes

L'Hôpital application errors

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Continuity means limits match

This is some text inside of a div block.

💡 Key Formula

Continuity conditions

📚 Problem Solving Strategy

Find limit as x→0 using L'Hôpital's rule

Equate with function value

Solve for k to get k=-1

⚠️ Common Mistakes

L'Hôpital application errors

If f(x) = [logₑx/(x-1)] for x≠1, k for x=1 is continuous at x=1, then k =

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

L'Hôpital's rule

📚 Problem Solving Strategy

Apply L'Hôpital's rule at x→1

Evaluate limit of logₑx/(x-1)

Get limit = 1, so k must equal 1

⚠️ Common Mistakes

L'Hôpital application errors

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use L'Hôpital for 0/0 form

This is some text inside of a div block.

💡 Key Formula

L'Hôpital's rule

📚 Problem Solving Strategy

Apply L'Hôpital's rule at x→1

Evaluate limit of logₑx/(x-1)

Get limit = 1, so k must equal 1

⚠️ Common Mistakes

L'Hôpital application errors

If A and B are finite sets and A ⊂ B, then

n(A ∪ B) = n(A)

n(A ∩ B) = n(B)

n(A ∪ B) = n(B)

n(A ∩ B) = ϕ

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

For A ⊂ B: n(A ∪ B) = n(B), n(A ∩ B) = n(A)

📚 Problem Solving Strategy

Since A ⊂ B, A is a proper subset of B, meaning all elements of A are in B

This means A ∩ B = A and A ∪ B = B

Therefore n(A ∪ B) = n(B)

⚠️ Common Mistakes

Confusing union and intersection properties

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Draw a Venn diagram to visualize subset relationship

This is some text inside of a div block.

💡 Key Formula

For A ⊂ B: n(A ∪ B) = n(B), n(A ∩ B) = n(A)

📚 Problem Solving Strategy

Since A ⊂ B, A is a proper subset of B, meaning all elements of A are in B

This means A ∩ B = A and A ∪ B = B

Therefore n(A ∪ B) = n(B)

⚠️ Common Mistakes

Confusing union and intersection properties

3 + 5 + 7 + ...... to n term is

n(n+2)

n(n-2)

n²

(n+1)²

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Sn = (n/2)[2a + (n-1)d]

📚 Problem Solving Strategy

Identify arithmetic sequence with a=3, d=2

Use arithmetic sequence sum formula Sn = (n/2)[2a + (n-1)d]

Simplify to get n(n+2)

⚠️ Common Mistakes

Not identifying AP correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern in first few terms

This is some text inside of a div block.

💡 Key Formula

Sn = (n/2)[2a + (n-1)d]

📚 Problem Solving Strategy

Identify arithmetic sequence with a=3, d=2

Use arithmetic sequence sum formula Sn = (n/2)[2a + (n-1)d]

Simplify to get n(n+2)

⚠️ Common Mistakes

Not identifying AP correctly

If [(1+i)/(1-i)]^m = 1, then the least positive integral value of m is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

i⁴ = 1, i² = -1

📚 Problem Solving Strategy

Convert complex fraction by rationalizing denominator

Find that [(1+i)/(1-i)] = i

Since i⁴ = 1, m = 4

⚠️ Common Mistakes

Not using complex number properties

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use properties of i

This is some text inside of a div block.

💡 Key Formula

i⁴ = 1, i² = -1

📚 Problem Solving Strategy

Convert complex fraction by rationalizing denominator

Find that [(1+i)/(1-i)] = i

Since i⁴ = 1, m = 4

⚠️ Common Mistakes

Not using complex number properties

If ⁿC₁₂ = ⁿC₈ then n is equal to

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

ⁿCᵣ = ⁿCₙ₋ᵣ

📚 Problem Solving Strategy

Use property ⁿCᵣ = ⁿCₙ₋ᵣ

Since ⁿC₁₂ = ⁿC₈, n-12 = 8

Solve to get n = 20

⚠️ Common Mistakes

Not using combination properties

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use combination formula symmetry

This is some text inside of a div block.

💡 Key Formula

ⁿCᵣ = ⁿCₙ₋ᵣ

📚 Problem Solving Strategy

Use property ⁿCᵣ = ⁿCₙ₋ᵣ

Since ⁿC₁₂ = ⁿC₈, n-12 = 8

Solve to get n = 20

⚠️ Common Mistakes

Not using combination properties

The total number of terms in expansion of (x+a)⁴⁷-(x-a)⁴⁷ after simplification is

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

(x+a)ⁿ-(x-a)ⁿ has (n+1)/2 terms when n is odd

📚 Problem Solving Strategy

Use odd power property difference of powers

Terms will be odd powers only up to 47

Count (47+1)/2 = 24 terms

⚠️ Common Mistakes

Not recognizing pattern

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern in expansion

This is some text inside of a div block.

💡 Key Formula

(x+a)ⁿ-(x-a)ⁿ has (n+1)/2 terms when n is odd

📚 Problem Solving Strategy

Use odd power property difference of powers

Terms will be odd powers only up to 47

Count (47+1)/2 = 24 terms

⚠️ Common Mistakes

Not recognizing pattern

Equation of line passing through (1,2) and perpendicular to y=3x-1 is

x+3y-7=0

x+3y+7=0

x+3y=0

x-3y=0

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1

📚 Problem Solving Strategy

Find slope of given line m₁=3

Perpendicular slope m₂=-1/3

Use point-slope form and simplify

⚠️ Common Mistakes

Wrong perpendicular slope

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Remember perpendicular slopes product = -1

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1

📚 Problem Solving Strategy

Find slope of given line m₁=3

Perpendicular slope m₂=-1/3

Use point-slope form and simplify

⚠️ Common Mistakes

Wrong perpendicular slope

The range of sec⁻¹x is

[-π/2,π/2]

(-π/2,π/2)

[0,π]

[0,π]-{π/2}

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

sec⁻¹x domain: (-∞,-1]∪[1,∞)

📚 Problem Solving Strategy

Consider domain of secx

Find principal values

Exclude π/2 as secπ/2 undefined

⚠️ Common Mistakes

Including undefined points

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Draw graph

This is some text inside of a div block.

💡 Key Formula

sec⁻¹x domain: (-∞,-1]∪[1,∞)

📚 Problem Solving Strategy

Consider domain of secx

Find principal values

Exclude π/2 as secπ/2 undefined

⚠️ Common Mistakes

Including undefined points

Binary operation * on R-{-1} defined by a*b=a/(b+1) is

* is associative and commutative

* is associative but not commutative

* is neither associative nor commutative

* is commutative but not associative

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Commutative: ab = ba, Associative: (ab)c = a(bc)

📚 Problem Solving Strategy

Check commutative: ab = ba: a/(b+1) ≠ b/(a+1)

Check associative: (ab)c ≠ a(bc)

Neither property holds

⚠️ Common Mistakes

Not checking both properties systematically

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Test with simple numbers

This is some text inside of a div block.

💡 Key Formula

Commutative: ab = ba, Associative: (ab)c = a(bc)

📚 Problem Solving Strategy

Check commutative: ab = ba: a/(b+1) ≠ b/(a+1)

Check associative: (ab)c ≠ a(bc)

Neither property holds

⚠️ Common Mistakes

Not checking both properties systematically

If f(x) is continuous at x=2, then value of K is

03-Apr

04-Mar

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

limₓ→c⁻f(x) = limₓ→c⁺f(x) = f(c)

📚 Problem Solving Strategy

Use continuity condition: limₓ→₂f(x)=f(2)

Evaluate limits from both sides

Solve for K to get 3/4

⚠️ Common Mistakes

Not checking both sided limits

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Remember continuity definition

This is some text inside of a div block.

💡 Key Formula

limₓ→c⁻f(x) = limₓ→c⁺f(x) = f(c)

📚 Problem Solving Strategy

Use continuity condition: limₓ→₂f(x)=f(2)

Evaluate limits from both sides

Solve for K to get 3/4

⚠️ Common Mistakes

Not checking both sided limits

If y=tan⁻¹[(sinx+cosx)/(cosx-sinx)], then dy/dx equals

1/2

π/4

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

d/dx[tan⁻¹(x)]=1/(1+x²)

📚 Problem Solving Strategy

Simplify fraction inside tan⁻¹

Use identities to show it equals tan(π/4+x)

Differentiate to get dy/dx=1

⚠️ Common Mistakes

Wrong chain rule application

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for standard angle forms

This is some text inside of a div block.

💡 Key Formula

d/dx[tan⁻¹(x)]=1/(1+x²)

📚 Problem Solving Strategy

Simplify fraction inside tan⁻¹

Use identities to show it equals tan(π/4+x)

Differentiate to get dy/dx=1

⚠️ Common Mistakes

Wrong chain rule application

If sinx=2t/(1+t²), tany=2t/(1-t²), then dy/dx equals

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

dy/dx=(dy/dt)÷(dx/dt)

📚 Problem Solving Strategy

Use parametric differentiation

Find dx/dt and dy/dt

Divide to get dy/dx=1

⚠️ Common Mistakes

Wrong parametric differentiation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use chain rule

This is some text inside of a div block.

💡 Key Formula

dy/dx=(dy/dt)÷(dx/dt)

📚 Problem Solving Strategy

Use parametric differentiation

Find dx/dt and dy/dt

Divide to get dy/dx=1

⚠️ Common Mistakes

Wrong parametric differentiation

Derivative of cos⁻¹(2x²-1) w.r.t cos⁻¹x is

-1/(2√1-x²)

2/x

1-x²

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

d/dx[cos⁻¹(x)]=-1/√(1-x²)

📚 Problem Solving Strategy

Let u=cos⁻¹(2x²-1) and v=1-cos⁻¹x

Find du/dx and dv/dx

Then du/dv = (du/dx)÷(dv/dx)=2

⚠️ Common Mistakes

Wrong derivative formulas

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use chain rule carefully

This is some text inside of a div block.

💡 Key Formula

d/dx[cos⁻¹(x)]=-1/√(1-x²)

📚 Problem Solving Strategy

Let u=cos⁻¹(2x²-1) and v=1-cos⁻¹x

Find du/dx and dv/dx

Then du/dv = (du/dx)÷(dv/dx)=2

⚠️ Common Mistakes

Wrong derivative formulas

If y=log(logx) then d²y/dx² equals

-(1+logx)/(x²log²x)

-(1+logx)/(x²logx)

(1+logx)/(x²log²x)

(1+logx)/(x²logx)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

d²y/dx²=d/dx(dy/dx)

📚 Problem Solving Strategy

First find dy/dx=1/(xlogx)

Then differentiate again

Simplify to get given option

⚠️ Common Mistakes

Wrong logarithm differentiation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use chain rule twice

This is some text inside of a div block.

💡 Key Formula

d²y/dx²=d/dx(dy/dx)

📚 Problem Solving Strategy

First find dy/dx=1/(xlogx)

Then differentiate again

Simplify to get given option

⚠️ Common Mistakes

Wrong logarithm differentiation

∫(cos2x-cos2θ)/(cosx-cosθ) dx equals

2(sinx+xcosθ)+c

2(sinx-xcosθ)+c

2(sinx+2xcosθ)+c

2(sinx-2xcosθ)+c

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

cos2A-cos2B=2(cosA+cosB)(cosA-cosB)

📚 Problem Solving Strategy

Use identity cos2A=2cos²A-1

Factor numerator

Integrate resulting form

⚠️ Common Mistakes

Not factoring correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Factor before integrating

This is some text inside of a div block.

💡 Key Formula

cos2A-cos2B=2(cosA+cosB)(cosA-cosB)

📚 Problem Solving Strategy

Use identity cos2A=2cos²A-1

Factor numerator

Integrate resulting form

⚠️ Common Mistakes

Not factoring correctly

Area of region bounded by y=x² and y=16 is

32/3 sq units

256/3 sq units

64/3 sq units

128/3 sq units

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Area = ∫(upper-lower) dx

📚 Problem Solving Strategy

Find intersection points by solving x²=16

Area = ∫₍₋₄⁾⁴(16-x²) dx

Evaluate to get 256/3

⚠️ Common Mistakes

Wrong limits

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Draw graph to visualize

This is some text inside of a div block.

💡 Key Formula

Area = ∫(upper-lower) dx

📚 Problem Solving Strategy

Find intersection points by solving x²=16

Area = ∫₍₋₄⁾⁴(16-x²) dx

Evaluate to get 256/3

⚠️ Common Mistakes

Wrong limits

The Set A has 4 elements and the set B has 5 elements then the number of injective mappings that can be defined from A to B is

144

120

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

P(n,r) = n!/(n-r)! where n=5, r=4

📚 Problem Solving Strategy

First identify that in injective mapping, each element of A must map to a unique element in B

Calculate number of choices: First element has 5 choices, second has 4, third has 3, and fourth has 2

Apply permutation formula: P(5,4) = 5!/(5-4)! = 5 × 4 × 3 × 2 = 120

⚠️ Common Mistakes

Students might confuse with total possible mappings or combinations

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Think of arranging 4 distinct elements in 5 positions

This is some text inside of a div block.

💡 Key Formula

P(n,r) = n!/(n-r)! where n=5, r=4

📚 Problem Solving Strategy

First identify that in injective mapping, each element of A must map to a unique element in B

Calculate number of choices: First element has 5 choices, second has 4, third has 3, and fourth has 2

Apply permutation formula: P(5,4) = 5!/(5-4)! = 5 × 4 × 3 × 2 = 120

⚠️ Common Mistakes

Students might confuse with total possible mappings or combinations

Let * be binary operation defined on R by a * b = (a+b)/4 ∀ a,b∈R then the operation is *

Commutative and Associative

Commutative but not Associative

Associative but not Commutative

Neither Associative nor Commutative

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

For commutativity: ab = ba For associativity: (ab)c = a(bc)

📚 Problem Solving Strategy

Check commutative property: ab = (a+b)/4 = (b+a)/4 = ba

Check associative property: (ab)c = ((a+b)/4 + c)/4 ≠ (a + (b+c)/4)/4 = a(bc)

Conclude that operation is commutative but not associative

⚠️ Common Mistakes

Not checking both properties independently

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Test with simple numbers to verify both properties

This is some text inside of a div block.

💡 Key Formula

For commutativity: ab = ba For associativity: (ab)c = a(bc)

📚 Problem Solving Strategy

Check commutative property: ab = (a+b)/4 = (b+a)/4 = ba

Check associative property: (ab)c = ((a+b)/4 + c)/4 ≠ (a + (b+c)/4)/4 = a(bc)

Conclude that operation is commutative but not associative

⚠️ Common Mistakes

Not checking both properties independently

The real part of (1-cosθ + i sinθ)⁻¹ is

1/2

1/(1+cosθ)

tanθ/2

cotθ/2

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

(a+bi)⁻¹ = (a-bi)/(a²+b²)

📚 Problem Solving Strategy

Multiply numerator and denominator by conjugate

Simplify (1-cosθ - i sinθ)/[(1-cosθ)² + sin²θ]

Real part is 1/2

⚠️ Common Mistakes

Not identifying conjugate method

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use complex conjugate method

This is some text inside of a div block.

💡 Key Formula

(a+bi)⁻¹ = (a-bi)/(a²+b²)

📚 Problem Solving Strategy

Multiply numerator and denominator by conjugate

Simplify (1-cosθ - i sinθ)/[(1-cosθ)² + sin²θ]

Real part is 1/2

⚠️ Common Mistakes

Not identifying conjugate method

The simplified form of i^n + i^(n+1) + i^(n+2) + i^(n+3) is

-1

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

i¹ = i, i² = -1, i³ = -i, i⁴ = 1

📚 Problem Solving Strategy

Write powers of i using cyclic pattern of 4

Group terms with same power

Sum equals 0

⚠️ Common Mistakes

Not using cyclic property

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use i⁴ = 1 pattern

This is some text inside of a div block.

💡 Key Formula

i¹ = i, i² = -1, i³ = -i, i⁴ = 1

📚 Problem Solving Strategy

Write powers of i using cyclic pattern of 4

Group terms with same power

Sum equals 0

⚠️ Common Mistakes

Not using cyclic property

The maximum value of (1/x)^x is

e^e

e^(1/e)

(1/e)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

d/dx(ln(x^n)) = n/x

📚 Problem Solving Strategy

Take ln of expression: ln((1/x)^x) = -x ln(x)

Differentiate and set equal to zero: -ln(x) - 1 = 0

Solve to get x = 1/e, max value = e^(1/e)

⚠️ Common Mistakes

Not using logarithmic differentiation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use logarithmic differentiation

This is some text inside of a div block.

💡 Key Formula

d/dx(ln(x^n)) = n/x

📚 Problem Solving Strategy

Take ln of expression: ln((1/x)^x) = -x ln(x)

Differentiate and set equal to zero: -ln(x) - 1 = 0

Solve to get x = 1/e, max value = e^(1/e)

⚠️ Common Mistakes

Not using logarithmic differentiation

The sum of 1st n terms of the series 1²/1 + 1²+2²/1+2 + 1²+2²+3²/1+2+3 +...... is

n(n+2)/3

n(n-2)/3

n(n-2)/6

n(n+2)/6

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Sum of squares = n(n+1)(2n+1)/6

📚 Problem Solving Strategy

Find pattern in numerator and denominator

Use sum of squares formula for numerator

Get n(n-2)/3

⚠️ Common Mistakes

Not identifying correct pattern

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for pattern in series

This is some text inside of a div block.

💡 Key Formula

Sum of squares = n(n+1)(2n+1)/6

📚 Problem Solving Strategy

Find pattern in numerator and denominator

Use sum of squares formula for numerator

Get n(n-2)/3

⚠️ Common Mistakes

Not identifying correct pattern

The 11th term in the expansion of [x + 1/√x]¹⁴ is

999/x

1001/x

1001

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

nCr = n!/(r!(n-r)!)

📚 Problem Solving Strategy

Use binomial expansion

Find general term: ¹⁴Cᵣ x^(r-r/2)

Put r=10 for 11th term

⚠️ Common Mistakes

Error in power calculation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use binomial theorem

This is some text inside of a div block.

💡 Key Formula

nCr = n!/(r!(n-r)!)

📚 Problem Solving Strategy

Use binomial expansion

Find general term: ¹⁴Cᵣ x^(r-r/2)

Put r=10 for 11th term

⚠️ Common Mistakes

Error in power calculation

If the straight lines 2x + 3y - 3 = 0 and x + ky + 7 = 0 are perpendicular, then the value of k is

2/3

3/2

-2/3

-3/2

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1 for perpendicular lines

📚 Problem Solving Strategy

Find slopes of both lines: m₁ = -2/3 and m₂ = -1/k

Use perpendicular lines condition: m₁m₂ = -1

Solve to get k = -2/3

⚠️ Common Mistakes

Not using negative reciprocal condition

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Perpendicular lines have negative reciprocal slopes

This is some text inside of a div block.

💡 Key Formula

m₁m₂ = -1 for perpendicular lines

📚 Problem Solving Strategy

Find slopes of both lines: m₁ = -2/3 and m₂ = -1/k

Use perpendicular lines condition: m₁m₂ = -1

Solve to get k = -2/3

⚠️ Common Mistakes

Not using negative reciprocal condition

Write the set builder form A = {-1, 1}

A = {x : x is a real number}

A = {x : x is an integer}

A = {x : x is a root of x² = 1}

A = {x : x is a root of x² + 1 = 0}

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Set builder notation rules • x² = 1 ⟺ x = ±1

📚 Problem Solving Strategy

Identify that -1 and 1 are roots of x² = 1

Check no other values satisfy

Verify set builder notation

⚠️ Common Mistakes

Choosing too broad conditions

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for equation that gives exactly these values

This is some text inside of a div block.

💡 Key Formula

• Set builder notation rules • x² = 1 ⟺ x = ±1

📚 Problem Solving Strategy

Identify that -1 and 1 are roots of x² = 1

Check no other values satisfy

Verify set builder notation

⚠️ Common Mistakes

Choosing too broad conditions

If α and β are roots of x² - ax + b = 0, then a² + b² = 0, then a² + b² equals

a² - 2b²

2a² - b²

a² - b²

a² + b²

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• α + β = a • αβ = b

📚 Problem Solving Strategy

Use Vieta's formulas: α + β = a, αβ = b

Square both relations

Substitute in expression

⚠️ Common Mistakes

Not using Vieta's formulas correctly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use relationships between roots and coefficients

This is some text inside of a div block.

💡 Key Formula

• α + β = a • αβ = b

📚 Problem Solving Strategy

Use Vieta's formulas: α + β = a, αβ = b

Square both relations

Substitute in expression

⚠️ Common Mistakes

Not using Vieta's formulas correctly

If 2nd and 5th terms of G.P. are 24 and 3 respectively, find sum of first 6 terms

189/2

189/5

179/2

2/189

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• ar = a₁r^(n-1) • Sn = a(1-r^n)/(1-r)

📚 Problem Solving Strategy

Use ratio formula: ar = 24, ar⁴ = 3

Find r = 1/2, a = 48

Sum formula: a(1-r⁶)/(1-r)

⚠️ Common Mistakes

Wrong ratio calculation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use geometric sequence properties

This is some text inside of a div block.

💡 Key Formula

• ar = a₁r^(n-1) • Sn = a(1-r^n)/(1-r)

📚 Problem Solving Strategy

Use ratio formula: ar = 24, ar⁴ = 3

Find r = 1/2, a = 48

Sum formula: a(1-r⁶)/(1-r)

⚠️ Common Mistakes

Wrong ratio calculation

Middle term of expansion of (10/x + x/10)¹⁰ is

⁷C₅

⁸C₅

⁹C₅

¹⁰C₅

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Middle term: r = n/2 • Binomial theorem

📚 Problem Solving Strategy

For middle term, r = n/2 = 5

Use binomial expansion

Find coefficient of x⁰

⚠️ Common Mistakes

Wrong power calculation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider even power expansion

This is some text inside of a div block.

💡 Key Formula

• Middle term: r = n/2 • Binomial theorem

📚 Problem Solving Strategy

For middle term, r = n/2 = 5

Use binomial expansion

Find coefficient of x⁰

⚠️ Common Mistakes

Wrong power calculation

The value of sin⁻¹(2√2/3) + sin⁻¹(1/3)

π/6

π/2

π/4

2π/3

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• sin⁻¹x + cos⁻¹x = π/2 • sin⁻¹(2√2/3)² + sin⁻¹(1/3)² = 1

📚 Problem Solving Strategy

Consider triangle with angles sin⁻¹(2√2/3) and sin⁻¹(1/3)

Use trigonometric identity sin⁻¹x + cos⁻¹x = π/2

Verify sum equals π/2

⚠️ Common Mistakes

Not recognizing complementary angles

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Draw triangle to visualize angles

This is some text inside of a div block.

💡 Key Formula

• sin⁻¹x + cos⁻¹x = π/2 • sin⁻¹(2√2/3)² + sin⁻¹(1/3)² = 1

📚 Problem Solving Strategy

Consider triangle with angles sin⁻¹(2√2/3) and sin⁻¹(1/3)

Use trigonometric identity sin⁻¹x + cos⁻¹x = π/2

Verify sum equals π/2

⚠️ Common Mistakes

Not recognizing complementary angles

Function x^x; x>0 is strictly increasing at

∀x∈ℝ

x<1/e

x>1/e

x<0

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

d/dx(x^x)=x^x(1+lnx)

📚 Problem Solving Strategy

Take ln, differentiate

Find where derivative positive

Get x>1/e condition

⚠️ Common Mistakes

Wrong differentiation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use logarithmic differentiation

This is some text inside of a div block.

💡 Key Formula

d/dx(x^x)=x^x(1+lnx)

📚 Problem Solving Strategy

Take ln, differentiate

Find where derivative positive

Get x>1/e condition

⚠️ Common Mistakes

Wrong differentiation

If f(x) continuous, f(x) = 3x-8 if x≤5, f(x) = 2k if x>5. Find k

2/7

3/7

4/7

7/2

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• Continuity: limx→a⁻ f(x) = limx→a⁺ f(x)

📚 Problem Solving Strategy

Use continuity at x=5

3(5)-8 = 2k

Solve k = 7/2

⚠️ Common Mistakes

Wrong continuity application

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider continuity condition

This is some text inside of a div block.

💡 Key Formula

• Continuity: limx→a⁻ f(x) = limx→a⁺ f(x)

📚 Problem Solving Strategy

Use continuity at x=5

3(5)-8 = 2k

Solve k = 7/2

⚠️ Common Mistakes

Wrong continuity application

If 1,w,w² are cube roots of unity, (1-w+w²)(1+w-w²) equals

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• w³ = 1 • 1+w+w² = 0

📚 Problem Solving Strategy

Use w³ = 1

Use 1+w+w² = 0

Multiply and simplify to 4

⚠️ Common Mistakes

Wrong roots of unity

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider properties of cube roots

This is some text inside of a div block.

💡 Key Formula

• w³ = 1 • 1+w+w² = 0

📚 Problem Solving Strategy

Use w³ = 1

Use 1+w+w² = 0

Multiply and simplify to 4

⚠️ Common Mistakes

Wrong roots of unity

Solve for x: tan⁻¹((1-x)/(1+x)) = (1/2)tan⁻¹x, x > 0

√3

-1

1/√3

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• tan⁻¹((1-x)/(1+x)) formula • tan(A/2) formula

📚 Problem Solving Strategy

Use tan(A+B) formula

Let left side = θ, right side = φ

Solve to get x = 1/√3

⚠️ Common Mistakes

Wrong inverse trig formulas

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider inverse tan properties

This is some text inside of a div block.

💡 Key Formula

• tan⁻¹((1-x)/(1+x)) formula • tan(A/2) formula

📚 Problem Solving Strategy

Use tan(A+B) formula

Let left side = θ, right side = φ

Solve to get x = 1/√3

⚠️ Common Mistakes

Wrong inverse trig formulas

Line through (2,2) perpendicular to 3x + y = 3, y-intercept

1/3

2/3

4/3

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

• m₁m₂ = -1 • Point-slope form

📚 Problem Solving Strategy

Find perpendicular slope

Write line equation

Find y-intercept

⚠️ Common Mistakes

Wrong perpendicular slope

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider perpendicular lines

This is some text inside of a div block.

💡 Key Formula

• m₁m₂ = -1 • Point-slope form

📚 Problem Solving Strategy

Find perpendicular slope

Write line equation

Find y-intercept

⚠️ Common Mistakes

Wrong perpendicular slope

If A = [2-k 2; 1 3-k] is singular 2x2 matrix, then the value of 5k - k² is equal to

-4

-6

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Det[2×2] = ad - bc

📚 Problem Solving Strategy

For singular matrix, determinant = 0. Calculate det(A) = (2-k)(3-k) - 2(1) = 0

Expand: (2-k)(3-k) - 2 = 6-5k+k² - 2 = 0

Solve k² - 5k + 4 = 0, therefore 5k - k² = 4

⚠️ Common Mistakes

Not recognizing singular matrix condition

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

For singular matrix, focus on determinant = 0

This is some text inside of a div block.

💡 Key Formula

Det[2×2] = ad - bc

📚 Problem Solving Strategy

For singular matrix, determinant = 0. Calculate det(A) = (2-k)(3-k) - 2(1) = 0

Expand: (2-k)(3-k) - 2 = 6-5k+k² - 2 = 0

Solve k² - 5k + 4 = 0, therefore 5k - k² = 4

⚠️ Common Mistakes

Not recognizing singular matrix condition

If Δ = [1 a a²; 1 b b²; 1 c c²] and Δ₁ = [1 1 1; bc ca ab; a b c], both are 3x3 matrix, then

Δ₁ ≠ Δ

Δ₁ = -Δ

Δ₁ = Δ

Δ₁ = 3Δ

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Determinant properties

📚 Problem Solving Strategy

Evaluate determinant of given matrices

Compare corresponding elements and properties

Show that Δ₁ = -Δ using determinant properties

⚠️ Common Mistakes

Not recognizing pattern in matrices

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use determinant properties and compare matrices

This is some text inside of a div block.

💡 Key Formula

Determinant properties

📚 Problem Solving Strategy

Evaluate determinant of given matrices

Compare corresponding elements and properties

Show that Δ₁ = -Δ using determinant properties

⚠️ Common Mistakes

Not recognizing pattern in matrices

The modulus of the complex number ((1+i²)(1+3i))/((2-6i)(2-2i)) is

1/√2

√2/4

4/√2

2/√2

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Complex number modulus formula

📚 Problem Solving Strategy

Simplify i² = -1 in numerator

Multiply numerator and denominator by conjugates

Calculate modulus

⚠️ Common Mistakes

Wrong conjugate multiplication

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use conjugate method

This is some text inside of a div block.

💡 Key Formula

Complex number modulus formula

📚 Problem Solving Strategy

Simplify i² = -1 in numerator

Multiply numerator and denominator by conjugates

Calculate modulus

⚠️ Common Mistakes

Wrong conjugate multiplication

Which of the following is an empty set?

{x:x²-9=0,x∈R}

{x:x²=x+2,x∈R}

{x:x²-1=0,x∈R}

{x:x²+1=0,x∈R}

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Quadratic equation properties

📚 Problem Solving Strategy

Check if equation x²+1=0 has real solutions

Since x²+1>0 for all real x

No real solutions exist, set is empty

⚠️ Common Mistakes

Not checking domain

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider domain restrictions

This is some text inside of a div block.

💡 Key Formula

Quadratic equation properties

📚 Problem Solving Strategy

Check if equation x²+1=0 has real solutions

Since x²+1>0 for all real x

No real solutions exist, set is empty

⚠️ Common Mistakes

Not checking domain

A line passes through (2,2) and is perpendicular to the line 3x+y=3. Its y-intercept is

4/3

1/3

2/3

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Perpendicular slopes product=-1

📚 Problem Solving Strategy

Use perpendicular slopes: m₁m₂=-1

Form equation using point-slope

Find y-intercept=4/3

⚠️ Common Mistakes

Wrong slope calculation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use perpendicular lines property

This is some text inside of a div block.

💡 Key Formula

Perpendicular slopes product=-1

📚 Problem Solving Strategy

Use perpendicular slopes: m₁m₂=-1

Form equation using point-slope

Find y-intercept=4/3

⚠️ Common Mistakes

Wrong slope calculation

If n is even and the middle term in the expansion of (x²+1/x)ⁿ is 924x⁶, then n is equal to

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Binomial expansion formula

📚 Problem Solving Strategy

In middle term, powers of x should balance

Use binomial theorem: ⁿCᵣ coefficient = 924

Solve to get n=12

⚠️ Common Mistakes

Wrong coefficient calculation

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Consider power balance in middle term

This is some text inside of a div block.

💡 Key Formula

Binomial expansion formula

📚 Problem Solving Strategy

In middle term, powers of x should balance

Use binomial theorem: ⁿCᵣ coefficient = 924

Solve to get n=12

⚠️ Common Mistakes

Wrong coefficient calculation

nᵗʰ term of the series 1+3/7+5/7²+1/7²+... is

(2n-1)/7ⁿ

(2n+1)/7ⁿ⁻¹

(2n-1)/7ⁿ⁻¹

(2n+1)/7ⁿ

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

Sequence general term formula

📚 Problem Solving Strategy

Find pattern in numerator: 2n-1

Find pattern in denominator: 7ⁿ⁻¹

General term = (2n-1)/7ⁿ⁻¹

⚠️ Common Mistakes

Wrong pattern recognition

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Look for arithmetic-geometric pattern

This is some text inside of a div block.

💡 Key Formula

Sequence general term formula

📚 Problem Solving Strategy

Find pattern in numerator: 2n-1

Find pattern in denominator: 7ⁿ⁻¹

General term = (2n-1)/7ⁿ⁻¹

⚠️ Common Mistakes

Wrong pattern recognition

If p(1/q+1/r), q(1/r+1/p), r(1/p+1/q) are in A.P., then p,q,r

are in A.P.

are not in G.P.

are not in A.P.

are in G.P.

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

A.P. properties

📚 Problem Solving Strategy

Let terms be a,b,c in A.P.

Express in terms of p,q,r

Show p,q,r form A.P.

⚠️ Common Mistakes

Wrong term arrangement

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Convert to common denominator

This is some text inside of a div block.

💡 Key Formula

A.P. properties

📚 Problem Solving Strategy

Let terms be a,b,c in A.P.

Express in terms of p,q,r

Show p,q,r form A.P.

⚠️ Common Mistakes

Wrong term arrangement

If 3x + i(4x - y) = 6 - i where x and y are real numbers, then the values of x and y respectively

3, 9

2, 4

2, 9

3, 4

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

z₁ = z₂ ⟺ Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)

📚 Problem Solving Strategy

Separate real and imaginary parts: 3x = 6 and 4x - y = -1

From real part, solve for x: x = 2

Substitute x in imaginary part to find y: y = 9

⚠️ Common Mistakes

Not separating real and imaginary parts properly

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Always separate complex equation into real and imaginary parts

This is some text inside of a div block.

💡 Key Formula

z₁ = z₂ ⟺ Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)

📚 Problem Solving Strategy

Separate real and imaginary parts: 3x = 6 and 4x - y = -1

From real part, solve for x: x = 2

Substitute x in imaginary part to find y: y = 9

⚠️ Common Mistakes

Not separating real and imaginary parts properly

If a₁,a₂,a₃,...a₁₀ is a geometric progression and a₃/a₁ = 25, then a₉/a₅ equals

3(5²)

5⁴

5³

2(5²)

🥳 Wohoo! Correct answer

This is some text inside of a div block.

💡 Key Formula

ar = a₁r^(r-1) for GP

📚 Problem Solving Strategy

Find r² = a₃/a₁ = 25, so r = 5

Note that a₉/a₅ = r⁴

Calculate r⁴ = 5⁴

⚠️ Common Mistakes

Not identifying power of common ratio

😢 Uh oh! Incorrect answer, Try again

🗝 Hint

Use GP property: ar/ar-n = r^n

This is some text inside of a div block.

💡 Key Formula

ar = a₁r^(r-1) for GP

📚 Problem Solving Strategy

Find r² = a₃/a₁ = 25, so r = 5

Note that a₉/a₅ = r⁴

Calculate r⁴ = 5⁴

⚠️ Common Mistakes

Not identifying power of common ratio