Knowledge Check : Algebra
Knowledge Check
Introduction to Algebra
Q1: In a school, triangular flower beds are arranged such that the 1st bed has 3 flowers, 2nd has 5, 3rd has 7 and so on. What is the general rule for the number of flowers in the nth bed?
A) 2n – 1
B) 2n + 1
C) n + 2
D) n² + 1
Q2: A farmer uses x litres of water per plant. If he has 25 plants, the algebraic expression for total water used is
A) x + 25
B) 25x
C) x – 25
D) x/25
Q3: How many terms are there in the expression 9a²b – 4ab + 7a – 3?
A) 2
B) 3
C) 4
D) 5
Q4: In the term –7xy², the coefficient of xy is
A) –7y
B) –7
C) –7x
D) 7
Q5: Which of the following is a monomial?
A) 5x + 3
B) 8x³
C) x² – 2x + 1
D) 4xy + z
Q6: If p = 3, q = –2, find the value of 2p² – 3pq + 5q.
A) 32
B) 28
C) 40
D) 44
Q7: The perimeter of a square with side length ‘s’ cm is expressed as
A) s²
B) 4s
C) s + 4
D) 2s
Q8: In the expression 5x²y + 3xy² – 2x + y, which are like terms?
A) 5x²y and 3xy²
B) 5x²y and –2x
C) 3xy² and y
D) None
Q9: A trinomial expression always has
A) 2 terms
B) 3 terms
C) 4 terms
D) 1 term
Q10: If x = 5, the value of expression (x + 3)(x – 2) is
A) 21
B) 24
C) 28
D) 15
Q11: The number of factors in the term 12xyz is
A) 3
B) 4
C) 5
D) 6
Q12: In a rangoli pattern, the number of dots are 4, 9, 16, 25, … The rule for nth figure is
A) n²
B) (n+1)²
C) 2n + 2
D) n(n+1)/2
Q13: The variable in the expression 7 + 5p – 3q is
A) 7
B) p and q
C) 5
D) 3
Q14: If the length of a rectangle is (l + 5) cm and breadth is (l – 3) cm, its perimeter is
A) 2l + 2
B) 2l + 10
C) 4l + 4
D) 4l + 2
Q15: Which is NOT a polynomial?
A) x² + 3x + 1
B) 5/x + 2
C) x³ – 2
D) 4x⁴
Q16: If a = 1, b = –1, then value of a³ + b³ + 3ab(a + b) is
A) 0
B) 1
C) –1
D) 2
Q17: The numerical coefficient in –9xyz is
A) –9
B) 9
C) –xyz
D) xyz
Q18: In a matchstick pattern, 1st figure uses 3, 2nd uses 6, 3rd uses 9 sticks. The rule for nth figure is
A) 3n
B) n + 2
C) 2n + 1
D) n(n+1)/2
Q19: If x = –3, then the value of x² – 5x + 4 is
A) 28
B) 22
C) 10
D) 4
Q20: An expression with exactly two unlike terms is called
A) Monomial
B) Binomial
C) Trinomial
D) Polynomial
Expressions and Operations
Q21: Simplify: 3x + 7x
A) 10x
B) 21x
C) 4x
D) 3x²
Q22: Find (2x)(3x)
A) 6x
B) 5x²
C) 6x²
D) 6x³
Q23: Expand: (x + 5)²
A) x² + 10x + 25
B) x² + 25
C) x² + 5x + 25
D) x² − 25
Q24: Simplify: 5x − 2x + 3
A) 3x + 3
B) 7x + 3
C) 5x + 1
D) 2x + 3
Q25: Find the value of 4a² when a = 3
A) 12
B) 18
C) 36
D) 48
Q26: Simplify: 6x ÷ 3
A) 2x
B) 3x
C) x
D) 6x
Q27: Simplify: (a + b)(a − b)
A) a² + b²
B) a² − b²
C) 2ab
D) a² + 2ab + b²
Q28: Which of the following represents a perfect square identity?
A) (a + b)²
B) a² − b²
C) 2ab
D) ab + ba
Q29: If 2⁻² = ?
A) 4
B) 1/4
C) -4
D) 2
Q30: Simplify: x³ × x²
A) x⁵
B) x⁶
C) x
D) x²
Q31: Simplify: (2x³) ÷ (x)
A) x²
B) 2x²
C) 2x³
D) x³
Q32: Simplify: 7x × 4
A) 28x
B) 11x
C) 21x
D) 7x²
Q33: Which expression is equivalent to (x + y)²?
A) x² + y²
B) 2xy + x² + y²
C) x² − 2xy + y²
D) 2x² + y²
Q34: Simplify: (6a + 2a)
A) 8a
B) 12a
C) 3a
D) 10a
Q35: Simplify: (x + 2)(x + 3)
A) x² + 5x + 6
B) x² + 6x + 5
C) x² + 2x + 3
D) x² − 1
Q36: Simplify: a⁵ ÷ a²
A) a³
B) a⁷
C) a²
D) a⁵
Q37: Simplify: (4x)(5y)
A) 20xy
B) 9xy
C) 20x²y²
D) 45xy
Q38: Simplify: (x − 3)²
A) x² − 6x + 9
B) x² + 9
C) x² − 9
D) x² + 6x + 9
Q39: What is the standard form of 500,000?
A) 5 × 10³
B) 5 × 10⁴
C) 5 × 10⁵
D) 5 × 10⁶
Q40: Simplify: (a²)³
A) a⁵
B) a⁶
C) a⁸
D) a⁹
Polynomials
Q41: The degree of the polynomial 7x⁵ – 2x³ + 9 is
A) 3
B) 4
C) 5
D) 7
Q42: If p(x) = x³ – 6x² + 11x – 6 and p(1) = 0, then (x–1) is a factor of p(x). This is an application of
A) Remainder theorem
B) Factor theorem
C) Division algorithm
D) Zeroes of polynomial
Q43: A cubic polynomial can have at most how many zeroes?
A) 1
B) 2
C) 3
D) 4
Q44: The zero of the polynomial 5x – 15 is
A) 3
B) –3
C) 15
D) 5
Q45: If one zero of the quadratic polynomial x² – 5x + k is 2, then the value of k is
A) 6
B) –6
C) 10
D) –10
Q46: The graph of a linear polynomial is
A) Parabola
B) Straight line
C) Curved line
D) Zig-zag line
Q47: For the polynomial p(x) = 2x³ – 3x² + 5x – 4, p(0) =
A) –4
B) 4
C) 0
D) –3
Q48: The number of zeroes that a quadratic polynomial can have is
A) 0, 1 or 2
B) Exactly 2
C) Exactly 1
D) Infinite
Q49: If α and β are zeroes of x² – 6x + 8, then α + β =
A) 6
B) –6
C) 8
D) –8
Q50: A polynomial of degree 0 is called
A) Zero polynomial
B) Constant polynomial
C) Linear polynomial
D) Quadratic polynomial
Q51: The geometrical meaning of zeroes of a cubic polynomial is
A) Number of times the graph intersects x-axis
B) Number of turning points
C) Area under curve
D) Slope of tangent
Q52: If p(x) = 4x² + 5x – 9, then p(–1) + p(1) =
A) –12
B) –10
C) –2
D) 0
Q53: The shape of the graph of y = x³ – x is
A) Parabola
B) Straight line
C) S-shaped cubic
D) V-shaped
Q54: If (x + 2) is a factor of x³ + 2x² – 5x – 6, then p(–2) must be
A) 0
B) 2
C) –2
D) 6
Q55: The value of the polynomial 3x⁴ – 4x³ + 2x – 7 at x = 0 is
A) –7
B) 7
C) 0
D) 2
Q56: A quadratic polynomial whose zeroes are 3 and –2 is
A) x² – x – 6
B) x² + x – 6
C) x² – x + 6
D) x² + x + 6
Q57: The graph of y = ax² + bx + c touches the x-axis at one point when discriminant is
A) Positive
B) Zero
C) Negative
D) Any value
Q58: Number of real zeroes of the polynomial x² + 1 is
A) 0
B) 1
C) 2
D) Infinite
Q59: If α, β, γ are zeroes of x³ – 6x² + 11x – 6, then αβγ =
A) 6
B) –6
C) 11
D) –11
Q60: The graph of a quadratic polynomial is always a
A) Straight line
B) Parabola
C) Hyperbola
D) Circle
Factorization and Division
Q61: Factorize: x² − 9
A) (x + 3)(x − 3)
B) (x + 9)(x − 1)
C) (x − 9)(x + 1)
D) (x − 3)²
Q62: Factorize: x² + 5x + 6
A) (x + 1)(x + 6)
B) (x + 2)(x + 3)
C) (x + 5)(x + 6)
D) (x − 2)(x − 3)
Q63: Factorize: 4a² − 9b²
A) (2a − 3b)(2a + 3b)
B) (4a − 9b)(a + b)
C) (2a − b)(2a + b)
D) (a − 3b)(a + 3b)
Q64: Factorize: x² − 5x
A) x(x − 5)
B) x(x + 5)
C) x²(1 − 5)
D) (x − 5)(x − 5)
Q65: Factorize: 9y² − 6y
A) 3y(3y − 2)
B) 3(3y − 2)
C) 9y(y − 6)
D) y(9y − 6y)
Q66: Factorize: x² + 7x + 10
A) (x + 2)(x + 5)
B) (x + 3)(x + 7)
C) (x − 2)(x − 5)
D) (x + 10)(x + 1)
Q67: Divide: (6x² ÷ 3x)
A) 2x
B) 3x
C) 2x²
D) x²
Q68: Divide: (8a³ ÷ 2a)
A) 4a²
B) 6a²
C) a³
D) 16a
Q69: Factorize: x³ − x
A) x(x − 1)(x + 1)
B) (x − 1)³
C) x(x − x)
D) (x + 1)(x + 2)
Q70: Factorize: a² − 4ab + 4b²
A) (a − 2b)²
B) (a + 2b)²
C) a(a − 4b)
D) (a − b)²
Q71: Factorize: 16x² − 1
A) (4x − 1)(4x + 1)
B) (16x − 1)(x + 1)
C) (8x − 1)(2x + 1)
D) (4x − 1)²
Q72: Factorize: y² + 8y + 16
A) (y + 4)²
B) (y + 8)(y + 2)
C) (y + 1)²
D) (y + 16)(y + 1)
Q73: Divide: (15x² ÷ 5x)
A) 3x
B) 3x²
C) 10x
D) 15x
Q74: Factorize: x² − 16
A) (x − 4)²
B) (x + 4)²
C) (x − 4)(x + 4)
D) (x + 4)(x + 4)
Q75: Factorize: 5x² − 20x
A) 5x(x − 4)
B) 5(x − 4)
C) x(5x − 20)
D) (x − 5)(x + 4)
Q76: Factorize: m² + 10m + 25
A) (m + 5)²
B) (m + 10)(m + 5)
C) m(m + 25)
D) (m + 1)²
Q77: Divide: (9a³ ÷ 3a²)
A) 3a
B) 3a²
C) a³
D) 27a
Q78: Factorize: 2x² + 6x
A) 2x(x + 3)
B) x(2x + 6)
C) (x + 2)(x + 3)
D) 2(x² + 3)
Q79: Factorize: x² + x − 6
A) (x + 3)(x − 2)
B) (x + 6)(x − 1)
C) (x − 6)(x + 1)
D) (x − 3)(x + 2)
Q80: Divide: (12y² ÷ 4y)
A) 3y
B) 3y²
C) 8y
D) 4y²
Progressions
Q81: In an arithmetic progression, the first term is 7 and the common difference is 4. The 10th term is
A) 39
B) 43
C) 47
D) 51
Q82: The sum of the first 20 terms of the AP 5, 9, 13, 17, … is
A) 840
B) 860
C) 880
D) 900
Q83: Which of the following is NOT an arithmetic progression?
A) 3, 7, 11, 15
B) 2, 5, 8, 11
C) 1, 4, 9, 16
D) –5, –2, 1, 4
Q84: In a school auditorium, seats are arranged such that the first row has 20 seats, and each next row has 3 more seats. How many seats are in the 15th row?
A) 62
B) 65
C) 68
D) 71
Q85: The nth term of the GP 3, 12, 48, 192, … is
A) 3×4ⁿ
B) 3×4ⁿ⁻¹
C) 12×4ⁿ⁻¹
D) 3×3ⁿ⁻¹
Q86: A farmer saves ₹100 in the first month, ₹110 in the second, ₹120 in the third, and so on. How much will he save in the 25th month?
A) ₹340
B) ₹350
C) ₹360
D) ₹370
Q87: The sum of the first 15 terms of an AP whose first term is 8 and common difference is –3 is
A) –195
B) –165
C) –135
D) –105
Q88: The common ratio of the GP 5, –10, 20, –40, … is
A) 2
B) –2
C) 1/2
D) –1/2
Q89: If three numbers 5, x, 45 are in GP, then x =
A) 15
B) 20
C) 25
D) 30
Q90: The 8th term of the AP 2, 7, 12, 17, … is
A) 37
B) 39
C) 41
D) 43
Q91: In a GP, if a = 4 and r = 3, the 6th term is
A) 2916
B) 1458
C) 729
D) 243
Q92: The sum of first n terms of the AP 10, 7, 4, … is 33. The value of n is
A) 6
B) 7
C) 8
D) 11
Q93: Which term of the GP 2, 6, 18, 54, … is 1458?
A) 7th
B) 8th
C) 9th
D) 10th
Q94: The common difference of an AP whose first term is –5 and 10th term is 22 is
A) 3
B) 4
C) 5
D) 6
Q95: In a village fair, a game gives prizes in GP: ₹10, ₹30, ₹90, ₹270, … What is the 5th prize?
A) ₹810
B) ₹2430
C) ₹7290
D) ₹21870
Q96: The sum of first 10 terms of AP 1 + 4 + 7 + 10 + … is
A) 145
B) 155
C) 165
D) 175
Q97: If a, b, c are in AP, then 2b =
A) a + c
B) a – c
C) a × c
D) a/c
Q98: The next term of the GP 16, –8, 4, –2, … is
A) 1
B) –1
C) 2
D) –2
Q99: A student scores 20, 25, 30, 35, … marks in monthly tests. His total marks after 12 tests will be
A) 390
B) 420
C) 450
D) 480
Q100: Which of the following sequences has common ratio 1?
A) 5, 5, 5, 5
B) 2, 4, 6, 8
C) 3, 9, 27, 81
D) 10, 7, 4, 1
Progressions
Q101: Find the 5th term of the AP: 3, 7, 11, …
A) 15
B) 19
C) 23
D) 27
Q102: Find the common difference of AP: 12, 9, 6, 3…
A) −1
B) −2
C) −3
D) −4
Q103: Find the sum of first 5 terms of AP: 2, 5, 8…
A) 30
B) 35
C) 40
D) 45
Q104: Find the 10th term of AP: 4, 9, 14…
A) 44
B) 45
C) 49
D) 50
Q105: Which is the correct formula for nth term of AP?
A) a + nd
B) a + (n−1)d
C) ad + n
D) a − nd
Q106: Find the 6th term of the GP: 2, 6, 18…
A) 486
B) 162
C) 54
D) 108
Q107: Find the common ratio of GP: 81, 27, 9…
A) 1/2
B) 1/3
C) 1/4
D) 1/6
Q108: In a GP, if a = 5 and r = 2, find the 4th term.
A) 20
B) 40
C) 80
D) 160
Q109: Find the sum of first 4 terms of GP: 3, 6, 12…
A) 45
B) 48
C) 60
D) 72
Q110: The 3rd term of a GP is 16 and r = 2. Find the first term.
A) 2
B) 4
C) 6
D) 8
Q111: Which of the following is a geometric progression?
A) 2, 4, 6, 8
B) 3, 6, 9, 12
C) 5, 10, 20, 40
D) 1, 3, 5, 7
Q112: If a=7 and d=3, find T₈ of AP.
A) 28
B) 31
C) 34
D) 37
Q113: If the common difference is 0, the AP is:
A) Increasing
B) Decreasing
C) Constant
D) Alternating
Q114: If r = 1, then GP is:
A) Increasing
B) Decreasing
C) Constant
D) Alternating
Q115: Find the 7th term of AP: 1, 4, 7…
A) 16
B) 19
C) 21
D) 22
Q116: What is the 5th term of GP: 3, 9, 27…?
A) 81
B) 162
C) 243
D) 729
Q117: In AP, if a=10 and d=−2, find T₆.
A) 0
B) −2
C) −5
D) −10
Q118: Ratio of two successive terms in GP is:
A) Common difference
B) Common ratio
C) First term
D) Mean
Q119: The 1st term of GP is 1 and r = 3. Find the 8th term.
A) 729
B) 2187
C) 6561
D) 59049
Q120: The difference between an AP and GP is:
A) AP increases by ratio, GP increases by difference
B) AP has constant difference, GP has constant ratio
C) Both use ratio
D) Both use difference
Progressions
Q121: The 12th term of an AP whose first term is 15 and common difference is –4 is
A) –29
B) –25
C) –21
D) 33
Q122: In a GP, the 3rd term is 36 and the 6th term is 972. The common ratio is
A) 3
B) 4
C) 6
D) 9
Q123: A school bus increases its speed by 5 km/h every hour. If it starts at 30 km/h, what is its speed in the 10th hour?
A) 70 km/h
B) 75 km/h
C) 80 km/h
D) 85 km/h
Q124: The sum of the first 25 terms of the AP 8, 5, 2, … is
A) –725
B) –700
C) –675
D) –650
Q125: Which term of the sequence 128, 64, 32, 16, … is 1/2?
A) 7th
B) 8th
C) 9th
D) 10th
Q126: If the nth term of a sequence is 5n – 2, then it is
A) AP with d = 5
B) GP with r = 5
C) AP with d = 3
D) Not an AP
Q127: A man saves ₹200 in January, ₹250 in February, ₹300 in March and so on. His savings form an AP. Total savings in first 12 months is
A) ₹5400
B) ₹5700
C) ₹6000
D) ₹6300
Q128: The 5th term of the GP 2, 6, 18, 54, … is
A) 162
B) 486
C) 1458
D) 4374
Q129: Three numbers are in AP and their sum is 24. If the middle number is 9, the numbers are
A) 6, 9, 12
B) 5, 9, 10
C) 7, 9, 8
D) 4, 9, 11
Q130: The first term of an AP is 5, last term is 45 and sum is 400. Number of terms is
A) 15
B) 16
C) 17
D) 18
Q131: In a GP of 7 terms, middle term is 16 and common ratio is 2. First term is
A) 1
B) 2
C) 4
D) 8
Q132: The sum of an infinite GP with |r| < 1 is 24 and first term is 18. Common ratio is
A) 1/4
B) 1/3
C) 1/2
D) 3/4
Q133: How many three-digit numbers are in AP?
A) 90
B) 180
C) 198
D) Infinite
Q134: The 20th term from the end of the AP 5, 9, 13, …, 101 is
A) 29
B) 33
C) 37
D) 41
Q135: If a, b, c are in GP, then b² =
A) ac
B) a + c
C) a – c
D) a/c
Q136: A student plants 2 trees in 1st year, 6 in 2nd, 18 in 3rd, … How many trees in 8th year?
A) 4374
B) 13122
C) 39366
D) 118098
Q137: The number of terms in the AP 7, 13, 19, …, 205 is
A) 33
B) 34
C) 35
D) 36
Q138: Which of the following cannot be the nth term of an AP?
A) 3n + 5
B) 4 – 2n
C) n² + 1
D) 7n – 3
Q139: The sum of all natural numbers from 1 to 100 is 5050. This is the sum of first 100 terms of the AP of
A) Odd numbers
B) Even numbers
C) Natural numbers
D) Prime numbers
Q140: In a GP, if sum of first three terms is 14 and their product is 64, the middle term is
A) 4
B) 6
C) 8
D) 10