CAT Questions | CAT Algebra Questions

CAT Quantitative Aptitude | CAT Algebra: Linear Equations; Quadratic Equations

CAT Algebra questions from Linear equations and Quadratic equations that appear in the Quantitative Aptitude section of the CAT Exam consists of concepts from Equations and Algebra. Get as much practice as you can in these two topics because the benefits of being good at framing equations can be enormous and useful in other CAT topics as well. In CAT Exam, one can generally expect to get 1~2 questions from Linear Equations and Quadratic Equations. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

  1. CAT Linear Equations - Counting

    3x + 4|y| = 33. How many integer values of (x, y) are possible?

    1. 6
    2. 3
    3. 4
    4. More than 6
    Choice D
    More than 6

  2. CAT Linear Equations - Counting

    (|x| - 3) (|y| + 4) = 12. How many pairs of integers (x, y) satisfy this equation?

    1. 4
    2. 6
    3. 10
    4. 8
    Choice C
    10

  3. CAT Linear Equations - Integer Values; Modulus

    x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations?

    1. 2
    2. 4
    3. 0
    4. 1
    Choice D
    1

  4. CAT Quadratic Equations - Counting.

    What is the number of real solutions of the equation x2 - 7|x| - 18 = 0?

    1. 2
    2. 4
    3. 3
    4. 1
    Choice A
    2

  5. CAT Quadratic Equations - Counting

    x2 - 9x + |k| = 0 has real roots. How many integer values can 'k' take?

    1. 40
    2. 21
    3. 20
    4. 41
    Choice D
    41

  6. CAT Quadratic Equations - Integer Roots

    x2 - 11x + |p| = 0 has integer roots. How many integer values can 'p' take?

    1. 6
    2. 4
    3. 8
    4. More than 8
    Choice D
    More than 8.

  7. CAT Linear Equations - Integer Solutions

    2x + 5y = 103. Find the number of pairs of positive integers x and y that satisfy this equation.

    1. 9
    2. 10
    3. 12
    4. 20
    Choice B
    10

  8. CAT Linear Equations - Maximum, Minimum

    Consider three numbers a, b and c. Max (a,b,c) + Min (a,b,c) = 13. Median (a,b,c) - Mean (a,b,c) = 2. Find the median of a, b, and c.

    1. 11.5
    2. 9
    3. 9.5
    4. 12
    Choice C
    9.5

  9. CAT Linear Equations - Unique Solutions

    a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.

    Which of the following statements if true would imply that the above system of equations does not have a unique solution?
    i. \\frac{a_{1}}{a_{2}}\\) = \\frac{b_{1}}{b_{2}}\\) = \\frac{c_{1}}{c_{2}}\\) ≠ \\frac{d_{1}}{d_{2}}\\)
    ii. \\frac{a_{1}}{a_{2}}\\) = \\frac{ a_{2} }{ a_{3} }\\) ; \\frac{ b_{1} }{ b_{2} }\\) = \\frac{ b_{2} }{ b_{3} }\\)
    iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers

    1. Statement i
    2. Statement ii
    3. Statement iii
    4. None
    Statement i

  10. CAT Quadratic Equations - Roots of Equation

    Equation x2 + 5x – 7 = 0 has roots a and b. Equation 2x2 + px + q = 0 has roots a + 1 and b + 1. Find p + q.

    1. 6
    2. 0
    3. -16
    4. 2
    Choice C
    -16

  11. CAT Quadratic Equations - Sum and product of the roots

    Sum of the roots of a quadratic equation is 5 less than the product of the roots. If one root is 1 more than the other root, find the product of the roots?

    1. 6 or 3
    2. 12 or 2
    3. 8 or 4
    4. 12 or 4
    Choice B
    12 or 2

  12. CAT Quadratic Equations - Real and imaginary Roots

    How many real solutions are there for the equation x2 – 7|x| - 30 = 0?

    1. 3
    2. 1
    3. 2
    4. none
    Choice C
    2

  13. CAT Quadratic Equations - Sum of the variables

    If (3x+2y-22)2 + (4x-5y+9)2 = 0 and 5x-4y = 0. Find the value of x+y.

    1. 7
    2. 9
    3. 11
    4. 13
    Choice B
    9

  14. CAT Cubic Equations - AP and roots of an equation

    Let x3- x2 + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true

    1. b=2,c=2
    2. b=1,c=1
    3. b= -1,c = 1
    4. b= -1,c= -1
    Choice B
    b=1,c=1

  15. CAT Linear Quadratic Equations - Roots of an Equation

    (3 + 2√2)(x2 - 3) + (3 - 2√2)(x2 - 3) = b which of the following can be the value of b?

    1. 2
    2. √2
    3. -√2
    4. All of the above
    Choice A
    2

  16. CAT Quadratic Equations - Real roots

    If f(y) = x2 + (2p + 1)x + p2 - 1 and  x is a real number, for what values of ‘p' the function becomes 0?

    1. p > 0
    2. p > -1
    3. p ≥ \\frac{-5}{4}\\)
    4. p ≤ \\frac{3}{4}\\)
    Choice C
    p ≥ \\frac{-5}{4}\\)

  17. CAT Linear Quadratic Equations - Medium

    A merchant decides to sell off 100 articles a week at a selling price of Rs. 150 each. For each 4% rise in the selling price he sells 3 less articles a week. If the selling price of each article is Rs x, then which of the below expression represents the number of articles sold by the merchant in that week?

    1. 175 - \\frac{x}{2}\\)
    2. \\frac{x}{3}\\) + 156
    3. 350 - \\frac{x^2}{2}\\)
    4. x2 - 2x + 75
    Choice A
    175 - \\frac{x}{2}\\)

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.


  1. CAT 2020 Question Paper Slot 3 - Linear & quadratic equations

    Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if

    1. |k| = 2
    2. k ≠ 2
    3. |k| ≠ 2
    4. k = 2
    Choice C
    |k| ≠ 2

  2. CAT 2020 Question Paper Slot 3 - Linear & quadratic equations

    Let m and n be positive integers, If x2 + mx + 2n = 0 and x2 + 2nx + m = 0 have real roots, then the smallest possible value of m + n is

    1. 8
    2. 6
    3. 5
    4. 7
    Choice B
    6

  3. CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

    The number of pairs of integers(x,y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99 is


  4. CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

    The number of integers that satisfy the equality (x2 - 5x + 7)x + 1 = 1 is

    1. 5
    2. 4
    3. 3
    4. 2
    Choice C
    3

  5. CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

    In how many ways can a pair of integers (x , a) be chosen such that x2 − 2 | x | + | a - 2 | = 0 ?

    1. 7
    2. 6
    3. 4
    4. 5
    Choice A
    7

  6. CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

    Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is

    1. 33
    2. 27
    3. 30
    4. 36
    Choice A
    33

  7. CAT 2020 Question Paper Slot 1 - Linear & quadratic equations

    The number of distinct real roots of the equation
    (x + \\frac{1}{x}))2 - 3(x + \\frac{1}{x})) + 2 = 0 equals


  8. CAT 2020 Question Paper Slot 1 - Linear & quadratic equations

    How many distinct positive integer-valued solutions exist to the equation (x2 - 7x + 11)(x2 - 13x + 42) = 1?

    1. 6
    2. 2
    3. 4
    4. 8
    Choice A
    6

  9. CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

    What is the largest positive integer such that \\frac{n^2+7n+12}{n^2-n-12}) is also positive integer?

    1. 6
    2. 8
    3. 16
    4. 12
    Choice D
    12

  10. CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

    Let A be a real number. Then the roots of the equation x2 - 4x – log2A = 0 are real and distinct if and only if

    1. A < \\frac{1}{16})
    2. A > \\frac{1}{8})
    3. A > \\frac{1}{16})
    4. A < \\frac{1}{8})
    Choice C
    \\frac{1}{16})

  11. CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

    The quadratic equation x2 + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b2 + c?

    1. 3721
    2. 549
    3. 361
    4. 427
    Choice B
    549

  12. CAT 2019 Question Paper Slot 1 - Quadratic equations

    The number of solutions of the equation |x|(6x2 + 1) = 5x2 is [TITA]


  13. CAT 2019 Question Paper Slot 1 - Quadratic equations

    The product of the distinct roots of ∣x2 - x - 6∣ = x + 2 is

    1. -4
    2. -16
    3. -8
    4. -24
    Choice B
    -16

  14. CAT 2018 Question Paper Slot 1 - Linear and Quadratic Equations

    If u2 + (u-2v-1)2 = -4v(u + v), then what is the value of u + 3v?

    1. \\frac{1}{4})
    2. \\frac{1}{2})
    3. 0
    4. -\\frac{1}{4})
    Choice D
    -\\frac{1}{4})

  15. CAT 2017 Question Paper Slot 2 - Quadratic Equations

    The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

    1. 1
    2. 2
    3. 3
    4. 4
    Choice C
    3

  16. CAT 2017 Question Paper Slot 1 - Algebra

    If x + 1 = x2 and x > 0, then 2x4 is:

    1. 6 + 4√5
    2. 3 + 5√5
    3. 5 + 3√5
    4. 7 + 3√5
    Choice D
    7 + 3√5

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.


  1. XAT 2018 Question Paper - QADI

    Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?

    1. The common root is 29.
    2. The smallest among the roots is 1.
    3. One of the roots is 5.
    4. Product of the roots of the other equation is 5.
    5. All of the above are possible, but none are definitely correct.
    Choice C
    One of the roots is 5.

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.


  1. IPMAT 2019 Question Paper - IPM Indore Quants

    If \|x| < 100\\) and \|y| < 100\\), then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is


  2. IPMAT 2019 Question Paper - IPM Indore Quants

    If a, b, c are real numbers a2 + b2 + c2 = 1, then the set of values ab + bc + ca can take is:

    1. [-1,2]
    2. [-\\frac{1}{2}\\), 2]
    3. [-1,1]
    4. [-\\frac{1}{2}\\), 1]
    Choice D
    [-\\frac{1}{2}\\), 1]

  3. IPMAT 2019 Question Paper - IPM Indore Quants

    Let \\alpha, \beta\\) be the roots of x2 - x + p = 0 and \\gamma, \delta\\) be the roots of x2 - 4x + q = 0 where p and q are integers. If \\alpha, \beta, \gamma, \delta\\) are in geometric progression then p + q is

    1. -34
    2. 30
    3. 26
    4. -38
    Choice A
    -34

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