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Tips To Solve CAT Quadratic Equations Polynomials

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

  • Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
  • A general form of two-variable linear equation is ax + by = c.
  • A general form of three-variable linear equation is ax + by + cz = d.

CAT Quadratic Equations Polynomials PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If am=bncp: This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If ambn: They have different slopes and hence must intersect at some point, resulting in a unique solution.

If am=bn=cp: The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

Question 1.

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The number of integral solutions of equation 2|x|(x2x^{2} + 1) =x2x^{2} is?

A
B
C
D

Question 2.

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Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has no real roots and the equation x2 + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

A
8
B
7
C
9
D
13

Question 3.

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x^4 – ax^3 + bx^2 – cx + 8 = 0 divided by x – 1 leaves a remainder of 4, divided by x + 1 leaves remainder 3, find b

A
2.5
B
-5.5
C
3.5
D
6.5

Question 4.

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Let f(x) = ax2 + bx + c, where, a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

What is the other root of f(x) = 0?

A
-7
B
-4
C
2
D
6

Question 5.

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

A
5
B
2
C
4
D
3

Question 6.

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If both a and b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax2 + bx + 1 = 0 having real roots is:

A
10
B
7
C
6
D
12

Question 7.

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The number of roots A2x+B2x1\frac{A^{2}}{x} + \frac{B^{2}}{x-1} = 1 is

A
1
B
2
C
3
D
None of these

Question 8.

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If (3 + 2√2) is a root of the equation ax2 + bx + c = 0, and (4 + 2√3) is a root of the equation ay2 + my + n = 0, where a, b, c, m and n are integers, then the value of

bm\frac{b}{m} + c2bn\frac{c-2b}{n} is

A
1
B
0
C
4
D
3

Question 9.

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What is the remainder when x^4 + 5x^3 – 3x^2 + 4x + 3 is divided by x + 2?

A
-41
B
-31
C
-18
D
41

Question 10.

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Let f(x) = ax2 + bx + c, where, a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

What is the value of a + b + c?

A
9
B
14
C
13
D
Cannot be determined

Question 11.

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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

A
8
B
7
C
5
D
6

Question 12.

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Let α and β be two distinct root of the equation 2x2x^{2} – 6x + k = 0, such that (α + β) and αβ are the two roots of the equation x2x^{2} + px + p = 0. Then the value of 8(k - p)?

A
B
C
D

Question 13.

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If x^4 – 8x^3 + ax^2 – bx + 16 = 0 has positive real roots, find a – b

A
-8
B
6
C
-12
D
-14

Question 14.

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The number of solutions to the equation |x|(6x^2 + 1) = 5 x^2 is

A
B
C
D

Question 15.

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If the roots of the equation x3− ax2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

A
1sqrt3\frac{-1}{sqrt{3}}
B
-1
C
0
D
1

Question 16.

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The equation x3x^{3} + (2r + 1)x2x^{2}+ (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other roots are real, then the minimum possible non-negative integer value of r is?

A
B
C
D

Question 17.

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If r is a constant such that |x2 – 4x - 13| = r has exactly three distinct real roots, then the value of r is?

A
21
B
15
C
17
D
18

Question 18.

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Davji shop sells samosas in boxes of different sizes. The samosas are priced at Rs. 2 per samosa upto 200 samosas. For every additional 20 samosas, the price of the whole lot goes down by 10 paisa per samosa. What should be the maximum size of the box that would maximize the revenue?

A
240
B
300
C
400
D
None of these

Question 19.

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

A
-16
B
-4
C
-8
D
-24

Question 20.

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Let k be the largest integer such that the equation (x - 1)2 + 2kx + 11 = 0 has no real roots. If y is a positive real number, then the least possible value of k/4y + 9y is?

A
B
C
D

Question 21.

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Suppose one of the roots of the equation ax2 – bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c3 then |a| equals

A
2
B
4
C
3
D
1

Question 22.

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Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?

A
1 ≤ m ≤ 3
B
4 ≤ m ≤ 6
C
7 ≤ m ≤ 9
D
10 ≤ m ≤ 12

Question 23.

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4x^3 + ax^2 – bx + 3 divided by x – 2 leaves remainder 2, divided by x + 3 leaves remainder 3. Find remainder when it is divided by x + 2.

A
26.8
B
29.2
C
32.2
D
35.2

Question 24.

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m is the smallest positive integer such that for any integer n > m, the quantity n3 – 7n2 + 11n – 5 is positive. What is the value of m?

A
4
B
5
C
8
D
None of these

Question 25.

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If x is a positive real number such that x^8 + (1/x )^8 = 47, then the value of x^9 +(1/x)^ 9 is

A
34√5
B
36√5
C
40√5
D
30√5

Question 26.

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For all real values of x, the range of the function f(x) = x2+2x+42x2+4x+9\frac{x^2+2x+4}{2x^2+4x+9} is

A
[3/7,1/2)
B
[3/7,8/9)
C
[4/9,1/2]
D
(3/7,1/2)

Question 27.

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x3 – 18x2 + bx – c = 0 has positive real roots, p, q and z. If geometric mean of the roots is 6, find b.

A
36
B
-216
C
108
D
-72

Question 28.

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The real root of the equation 26x + 23x+2 - 21 = 0 is

A
(log23\log_{2}3)/3
B
log29\log_{2}9
C
(log27\log_{2}7)/3
D
log227\log_{2}27

Question 29.

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Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the root as (3, 2). What will be the exact roots of the original quadratic equation?

A
(6, 1)
B
(–3, –4)
C
(4, 3)
D
(–4, –3)

Question 30.

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A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f(x) at x = 10?

A
-119
B
-159
C
-110
D
-180

Question 31.

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

A
A < 1/16
B
A < 1/8
C
A > 1/8
D
A > 1/16

Question 32.

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If the equation x3 – ax2 + bx – a = 0 has three real roots, then it must be the case that,

A
b = 1
B
b ≠ 1
C
a = 1
D
a ≠ 1

Question 33.

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A quadratic equation x2 + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

A
B
C
D

Question 34.

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What is the value of 27x^3 + 18x^2y + 12xy^2 + y^3 when x = 4, y = – 8?

A
64
B
256
C
512
D
1984

Question 35.

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Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx2, where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.

How many units should Mr. David produce daily?

A
130
B
100
C
70
D
150

Question 36.

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A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

A
B
C
D

Question 37.

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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?

A
3721
B
549
C
361
D
427

Question 38.

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Let a, b and c be non-zero real numbers such that b2 < 4ac, and f(x) = ax2 + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

A
the empty set
B
the set of all integers
C
either the empty set or the set of all integers
D
the set of all positive integers

Question 39.

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x^3 – 4x^2 + mx – 2 = 0 has 3 positive roots, two of which are p and 1/p Find m.

A
5
B
-10
C
-5
D
0

Question 40.

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Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx2, where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.

What is the maximum daily profit, in rupees, that Mr. David can realize from his business?

A
620
B
920
C
840
D
760

Question 41.

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If the roots x1 and x2 of the quadratic equation x2 − 2x + c = 0 also satisfy the equation 7x2 – 4x1 = 47, then which of the following is true?

A
c = -15
B
x1x_{1} = -5, x2x_{2} = 3
C
x1x_{1} = 4.5, x2x_{2} = -2.5
D
None of these

Question 42.

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A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

A
175
B
200
C
150
D
125

Question 43.

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If x is a positive real number such that x8+(1x)8x^{8}+\left( \frac{1}{x} \right)^{8}=47 , then the value of x9+(1x)9x^{9}+\left( \frac{1}{x} \right)^{9} is

A
40540\sqrt{5}
B
36536\sqrt{5}
C
30530\sqrt{5}
D
34534\sqrt{5}

Question 44.

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How many distinct positive integer-valued solutions exist to the equation (x27x+11)(x213x+42)(x^{2}-7x+11)^{(x2-13x+42)} = 1?

A
2
B
8
C
4
D
6

Question 45.

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Given the quadratic equation x2 – (A – 3)x – (A – 2), for what value of A will the sum of the squares of the roots be zero?

A
-2
B
3
C
6
D
None of these

Question 46.

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What values of x satisfy x2/3+x1/3x^{2/3}+x^{1/3} - 2 \le 0?

A
–8 ≤ x ≤ 1
B
–1 ≤ x ≤ 8
C
1 < x < 8
D
1 ≤ x ≤ 8

Question 47.

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Let f(x) be a quadratic ploynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f(4) = 6, then f(-2) is equal to

A
24
B
6
C
36
D
12

Question 48.

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

A
1/2
B
-1/4
C
0
D
1/4

Question 49.

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One root of x2 + kx – 8 = 0 is square of the other. Then the value of k is

A
2
B
8
C
-8
D
-2

Question 50.

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Let r and c be real numbers. If r and -r are roots of 5x3 + cx2 - 10x + 9 = 0, then c equals

A
4
B
-9/2
C
-4
D
9/2

Question 51.

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For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

x2 – y2 = 0

(x – k) 2 + y2 = 1

A
2
B
0
C
2\sqrt{2}
D
- 2\sqrt{2}

Question 52.

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

A
B
C
D

Question 53.

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For real x, the maximum possible value of x(1+x4)\frac{x}{\sqrt{(1+x^{4})}} is

A
1
B
1/2
C
12\frac{1}{\sqrt{2}}
D
13\frac{1}{\sqrt{3}}

Question 54.

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If a and b are integers such that 2x2 − ax + 2 > 0 and x2 − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

A
B
C
D

Question 55.

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The number of integral solutions of the equation (x210)x23x10(x^{2}-10)^{x2-3x-10} = 1 is

A
B
C
D

Question 56.

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

A
4
B
5
C
6
D
7

Question 57.

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The smallest integer n such that n3 – 11n+ 32n – 28 > 0 is

A
B
C
D

Question 58.

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Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

What are the unique values of b and c in the equation 4x2 + bx + c = 0 if one of the roots of the equation is (−1/2)?

A. The second root is 1/2
B. The ratio of c and b is 1

A
1
B
2
C
3
D
4

Question 59.

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Two positive integers differ by 4 and sum of their reciprocals is 1021\frac{10}{21}. Then one of the numbers is

A
3
B
1
C
5
D
21

Question 60.

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If f1(x) = x2 + 11x + n and f2(x) = x, then the largest positive integer n for which the equation f1(x) = f2(x) has two distinct real roots, is :

A
B
C
D

Question 61.

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Let p and q be the roots of the quadratic equation x− (α − 2)x − α − 1 = 0. What is the minimum possible value of p2 + q2?

A
0
B
3
C
4
D
5

Question 62.

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If one root of x2 − px + 12 = 0 is 4, while the equation x2 − px + q = 0 has equal roots, then the value of q

A
49/4
B
4/49
C
4
D
1/4

Question 63.

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The minimum possible value of the squares of the roots of the equation: x^2 + (a + 3)x – (a + 5) = 0 is

A
1
B
2
C
3
D
4

Question 64.

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The number of roots common between the two equations x+ 3x+ 4x + 5 = 0 and x+ 2x+ 7x + 3 = 0 is:

A
0
B
1
C
2
D
3

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