Quadratic Equations Polynomials

Test Name	No. of Questions	Marks (of each)	Time
Test-1	11	3	40 Minutes
Test-2	12	3	40 Minutes
Test-3	12	3	40 Minutes
Test-4	10	3	40 Minutes
Test-5	10	3	40 Minutes
Test-6	9	3	40 Minutes

Question 1.

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

Question 2.

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.

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.

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.

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.

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.

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

A

1

B

2

C

3

D

None of these

Question 8.

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

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

A

1

B

0

C

4

D

3

Question 9.

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.

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.

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.

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

Question 13.

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.

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

Question 15.

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

\frac{-1}{sqrt{3}}

B

-1

C

0

D

1

Question 16.

The equation $x^{3}$ + (2r + 1) $x^{2}$ + (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?

Question 17.

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.

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.

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

A

-16

B

-4

C

-8

D

-24

Question 20.

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?

Question 21.

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.

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.

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.

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.

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.

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

A

[3/7,1/2)

B

[3/7,8/9)

C

[4/9,1/2]

D

(3/7,1/2)

Question 27.

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.

The real root of the equation 2^6x + 2^3x+2 - 21 = 0 is

A

(

\log_{2}3

)/3

B

\log_{2}9

C

(

\log_{2}7

)/3

D

\log_{2}27

Question 29.

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.

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.

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.

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.

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

Question 34.

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.

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.

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

Question 37.

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.

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.

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.

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.

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

x_{1}

= -5,

x_{2}

= 3

C

x_{1}

= 4.5,

x_{2}

= -2.5

D

None of these

Question 42.

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.

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

A

40\sqrt{5}

B

36\sqrt{5}

C

30\sqrt{5}

D

34\sqrt{5}

Question 44.

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

A

2

B

8

C

4

D

6

Question 45.

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.

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

A

–8 ≤ x ≤ 1

B

–1 ≤ x ≤ 8

C

1 < x < 8

D

1 ≤ x ≤ 8

Question 47.

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.

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.

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.

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.

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

\sqrt{2}

D

-

\sqrt{2}

Question 52.

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

Question 53.

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

A

1

B

1/2

C

\frac{1}{\sqrt{2}}

D

\frac{1}{\sqrt{3}}

Question 54.

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

Question 55.

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

Question 56.

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.

The smallest integer n such that n³ – 11n²+ 32n – 28 > 0 is

Question 58.

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.

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

A

3

B

1

C

5

D

21

Question 60.

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 :

Question 61.

Let p and q be the roots of the quadratic equation x²− (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.

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.

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.

The number of roots common between the two equations x³+ 3x²+ 4x + 5 = 0 and x³+ 2x²+ 7x + 3 = 0 is:

A

0

B

1

C

2

D

3

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Question 1.

The number of integral solutions of equation 2|x|(x2x^{2}x2 + 1) =x2x^{2}x2 is?

Question 2.

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

Question 3.

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

Question 4.

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?

Question 5.

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

Question 6.

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:

Question 7.

The number of roots A2x+B2x−1\frac{A^{2}}{x} + \frac{B^{2}}{x-1}xA2​+x−1B2​ = 1 is

Question 8.

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}mb​ + c−2bn\frac{c-2b}{n}nc−2b​ is

Question 9.

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

Question 10.

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?

Question 11.

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

Question 12.

Let α and β be two distinct root of the equation 2x2x^{2}x2 – 6x + k = 0, such that (α + β) and αβ are the two roots of the equation x2x^{2}x2 + px + p = 0. Then the value of 8(k - p)?

Question 13.

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

Question 14.

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

Question 15.

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

Question 16.

The equation x3x^{3}x3 + (2r + 1)x2x^{2}x2+ (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?

Question 17.

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

Question 18.

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?

Question 19.

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

Question 20.

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?

Question 21.

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

Question 22.

Question 23.

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.

Question 24.

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?

Question 25.

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

Question 26.

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}2x2+4x+9x2+2x+4​ is

Question 27.

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

Question 28.

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

Question 29.

Question 30.

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?

Question 31.

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

Question 32.

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

Question 33.

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

Question 34.

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

Question 35.

How many units should Mr. David produce daily?

Question 36.

Question 37.

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?

Question 38.

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

Question 39.

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

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

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

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.

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

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:

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

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

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

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.

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

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

The equation $x^{3}$ + (2r + 1) $x^{2}$ + (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?

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

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

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?

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

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?

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

The real root of the equation 2^6x + 2^3x+2 - 21 = 0 is

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

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

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

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?

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

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?

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

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

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?

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

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

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

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

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

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

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

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

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

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