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Question 1:

Let \(p(x)\) be a polynomial. When \(p(x)\) is divided by \((x-1)\), it leaves 2 as the remainder. When \(p(x)\) is divided by \((x-2)\), it leaves 1 as the remainder. What is the remainder when \(p(x)\) is divided by \((x-1)(x-2)\)?

-3

3 - x

3 - 2 x

Question 2:

-3

3 - x

3 - 2 x

Question 3:

-3

3 - x

3 - 2 x

Question 4:

Consider the following in respect of a positive real number \( x \): I. \( x+\frac{1}{x} > 1 \) II. \( \bigl(x+\frac{1}{x}\bigr)^2 > 2 \) III. \( \bigl(x+\frac{1}{x}\bigr)^4 > 9 \)
Which of the above are correct?

I and II only

II and III only

I and III only

I, II and III

Question 5:

Let \(p\) and \(q\) be natural numbers such that \(q>p\). What is the largest value of \(p\) such that \(q^{2}-5p-4\) is negative?

N/A

Question 6:

Let \( x \) and \( y \) be natural numbers, each less than 20, such that \( x \), \( y \), \( x+y \) and \( x-y \) are prime numbers. How many such combinations of \( (x, y, x+y, x-y) \) are possible?

One

Two

Three

None

Question 7:

If \( (x+1)(x+p)\left(x^{2}+p^{2}\right)=x^{4}-1 \), then what is the value of \( p \) ?

-1

Cannot be determined

Question 8:

If \( (2+\sqrt{3})^{x} + (2-\sqrt{3})^{x} = 2 \), then what is \( (2+\sqrt{3})^{x} - (2-\sqrt{3})^{x} \) equal to?

0.5

1.5

Question 9:

If \(\tfrac{1}{a}+\tfrac{1}{b}=\tfrac{5}{6}\) and \(\tfrac{1}{a^{2}}+\tfrac{1}{b^{2}}=\tfrac{13}{36}\), then what is \(\tfrac{1}{a^{3}}+\tfrac{1}{b^{3}}\) equal to?

31/216

35/216

37/216

41/216

Question 10:

What is the remainder when \( x^{6} \) is divided by \( x^{2}+1 \) ?

-1

\( x+1 \)

Question 11:

If \( \log_{10}2=0.301 \) and \( \log_{10}3=0.477 \), then what is the number of digits in the expansion of \(60^{60}\)?

105

106

107

108

Question 12:

What is the remainder when \(17^{25} + 19^{25}\) is divided by 18?

Question 13:

The HCF of \( x \) and \( y \) is \( H \). Consider the following statements in respect of the HCF of \( p=\frac{x^{3}+y^{3}}{x^{2}-x y+y^{2}} \) and \( q=\frac{x^{3}-y^{3}}{x^{2}+x y+y^{2}} \). I. The HCF of \( p \) and \( q \) can be \( H \). II. The HCF of \( p \) and \( q \) can be \( 2 H \). Which of the statements given above is/are correct?

I only

II only

Both I and II

Neither I nor II

Question 14:

If \( x^{4}=x^{2}+1 \), where \( x>0 \), then what is \( 2x^{4} \) equal to?

\( 2+\sqrt{3} \)

\( 3+\sqrt{5} \)

\( 5-2\sqrt{3} \)

\( 3-\sqrt{5} \)

Question 15:

If x is a positive real number, which of the following statements is always true?

x + 1/x < 1

x + 1/x = 1

x + 1/x > 1

x + 1/x = 2

Question 16:

What is the minimum value of the expression (x + 1/x)^2 for x > 0?

Question 17:

For x > 0, which of the following is true about the expression (x + 1/x)^4?

It is always greater than 16

It equals 16 when x = 1

It is less than 9 for some x

It can never exceed 100

Question 18:

Which of the following options correctly states when the equality (x + 1/x)^4 = 81 holds true?

x = 3 or x = 1/3

x = 2 or x = 1/2

x = 4 or x = 1/4

x = 1

Question 19:

If x is a positive real number, which of the following expressions is always greater than 1?

x + 1/x

x - 1/x

x^2 + 1/x^2

1/x

Question 20:

For x > 0, which inequality is always true?

(x + 1/x)^2 > 2

(x + 1/x)^2 < 2

(x + 1/x)^2 = 2

(x + 1/x)^2 > 4

Question 21:

Which statement is true for the expression (x + 1/x)^4 when x is a positive real number?

(x + 1/x)^4 > 16

(x + 1/x)^4 < 16

(x + 1/x)^4 = 16

(x + 1/x)^4 > 81

Question 22:

What is the minimum value of the expression (x + 1/x) for x > 0?

Question 23:

If x is a positive real number, which of the following inequalities is always true?

x + 1/x > 1

x + 1/x < 1

x + 1/x = 1

x + 1/x = 2

Question 24:

For x > 0, which inequality correctly represents the minimum value of (x + 1/x)^2?

(x + 1/x)^2 >= 4

(x + 1/x)^2 <= 4

(x + 1/x)^2 > 5

(x + 1/x)^2 = 3

Question 25:

What is the minimum value of (x + 1/x)^4 for x being a positive real number?

(x + 1/x)^4 >= 16

(x + 1/x)^4 <= 16

(x + 1/x)^4 > 20

(x + 1/x)^4 = 15

Question 26:

Which statement is true for all positive real numbers x?

x + 1/x is greater than or equal to 2

x + 1/x is less than 2

x + 1/x equals 0

x + 1/x equals 1

Question 27:

For any positive real number x, which of the following statements is always true?

x + 1/x < 1

x + 1/x = 1

x + 1/x > 1

x + 1/x = 2

Question 28:

What is the minimum value of the expression (x + 1/x)^2 for x > 0?

Question 29:

Which of the following is a possible value of (x + 1/x)^4 for some x > 0?

Question 30:

If x is a positive real number, which of the following inequalities must be true?

(x + 1/x)^2 > 3

(x + 1/x)^2 < 3

(x + 1/x)^2 = 3

(x + 1/x)^2 = 5

Question 31:

For any positive real number x, which of the following inequality is always true?

x + 1/x > 1

x + 1/x < 1

x + 1/x = 1

x + 1/x > 0

Question 32:

Which of the following is a correct statement about the expression (x + 1/x)^2 for any positive real number x?

(x + 1/x)^2 > 4

(x + 1/x)^2 < 4

(x + 1/x)^2 = 4

(x + 1/x)^2 > 2

Question 33:

If x is a positive real number, then (x + 1/x)^4 is always greater than:

Question 34:

Which of the following options correctly lists the inequalities that are true for all positive real numbers x?

I and II only

II and III only

I and III only

I, II and III

Question 35:

For a positive real number x, which of the following inequalities is always true?

x + 1/x > 1

x + 1/x < 1

x + 1/x = 1

x + 1/x >= 2

Question 36:

If x is a positive real number, which of the following statements is correct regarding (x + 1/x)^2?

(x + 1/x)^2 > 2

(x + 1/x)^2 < 2

(x + 1/x)^2 = 2

(x + 1/x)^2 <= 1

Question 37:

What is the minimum value of the expression (x + 1/x) for x being a positive real number?

Question 38:

Which of the following is true for the expression (x + 1/x)^4 when x is a positive real number?

(x + 1/x)^4 > 9

(x + 1/x)^4 >= 16

(x + 1/x)^4 = 8

(x + 1/x)^4 < 16

Question 39:

For a positive real number x, which statement is true about the expression x + 1/x?

x + 1/x is always greater than 1

x + 1/x equals 1

x + 1/x is less than 1

x + 1/x can be zero

Question 40:

If x is a positive real number, which of the following is true for the expression (x + 1/x)^2?

(x + 1/x)^2 is always greater than 4

(x + 1/x)^2 equals 4

(x + 1/x)^2 is less than 4

(x + 1/x)^2 can be negative

Question 41:

What is the minimum value of the expression (x + 1/x)^4 for any positive real number x?

Question 42:

Which of the following statements is correct for any positive real number x?

x + 1/x is always greater than or equal to 2

x + 1/x is always less than 2

x + 1/x can be less than 1

x + 1/x equals 0

Question 43:

If x is a positive real number, which of the following statements is always true?

x + 1/x > 1

x + 1/x = 1

x + 1/x < 1

x + 1/x = 2

Question 44:

For x > 0, which inequality is correct for the expression (x + 1/x)^2?

(x + 1/x)^2 > 4

(x + 1/x)^2 < 4

(x + 1/x)^2 = 4

(x + 1/x)^2 > 9

Question 45:

What is the minimum value of the expression (x + 1/x)^4 for x > 0?

Question 46:

Which of the following is a possible value of (x + 1/x) when x is a positive real number?

1.5

-1

Question 47:

For a positive real number x, which of the following statements is true about the expression x + 1/x?

It is always greater than 1.

It equals 1 when x = 1.

It is less than 1 when x is greater than 1.

It is always less than 2.

Question 48:

If x is a positive real number, which inequality is true for the expression (x + 1/x)^2?

It is always greater than or equal to 4.

It is less than 4 when x = 1.

It decreases as x increases.

It is always less than 9.

Question 49:

Which of the following is a possible value for the expression (x + 1/x)^4 when x is a positive real number?

Question 50:

What is the minimum value of the expression (x + 1/x)^4 for any positive real number x?

Question 51:

If x is a positive real number, which of the following statements is true about the expression x + 1/x?

It is always less than 1

It is always equal to 1

It is always greater than 1

It can be less than 1

Question 52:

For x > 0, the minimum value of (x + 1/x)^2 is:

Question 53:

What is the minimum value of (x + 1/x)^4 for x > 0?

Question 54:

Which of the following is a correct inequality involving x + 1/x for x > 0?

(x + 1/x) > 1

(x + 1/x) < 2

(x + 1/x) = 1

(x + 1/x) < 1

Question 55:

For any positive real number x, which of the following inequalities is always true?

x + 1/x > 1

(x + 1/x)^2 > 2

(x + 1/x)^4 > 9

(x + 1/x)^2 > 4

Question 56:

If x is a positive real number, which statement about the expression (x + 1/x)^2 is true?

It is always less than 4

It is equal to 4 when x equals 1

It decreases as x increases

It is always greater than 2

Question 57:

Which of the following is a correct inequality involving the expression (x + 1/x)^4 for any positive real number x?

(x + 1/x)^4 > 16

(x + 1/x)^4 < 16

(x + 1/x)^4 = 81 when x = 3

(x + 1/x)^4 > 8

Question 58:

For x > 0, which of the following statements is true regarding the minimum value of the expression x + 1/x?

The minimum value is 1

The minimum value is 2, achieved when x = 1

The minimum value is 0

There is no minimum value

Question 59:

If x is a positive real number, which of the following statements is true for the expression x + 1/x?

x + 1/x < 1

x + 1/x = 1

x + 1/x > 1

x + 1/x = 2

Question 60:

For x > 0, what is the minimum value of (x + 1/x)^2?

Question 61:

Which of the following is a possible value of (x + 1/x)^4 for some x > 0?

Question 62:

What is the minimum value of (x + 1/x)^4 for x > 0?

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