MCQs of Calculus

By: Prof. Dr. Fazal Rehman | Last updated: February 12, 2025

Solution:

We are given the polynomial:
\[ P(x) = x^3 – 4x^2 + 3x + 5 \]
Substituting \( x = 2 \):
\[ P(2) = (2)^3 – 4(2)^2 + 3(2) + 5 \]
Calculating step by step:
\[ 2^3 = 8 \]
\[ 4(2^2) = 4(4) = 16 \]
\[ 3(2) = 6 \]
Now, substitute these values:
\[ P(2) = 8 – 16 + 6 + 5 \]
Simplifying:
\[ 8 – 16 = -8 \]
\[ -8 + 6 = -2 \]
\[ -2 + 5 = 3 \]
Thus, the correct answer is:
\[ \boxed{3} \]

Solution:

A polynomial is an algebraic expression consisting of variables and coefficients with non-negative integer exponents. Let’s analyze each option:

– **Option (a):** \( x^2 + \frac{1}{x} \) contains \( \frac{1}{x} = x^{-1} \), which has a negative exponent. Not a polynomial.
– **Option (b):** \( 3x^4 + 2x – 7 \) consists of terms with non-negative integer exponents. This is a polynomial.
– **Option (c):** \( \sqrt{x} + 5 = x^{1/2} + 5 \) contains a fractional exponent. Not a polynomial.
– **Option (d):** \( x^{-3} + x \) contains \( x^{-3} \), which has a negative exponent. Not a polynomial.

Thus, the correct answer is:
\[ \boxed{3x^4 + 2x – 7} \]

Solution:

Since \( x-2 \) is a factor of \( P(x) \), we know that \( P(2) = 0 \).

Substituting \( x = 2 \) in \( P(x) = x^3 – 5x^2 + ax + 10 \):
\[
(2)^3 – 5(2)^2 + a(2) + 10 = 0
\]
\[
8 – 20 + 2a + 10 = 0
\]
\[
-2 + 2a = 0
\]
\[
2a = 2
\]
\[
a = 1
\]

Thus, the correct answer is:
\[ \boxed{1} \]

Solution:

The **degree** of a polynomial is the highest exponent of the variable in the given expression.

The given polynomial is:
\[
7x^5 – 3x^3 + 2x^8 + 4
\]
The exponents of \( x \) in each term are:
– \( 5 \) in \( 7x^5 \)
– \( 3 \) in \( -3x^3 \)
– \( 8 \) in \( 2x^8 \)
– \( 0 \) in \( 4 \) (constant term)

The highest exponent is **8**, so the degree of the polynomial is **8**.

Thus, the correct answer is:
\[ \boxed{8} \]

Solution:

To find the remainder when \( P(x) \) is divided by \( x – 1 \), we use the **remainder theorem**, which states that the remainder is \( P(1) \).

Given polynomial:
\[
P(x) = x^4 + 3x^3 – 2x + 7
\]
Substituting \( x = 1 \):
\[
P(1) = (1)^4 + 3(1)^3 – 2(1) + 7
\]
\[
= 1 + 3 – 2 + 7
\]
\[
= 9
\]

Thus, the correct answer is:
\[ \boxed{9} \]

Solution:

The standard identity for the **cube of a binomial** is:
\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Now, let’s check the given options:

– **Option (a):** \( a^3 + 3a^2b + 3ab^2 + b^3 \) **(Correct identity)**
– **Option (b):** \( a^3 – b^3 \) (Incorrect; this is the difference of cubes formula)
– **Option (c):** \( (a+b)(a-b) \) (Incorrect; this represents the difference of squares)
– **Option (d):** \( a^3 – 3a^2b + 3ab^2 – b^3 \) (Incorrect; this is the expansion of \( (a-b)^3 \))

Thus, the correct answer is:
\[ \boxed{a^3 + 3a^2b + 3ab^2 + b^3} \]

Solution:

A **monomial** is an algebraic expression that consists of only **one term**.

Let’s analyze each option:
– **Option (a):** \( x^2 + 2x \) has two terms (\( x^2 \) and \( 2x \)), so it is not a monomial.
– **Option (b):** \( 4x^3 \) is a **single term** with a coefficient of 4 and exponent 3, so it is a monomial.
– **Option (c):** \( 3x^2 + 5x + 7 \) has three terms, so it is not a monomial.
– **Option (d):** \( x – 1 \) has two terms (\( x \) and \( -1 \)), so it is not a monomial.

Thus, the correct answer is:
\[ \boxed{4x^3} \]

Solution:

By Vieta’s formulas, for a polynomial of the form:
\[
P(x) = x^3 + ax^2 + bx + c
\]
where the roots are \( 1, 2, 3 \), we use the sum of the roots formula:
\[
1 + 2 + 3 = -a
\]
\[
6 = -a
\]
Thus, \( a = -6 \).

By Vieta’s product of roots in pairs formula:
\[
1 \cdot 2 + 1 \cdot 3 + 2 \cdot 3 = b
\]
\[
2 + 3 + 6 = b
\]
\[
b = 11
\]

Finding \( a + b \):
\[
-6 + (-6) = -6
\]

Thus, the correct answer is:
\[ \boxed{-6} \]

Solution:

The coefficient of **\( x \)** in a polynomial is the number that multiplies \( x \).

Given polynomial:
\[
P(x) = x^3 – 2x + 5
\]
Identifying the term with \( x \), we see that \( -2x \) is the term involving \( x \), and its **coefficient is -2**.

Thus, the correct answer is:
\[ \boxed{-2} \]

Solution:

The **leading coefficient** of a polynomial is the coefficient of the highest-degree term.

Given polynomial:
\[
P(x) = x^4 – 3x^3 + 2x^2 – 4x + 7
\]
The term with the highest exponent is \( x^4 \), and its coefficient is **1**.

Thus, the correct answer is:
\[ \boxed{1} \]