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Q1 Indices and Surds Evaluating Expressions Using Index Laws View

$$\left[ \left( \frac { - 1 } { 2 } \right) ^ { - 2 } \right] ^ { 3 }$$
What is the result of this operation?
A) - 32
B) - 16
C) 12
D) 32
E) 64

Q2 Partial Fractions Evaluating an algebraic expression given a constraint View

$$a \cdot b = \frac { 3 } { 2 }$$
Given that, what is the value of the expression $\left( a + \frac { 1 } { 2 b } \right) \left( b - \frac { 1 } { a } \right)$?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 2 } { 5 }$

Q3 Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View

$$4 ^ { x - 2 } = 6 ^ { 2 x - 2 }$$
Given that, what is the value of the expression $9 ^ { \mathbf { x } }$?
A) $\frac { 9 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 5 } { 4 }$
E) $\frac { 9 } { 4 }$

Q4 Indices and Surds Solving Equations Involving Surds View

For a real number $\mathbf { x }$,
$$\sqrt { \frac { \sqrt { x } + 2 } { \sqrt { x } - 2 } } = \sqrt { x } + 2$$
Given that, what is x?
A) 3
B) 4
C) 5
D) 6
E) 7

Q5 Arithmetic Sequences and Series Arithmetic-Geometric Sequence Interplay View

The geometric mean of real numbers a and b is 4, and the geometric mean of $a - 1$ and $b + 1$ is 6.
Accordingly, what is the difference $\mathbf { a - b }$?
A) 20
B) 21
C) 22
D) 23
E) 24

Q6 Number Theory Congruence Reasoning and Parity Arguments View

Let $n$ be a positive integer with $n \leq 20$. The sum
$$1 + 2 + 3 + \cdots + n$$
is divisible by 9. Accordingly, what is the sum of the possible values of n?
A) 50
B) 52
C) 56
D) 60
E) 64

Q8 Partial Fractions Finding a ratio or relationship between variables from an equation View

For distinct positive real numbers $x$ and $y$,
$$\left( \frac { x } { y } - \frac { y } { x } \right) \cdot \frac { x y } { 4 } = ( x - y ) ^ { 2 }$$
Given that, what is the ratio $\frac { x } { y }$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

Q9 Simultaneous equations View

$$\frac { x ^ { 3 } - x ^ { 2 } y - x y ^ { 2 } + y ^ { 3 } } { 2 x ^ { 2 } - 4 x y + 2 y ^ { 2 } } = \frac { 1 } { 2 }$$
Given that, what is the sum $\mathbf { x } + \mathbf { y }$?
A) 1
B) 2
C) 4
D) $\frac { 3 } { 2 }$
E) $\frac { 4 } { 3 }$

Q10 Simultaneous equations View

Let $\mathbf { k }$ be a nonzero real number such that
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } = ( 6 k ) ^ { 2 } \\ & ( x - 2 k ) ^ { 2 } + y ^ { 2 } = ( 2 k \sqrt { 5 } ) ^ { 2 } \end{aligned}$$
Accordingly, which of the following is the equivalent of $x ^ { 2 } - y ^ { 2 }$ in terms of $k$?
A) $13 \mathrm { k } ^ { 2 }$
B) $14 \mathrm { k } ^ { 2 }$
C) $15 k ^ { 2 }$

Q11 Inequalities Solving equations involving modulus View

$$| x - 2 | \cdot | x - 3 | = 3 - x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) - 3
B) - 2
C) 0
D) 2
E) 4

Q12 Inequalities Algebraic identities and properties of modulus View

For real numbers $a$ and $b$, it is known that $( | a | - a ) ( | b | + b ) > 0$.
Accordingly, I. $a + b < 0$ II. $a - b < 0$ III. $\mathrm { a } \cdot \mathrm { b } < 0$ Which of the following statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Q13 Principle of Inclusion/Exclusion View

A tour company has organized tours to three different museums. The following is known about those who participated in these tours.

30 people participated in each tour.
10 people participated in all three tours.
33 people participated in at least two tours.

Accordingly, how many people participated in only one of the tours?
A) 10
B) 11
C) 12
D) 13
E) 14

Q14 Combinations & Selection Partitioning into Teams or Groups View

Four distinct marbles will be distributed to 3 siblings such that each sibling receives at least 1 marble.
In how many different ways can this distribution be done?
A) 24
B) 32
C) 36
D) 40
E) 48

Q15 Inequalities Solve Polynomial/Rational Inequality for Solution Set View

Let $f : \mathbf { R } \backslash \{ 0 \} \rightarrow \mathbf { R }$ with
$$f ( x ) = \frac { 2 } { x } - x + 1$$
For this function, which of the following is the set of all $x$ points such that $f ( x ) \in ( 0 , \infty )$?
A) $( - \infty , 0 )$
B) $( - 1 , \infty )$
C) $( 0,1 ) \cup ( 2 , \infty )$
D) $( - 2,0 ) \cup ( 2 , \infty )$
E) $( - \infty , - 1 ) \cup ( 0,2 )$

Q16 Permutations & Arrangements Counting Functions with Composition or Mapping Constraints View

Let $A = \{ 1,2,3 \}$ and $f : A \rightarrow A$ be a function. How many one-to-one functions $f$ satisfy the condition
$$f ( n ) \neq n$$
for every $n \in A$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q18 Composite & Inverse Functions Graphical Interpretation of Inverse or Composition View

Below are the graph of the line $y = x$ and the graph of the function $y = f ( x )$.
Starting from point $\mathbf { Q } ( \mathbf { a } , \mathbf { 0 } )$ and following the arrows, point $\mathbf { P } ( \mathbf { a } , \mathbf { b } )$ is reached. Accordingly, $\mathbf { b }$ is equal to which of the following?
A) $a + f ( a )$
B) $a \cdot f ( a )$
C) $f ( a ) - a$
D) $f ( f ( a ) )$
E) $f ( a + f ( a ) )$

Q19 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

$$P ( x ) = \left( x ^ { 2 } + 2 \right) ^ { 3 } + ( x - 3 ) ^ { 5 }$$
In this polynomial, what is the coefficient of the $x ^ { 4 }$ term?
A) - 9
B) - 3
C) 1
D) 11
E) 21

Q20 Factor & Remainder Theorem Remainder Theorem with Composed or Shifted Arguments View

$$P ( x ) = x ^ { 2 } - 3 x + 2$$
Given that, when $P ( x - 1 ) + P ( 3 x - 3 )$ is divided by $x - 1$, which of the following is the quotient obtained?
A) $4 x - 10$
B) $4 x - 22$
C) $10 x - 16$
D) $10 x - 18$
E) $10 x - 22$

Q21 Proof by induction Determine unknown parameters from tangent conditions View

A third-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with leading coefficient 1 and its derivative $P ^ { \prime } ( x )$ satisfy
$$P ( 0 ) = P ( 1 ) = P ^ { \prime } ( 1 ) = 0$$
Accordingly, what is the value of $P ( - 1 )$?
A) 3
B) 1
C) 0
D) - 2
E) - 4

Q22 Solving quadratics and applications Quadratic equation with parametric or self-referential conditions View

Let k be a positive real number. If one root of the equation
$$3 x ^ { 2 } + k x - 2 = 0$$
is k, what is the other root?
A) $\frac { \sqrt { 2 } } { 3 }$
B) $\frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { - 2 \sqrt { 2 } } { 3 }$
D) $\frac { - \sqrt { 2 } } { 6 }$
E) $\frac { - \sqrt { 3 } } { 6 }$

Q23 Trig Proofs Simplification of Trigonometric Expressions with Specific Angles View

$$\frac { \sin 48 ^ { \circ } } { \sin 16 ^ { \circ } } - \frac { \cos 48 ^ { \circ } } { \cos 16 ^ { \circ } }$$
Which of the following is this expression equal to?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 2
E) 3

Q24 Trig Proofs Evaluate trigonometric expression given a constraint View

ABCD is a square $| \mathrm { AB } | = 3$ units $| \mathrm { BE } | = | \mathrm { CF } | = 1$ unit $m ( \widehat { F A E } ) = x$
According to the given information above, what is the value of $\cot \mathrm { x }$?
A) $\frac { 6 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 7 } { 6 }$
D) $\frac { 9 } { 7 }$
E) $\frac { 11 } { 8 }$

Q25 Trigonometric equations in context View

For $0 \leq x \leq 2 \pi$, $\cos \mathrm { x } + \sin 2 \mathrm { x } = \cot \mathrm { x }$ What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) $2 \pi$
B) $3 \pi$
C) $4 \pi$
D) $\frac { 5 \pi } { 2 }$
E) $\frac { 7 \pi } { 2 }$

Q26 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View

The functions $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } + \mathrm { xi }$ and $\mathrm { g } ( \mathrm { x } ) = 2 \mathrm { x } - \mathrm { xi }$ are defined from the set of real numbers to the set of complex numbers and satisfy
$$f ( a ) + g ( b ) = 4 + 2 i$$
Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 7 } { 2 }$
B) $\frac { 9 } { 2 }$
C) $\frac { 10 } { 3 }$
D) $\frac { 13 } { 3 }$
E) $\frac { 15 } { 4 }$

Q27 Complex Numbers Arithmetic Modulus Computation View

Let $z$ be a complex number and
$$z \cdot | \operatorname { Re } ( z ) | = - 4 + 3 i$$
Accordingly, what is $| \mathbf { z } |$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$

Q28 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View

$$\alpha = \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 }$$
Given that, which of the following is $\alpha ^ { 23 }$ equal to?
A) $\cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 }$
B) $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$
C) $\cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 }$
D) $\cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 }$
E) $\cos \pi + \mathrm { i } \sin \pi$

Q29 Laws of Logarithms Find or Apply an Inverse Function Formula View

For the function $f ( x ) = \log _ { x } 2$,
$$f \left( 4 ^ { a } \right) \cdot f ^ { - 1 } \left( \frac { 1 } { 3 } \right) = 6$$
What is the value of a that satisfies this equation?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 4 } { 3 }$

Q30 Laws of Logarithms Solve a Logarithmic Equation View

$$\log _ { 2 } \left( \frac { 1 } { \sqrt { x } } \right) + \log _ { 4 } \left( \frac { 4 } { y } \right) = 3$$
Given that, what is the product $x \cdot y$?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 1 } { 16 }$

Q31 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View

The first three terms of a geometric sequence are $\mathbf { a } + \mathbf { 3 }$, a, and $\mathbf { a } - \mathbf { 2 }$ respectively. Accordingly, what is the fourth term?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 6 }$

Q32 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

An equilateral triangle is inscribed in circle $\mathrm { C } _ { 1 }$ with radius 1 unit as shown in the figure. Let $\mathrm { C } _ { 2 }$ be the circle passing through the midpoints of the sides of this triangle. The same operation is performed for circle $\mathrm { C } _ { 2 }$ and circle $\mathrm { C } _ { 3 }$ is obtained. By repeating this process infinitely many times, the sequence of circles $C _ { 1 } , C _ { 2 } , C _ { 3 } , \cdots$ is obtained.
For each positive integer $\mathbf { k }$, let $\mathbf { A } _ { \mathbf { k } }$ be the area bounded by circle $\mathbf { C } _ { \mathbf { k } }$ in square units. Accordingly, what is the result of the sum
$$\sum _ { k = 1 } ^ { \infty } A _ { k }$$
?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\frac { 6 \pi } { 5 }$
E) $\frac { 9 \pi } { 8 }$

Q33 Probability Definitions Set Operations Using Inequality-Defined Sets View

For positive integers $n$, the subsets of the set $R$ of real numbers are defined as
$$A _ { n } = \left\{ x \in R : \frac { ( - 1 ) ^ { n } } { n } < x < \frac { 2 } { n } \right\}$$
Accordingly, $$A _ { 1 } \cap A _ { 2 } \cap A _ { 3 }$$
the intersection set is equal to which of the following?
A) $\left( \frac { 1 } { 2 } , \frac { 2 } { 3 } \right)$
B) $\left( \frac { 1 } { 2 } , 2 \right)$
C) $\left( \frac { - 1 } { 3 } , \frac { 2 } { 3 } \right)$
D) $\left( \frac { - 1 } { 3 } , 1 \right)$
E) $\left( - 1 , \frac { 2 } { 3 } \right)$

Q34 Matrices Linear System and Inverse Existence View

Let I be the $2 \times 2$ identity matrix and
$$A = \left[ \begin{array} { l l } 4 & 5 \\ 1 & 3 \end{array} \right]$$
Accordingly, which of the following is $( \mathbf { A } - \mathbf { I } ) ^ { - \mathbf { 1 } }$ equal to?
A) $\left[ \begin{array} { r r } 2 & - 5 \\ - 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r } 1 & - 4 \\ - 2 & 3 \end{array} \right]$
C) $\left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right]$
E) $\left[ \begin{array} { l l } 2 & - 5 \\ 0 & - 3 \end{array} \right]$

Q35 Matrices Determinant and Rank Computation View

$$\left[ \begin{array} { l l } 3 & 2 \\ 1 & 0 \end{array} \right] \cdot \mathrm { A } = \left[ \begin{array} { c c } - 2 & 4 \\ 1 & 5 \end{array} \right]$$
What is the determinant of matrix A that satisfies this equation?
A) 4
B) 5
C) 6
D) 7
E) 8

Q36 Curve Sketching Evaluate Composition from Algebraic Definitions View

The graph of the function $f : R \rightarrow R$ is given below.
Using the function f, the function g is defined for every $\mathrm { x } _ { 0 } \in \mathrm { R }$ as
$$g \left( x _ { 0 } \right) = f \left( x _ { 0 } \right) + \lim _ { x \rightarrow x _ { 0 } + } f ( x )$$
Accordingly, what is the value of (gof)(2)?
A) - 2
B) - 1
C) 0
D) 1
E) 2

Q37 Differentiation from First Principles View

$$\lim _ { x \rightarrow 1 } \frac { f ( x + 1 ) - 3 } { x - 1 } = 2$$
Given that, what is the value of the limit $\lim _ { x \rightarrow 2 } \frac { x \cdot f ( x ) - 6 } { x - 2 }$?
A) 5
B) 6
C) 7
D) 8
E) 9

Q38 Laws of Logarithms Limit involving transcendental functions View

$$\lim _ { x \rightarrow \infty } \frac { \ln ( x - 3 ) } { \ln \sqrt { x } }$$
What is the value of this limit?
A) 1
B) 2
C) 3
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

Q39 Curve Sketching Asymptote Determination View

$$f ( x ) = \frac { - k x ^ { 3 } + k ^ { 2 } x } { k ^ { 3 } x ^ { 2 } + x - ( k + 1 ) }$$
The function has a vertical asymptote at $x = 1$. Accordingly, what is the value of $f ( 2 )$?
A) - 5
B) - 4
C) - 3
D) - 2
E) - 1

Q40 Exponential Functions True/False or Multiple-Statement Verification View

A function f is defined on the set of real numbers as
$$f ( x ) = 1 + e ^ { - x }$$
Accordingly, I. The range of function f is $( 1 , \infty )$. II. Function f is decreasing on its domain. III. The line $y = 0$ is a horizontal asymptote of function f. Which of the following statements are true?
A) Only II
B) Only III
C) I and II

Q41 Tangents, normals and gradients Find tangent line equation at a given point View

The tangent line drawn to the graph of the function $y = f ( x )$ at the point $( 2,4 )$ passes through the point $( - 1,3 )$.
Accordingly, what is the value of $f ^ { \prime } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 5 }$

Q42 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

The derivative of a function f that is defined and differentiable on the set of real numbers is given as
$$f ^ { \prime } ( x ) = \begin{cases} 1 , & \text{if } x \leq 1 \\ x , & \text{if } x > 1 \end{cases}$$
Given that $f ( 1 ) = 1$, what is the value of $f ( 0 ) + f ( 3 )$?
A) 2
B) 3
C) 4
D) 5
E) 6

Q43 Chain Rule MCQ on derivative and graph interpretation View

$$f ( x ) = 2 x ( x - 1 ) ^ { 3 } + ( x - 1 ) ^ { 4 }$$
What is the value of the third derivative of the function at the point $x = 1$?
A) 10
B) 12
C) 14
D) 16
E) 18

Q44 Tangents, normals and gradients Find tangent line with a specified slope or from an external point View

$$x ^ { 2 } - y ^ { 2 } = 1$$
What is the distance between the points where the lines tangent to the hyperbola curve and having slope 3 intersect the y-axis, in units?
A) $\sqrt { 2 }$
B) $2 \sqrt { 2 }$
C) $4 \sqrt { 2 }$
D) $\sqrt { 3 }$
E) $2 \sqrt { 3 }$

Q45 Indefinite & Definite Integrals Evaluate a Closed-Form Expression Using the Reduction Formula View

For every integer $m$ greater than 1
$$\int \tan ^ { m } x d x = \frac { 1 } { m - 1 } \tan ^ { m - 1 } x - \int \tan ^ { m - 2 } x d x$$
the equality is satisfied. Accordingly, what is the value of the integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { 4 } \mathrm { xdx }$?
A) $\frac { 2 \pi + 3 } { 4 }$
B) $\frac { 4 \pi - 3 } { 8 }$
C) $\frac { 3 \pi - 8 } { 12 }$
D) $\pi + 2$
E) $2 \pi + 1$

Q46 Integration by Parts Definite Integral Evaluation by Parts View

f is a differentiable function on the set of real numbers and
$$\begin{aligned} & \int _ { 0 } ^ { 3 } f ( x ) d x = 2 \\ & \int _ { 0 } ^ { 3 } x f ^ { \prime } ( x ) d x = 1 \end{aligned}$$
Given this, what is the value of $\mathbf { f } \boldsymbol { ( } \mathbf { 3 } \boldsymbol { ) }$?
A) 0
B) 1
C) 2
D) 3
E) 4

Q47 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

$f ( x ) = \begin{cases} 2 x - 4 , & \text{if } 0 \leq x < 1 \\ - 2 , & \text{if } 1 \leq x < 4 \\ x - 6 , & \text{if } 4 \leq x \leq 6 \end{cases}$
Given this, what is the value of the integral $\int _ { 0 } ^ { 6 } f ( x ) d x$?
A) - 11
B) - 10
C) - 9
D) - 8
E) - 7

Q48 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

$$\int _ { 4 } ^ { 9 } \frac { \sqrt { x } } { x - 1 } d x$$
If the substitution $\mathbf { u } = \sqrt { \mathbf { x } }$ is made in the integral, which of the following integrals is obtained?
A) $\int _ { 4 } ^ { 9 } \frac { u } { u ^ { 2 } - 1 } d u$
B) (missing option)
C) $\int _ { 2 } ^ { 3 } \frac { u } { 2 \left( u ^ { 2 } - 1 \right) } d u$
D) $\int _ { 2 } ^ { 3 } \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$
E) $\int _ { 2 } ^ { 3 } \frac { u } { u ^ { 2 } - 1 } d u$

Q49 Areas Between Curves Find Parameter Given Area Condition View

In the rectangular coordinate plane; the region between the curve $y = 3 \sqrt { x }$, the line $x = 1$, and the line $y = 0$ is divided into two regions of equal area by the line $y = m x$.
Accordingly, what is m?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 4 }$
D) 1
E) 2

Q50 Volumes of Revolution Volume of Revolution about a Horizontal Axis (Set Up Only) View

In the first quadrant; the region between the x-axis, the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, and the circle $x ^ { 2 } + y ^ { 2 } = 7$ is rotated $360 ^ { \circ }$ around the x-axis.
Which of the following is the integral expression of the volume of the solid of revolution obtained?
A) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 - x ^ { 2 } \right) d x$
B) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } + 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
C) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$
D) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
E) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$

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