Find the value of $$\int _ { 1 } ^ { 4 } \left( 3 \sqrt { x } + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$$ A - 0.75 B 7.125 C 11 D 17 E 18 F 21.875 G 34.5
$A ( 0,2 )$ and $C ( 4,0 )$ are opposite vertices of the square $A B C D$. What is the equation of the straight line through $B$ and $D$ ? A $y = - 2 x + 5$ B $y = - \frac { 1 } { 2 } x - 3$ C $y = - \frac { 1 } { 2 } x + 2$ D $y = x$ E $y = 2 x - 3$ F $y = 2 x + 2$
A student is chosen at random from a class. Each student is equally likely to be chosen. Which of the following conditions is/are necessary for the probability that the student wears glasses to equal $\frac { 4 } { 15 }$ ? I Exactly 11 students in the class do not wear glasses. II The number of students in the class is divisible by 3 . III The class contains 30 students, and 8 of them wear glasses. A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
Consider the following claim about positive integers $a , b$ and $c$ : if $a$ is a factor of $b c$, then $a$ is a factor of $b$ or $a$ is a factor of $c$ Which of the following provide(s) a counterexample to this claim? I $a = 5 , b = 10 , c = 20$ II $\quad a = 8 , b = 4 , c = 4$ III $a = 6 , b = 7 , c = 12$ A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
On which line is the first error in the following argument? A $\sin ^ { 2 } x + \cos ^ { 2 } x = 1$ for all values of $x$. B Therefore $\cos x = \sqrt { 1 - \sin ^ { 2 } x }$ for all values of $x$. C Hence $1 + \cos x = 1 + \sqrt { 1 - \sin ^ { 2 } x }$ for all values of $x$. D Thus $( 1 + \cos x ) ^ { 2 } = \left( 1 + \sqrt { 1 - \sin ^ { 2 } x } \right) ^ { 2 }$ for all values of $x$. E Substituting $x = \pi$ gives $0 = 4$.
Consider the following two statements about the polynomial $\mathrm { f } ( x )$ : $P : \quad \mathrm { f } ( x ) = 0$ for exactly three real values of $x$ $Q : \quad \mathrm { f } ^ { \prime } ( x ) = 0$ for exactly two real values of $x$ Which one of the following is correct? A $P$ is necessary but not sufficient for $Q$. B $P$ is sufficient but not necessary for $Q$. C $P$ is necessary and sufficient for $Q$. D $P$ is not necessary and not sufficient for $Q$.
A circle has equation $( x - 9 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$ A square has vertices at $( 1,0 ) , ( 1,2 ) , ( - 1,2 )$ and $( - 1,0 )$. A straight line bisects both the area of the circle and the area of the square. What is the $x$-coordinate of the point where this straight line meets the $x$-axis? A 2 B 3 C 4 D 4.5 E 5 F 6 G The straight line is not uniquely determined by the information given, so there is more than one possible point of intersection. $\mathbf { H }$ There is no straight line that bisects both the area of the circle and the area of the square.
Consider the following statement about the polynomial $\mathrm { p } ( x )$, where $a$ and $b$ are real numbers with $a < b$ : (*) There exists a number $c$ with $a < c < b$ such that $\mathrm { p } ^ { \prime } ( c ) = 0$. Which one of the following is true? A The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary and sufficient for ( $*$ ) B The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary but not sufficient for (*) C The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is sufficient but not necessary for ( $*$ ) D The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is not necessary and not sufficient for ( $*$ )
Consider the following statements about a polynomial $\mathrm { f } ( x )$ : I $\mathrm { f } ( x ) = p x ^ { 3 } + q x ^ { 2 } + r x + s$, where $p \neq 0$. II There is a real number $t$ for which $\mathrm { f } ^ { \prime } ( t ) = 0$. III There are real numbers $u$ and $v$ for which $\mathrm { f } ( u ) \mathrm { f } ( v ) < 0$. Which of these statements is/are sufficient for the equation $\mathrm { f } ( x ) = 0$ to have a real solution?
The first seven terms of a sequence of positive integers are: $$\begin{aligned}
& u _ { 1 } = 15 \\
& u _ { 2 } = 21 \\
& u _ { 3 } = 30 \\
& u _ { 4 } = 37 \\
& u _ { 5 } = 44 \\
& u _ { 6 } = 51 \\
& u _ { 7 } = 59
\end{aligned}$$ Consider the following statement about this sequence: (*) If $n$ is a prime number, then $u _ { n }$ is a multiple of 3 or $u _ { n }$ is a multiple of 5 . What is the smallest value of $n$ that provides a counterexample to $( * )$ ? A 1 B 2 C 3 D 4 E 5 F 6 G 7
A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers: Show that if $a ^ { 2 } - 4 b ^ { 3 } \geq 0$ then there exist real numbers $x$ and $y$ such that $a = x y ( x + y )$ and $b = x y$. Consider the following attempt: $$\begin{aligned}
& ( x - y ) ^ { 2 } \geq 0 \\
& \text { so } \quad x ^ { 2 } + y ^ { 2 } - 2 x y \geq 0 \\
& \text { so } \quad ( x + y ) ^ { 2 } - 4 x y \geq 0 \\
& \text { so } \quad x ^ { 2 } y ^ { 2 } ( x + y ) ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } \geq 0 \\
& \text { so } \quad a ^ { 2 } - 4 b ^ { 3 } \geq 0
\end{aligned}$$ Which of the following best describes this attempt? A It is completely correct. B It is incorrect, but it would be correct if written in the reverse order. C It is incorrect, but the student has correctly proved the converse. D It is incorrect because there is an error in line (II). $\mathbf { E }$ It is incorrect because there is an error in line (III). F It is incorrect because there is an error in line (IV).
Which of the following statements about polynomials $f$ and $g$ is/are true? I If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\int _ { 0 } ^ { x } \mathrm { f } ( t ) \mathrm { d } t \geq \int _ { 0 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t$ for all $x \geq 0$. II If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$. III If $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$, then $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$. A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
A region $R$ in the ( $x , y$ )-plane is defined by the simultaneous inequalities $$\begin{array} { r }
y - x < 3 \\
y - x ^ { 2 } < 1
\end{array}$$ Which of the following statements is/are true for every point in $R$ ? I $- 1 < x < 2$ II $\quad ( y - x ) \left( y - x ^ { 2 } \right) < 3$ III $y < 5$ A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
Consider the following simultaneous equations, where $p$ is a real number: $$\begin{array} { r }
p 2 ^ { x } + \log _ { 2 } y = 2 \\
2 ^ { x } + \log _ { 2 } y = 1
\end{array}$$ What is the complete range of $p$ for which these simultaneous equations have a real solution $( x , y )$ ? A $p < 1$ B $p \neq 1$ C $p > 1$ D $p < 1$ or $p > 2$ E $\quad p \neq 1$ and $p < 2$ F $p > 1$ and $p < 2$ G $p > 2$ H All real values of $p$
A circle has equation $$x ^ { 2 } + a x + y ^ { 2 } + b y + c = 0$$ where $a , b$ and $c$ are non-zero real constants. Which one of the following is a necessary and sufficient condition for the circle to be tangent to the $y$-axis? A $a ^ { 2 } = 4 c$ B $b ^ { 2 } = 4 c$ C $\frac { a } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$ D $\frac { b } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$ $\mathbf { E } \quad - \frac { a } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$ $\mathbf { F } \quad - \frac { b } { 2 } = \sqrt { \frac { a ^ { 2 } + b ^ { 2 } } { 4 } - c }$
$\quad p$ and $q$ are real numbers, and the equation $$x | x | = p x + q$$ has exactly $k$ distinct real solutions for $x$. Which one of the following is the complete list of possible values for $k$ ? A $0,1,2$ B $0,1,2,3$ C $0,1,2,3,4$ D 0, 2, 4 E 1, 2, 3 F 1,2,3,4
Consider the following functions defined for $x > 1$ : $$\begin{aligned}
& \mathrm { f } ( x ) = \log _ { 2 } \left( \log _ { 2 } \sqrt { x } \right) \\
& \mathrm { g } ( x ) = \log _ { 2 } \left( \sqrt { \log _ { 2 } x } \right)
\end{aligned}$$ Which one of the following is true for all values of $x > 1$ ? A $0 \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq 0$ B $0 \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq 0$ C $\frac { 1 } { 2 } \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq \frac { 1 } { 2 }$ D $\frac { 1 } { 2 } \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq \frac { 1 } { 2 }$ E $1 \leq \mathrm { f } ( x ) \leq \mathrm { g } ( x )$ or $\mathrm { g } ( x ) \leq \mathrm { f } ( x ) \leq 1$ F $\quad 1 \leq \mathrm { g } ( x ) \leq \mathrm { f } ( x )$ or $\mathrm { f } ( x ) \leq \mathrm { g } ( x ) \leq 1$
A student chooses two distinct real numbers $x$ and $y$ with $0 < x < y < 1$. The student then attempts to draw a triangle $A B C$ with: $$\begin{aligned}
A B & = 1 \\
\sin A & = x \\
\sin B & = y
\end{aligned}$$ Which of the following statements is/are correct? I For some choice of $x$ and $y$, there is exactly one triangle the student could draw. II For some choice of $x$ and $y$, there are exactly two different triangles the student could draw. III For some choice of $x$ and $y$, there are exactly three different triangles the student could draw. (Note that congruent triangles are considered to be the same.) A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
The angle $\theta$ can take any of the values $1 ^ { \circ } , 2 ^ { \circ } , 3 ^ { \circ } , \ldots , 359 ^ { \circ } , 360 ^ { \circ }$. For how many of these values of $\theta$ is it true that $$\sin \theta \sqrt { 1 + \sin \theta } \sqrt { 1 - \sin \theta } + \cos \theta \sqrt { 1 + \cos \theta } \sqrt { 1 - \cos \theta } = 0$$ A 0 B 1 C 2 D 4 E 93 F 182 G 271 H 360
A sequence of functions $f _ { 1 } , f _ { 2 } , f _ { 3 } , \ldots$ is defined by $$\begin{aligned}
\mathrm { f } _ { 1 } ( x ) & = | x | \\
\mathrm { f } _ { n + 1 } ( x ) & = \left| \mathrm { f } _ { n } ( x ) + x \right| \quad \text { for } n \geq 1
\end{aligned}$$ Find the value of $$\int _ { - 1 } ^ { 1 } \mathrm { f } _ { 99 } ( x ) \mathrm { d } x$$ A 0 B 0.5 C 1 D 49.5 E 50 F 99 G 99.5 H 100