Đề thi HSG Toán tiếng Anh lớp 10 Nam Định 2021

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• Đề thi HSG Toán tiếng Anh Lớp 11 SGD Nam Định năm 2018
• Đề thi HSG Toán tiếng Anh Lớp 11 năm 2019 SGD Nam Định

PART 1. MULTIPLE CHOICE QUESTIONS (7,0 points)

Question 1. Given three distinct points $A$, $B$ and $C$. Which of the following statements is true?
A. \( \overrightarrow{CA}-\overrightarrow{AB}=\overrightarrow{CB} \).
B. \( \overrightarrow{AB}=\overrightarrow{CA}=-\overrightarrow{BC} \).
C. \( \overrightarrow{BA}=\overrightarrow{AC}=\overrightarrow{CB} \).
D. \( \overrightarrow{BA}-\overrightarrow{BC}=\overrightarrow{AC} \).

Question 2. In the \( Oxy \) coordinate plane, given \( \triangle ABC \) with \( A(-1;-4) ,B(6;7)\) and \( C(-2;9) \). Let \( G \) be the centroid of \( \triangle ABC \). The coordinates of \( G \) are
A. \( G(1;4) \).
B. \( G(-1;4) \).
C. \( G(1;-4) \).
D. \( G(3;12) \).

Question 3. Given a right triangle \( ABC \) at A. Which of the following statements is false?
A. \( \overrightarrow{AB}\cdot \overrightarrow{AC} < \overrightarrow{BA}\cdot \overrightarrow{BC} \).
B. \( \overrightarrow{AC}\cdot \overrightarrow{BC} < \overrightarrow{BC}\cdot \overrightarrow{AB} \).
C. \( \overrightarrow{AB}\cdot \overrightarrow{BC} < \overrightarrow{CA}\cdot \overrightarrow{CB} \).
D. \( \overrightarrow{AC}\cdot \overrightarrow{CB} < \overrightarrow{AC}\cdot \overrightarrow{BC} \).

Question 4. Given \( A=\{1;2;3;4\}\). How many subsets does the set \( A \) have?
A. $18$.
B. $16$.
C. $15$.
D. $14$.

Question 5. Given equation \( (x^2-x+1)(x-1)(x+1)=0 \). Which of the following equations is equivalent to the given equation?
A. \( x+1=0 \).
B. \( x-1=0 \).
C. \( x^2-x+1=0 \).
D. \( (x-1)(x+1)=0 \).

Question 6. Find all values of \( m \) such that function \( y=(m-1)x+2-21 \) is decreasing on its domain.
A. \(m>1\).
B. \( m \geqslant 1 \).
C. \( m \leqslant 1 \).
D. \( m<1 \).

Question 7. Let $a, b, c$ be three positive real numbers satisfying $a+b+c=3$. Determine the maximum value of $T=\sqrt{a b}+\sqrt{b c}+\sqrt{c a}$.
A. $3$.
B. $4$.
C. $2$.
D. $6$.

Question 8. Given the fact that the system of equations $\left\{\begin{array}{l}x^{3}(2+3 y)=8 \\ \left(y^{3}-2\right) x=6\end{array}\right.$ has exactly two distinct solutions $\left(x_{1}, y_{1}\right) ;\left(x_{2}, y_{2}\right)$. The value of $S=x_{1}^{4}+y_{1}^{4}+x_{2}^{4}+y_{2}^{4}$ is
A. $34$.
B. $40$.
C. $28$.
D. $36$.

Question 9. Find all parameters $m$ such that equation $x^{2}+(m-1) x+m^{2}-1=0$ has two distinct roots and these roots have the same sign.
A. $m<-1$ or $m>1$.
B. $1<m<\frac{5}{3}$.
C. $-1<m<1$.
D. $\frac{-5}{3}<m<-1$.

Question 10. Given two equations $m x^{2}-2(m-1) x+m-2=0$ and $(m-2) x^{2}-3 x+m^{2}-15=0$. How many values of $m$ which make these above equations equivalent?
A. $2$.
B. $3$.
C. $0$.
D. $1$.

Question 11. Given $\triangle A B C$ with $A B=13, B C=2 \sqrt{33}, C A=17$. Compute the length of the median $A M$ of $\triangle A B C$.
A. $A M=2 \sqrt{35}$.
B. $A M=15$.
C. $A M=\sqrt{194}$.
D. $A M=14$.

Question 12. A ball is thrown straight up from 60 meters above the ground with a velocity of 20 meters per second $(20 \mathrm{~m} / \mathrm{s})$. The height of the ball at second $t$ after throwing can be computed by the quadratic function $s(t)=-5 t^{2}+20 t+60$, where $s(t)$ is in meters. After how many seconds does the ball hit the ground?
A. $t=2$.
B. $t=1$.
C. $t=4$.
D. $t=6$.

Question 13. Given $\triangle A B C$ with the sides $A C=3 \sqrt{3}$, side $B C=3 \sqrt{2}, A=45^{\circ}$ and $B>A+C$. Compute the degree measure of $\widehat{A B C}$.
A. $\widehat{A B C}=60^{\circ}$.
B. $\widehat{A B C}=150^{\circ}$.
C. $\widehat{A B C}=30^{\circ}$.
D. $\widehat{A B C}=120^{\circ}$.

Question 14. In the $O x y$ coordinate plane, given Parabol $(P): y=x^{2}-5 x+2 m$. Let $S$ be the set of all values of $m$ such that the Parabol $(P)$ cuts $O x$ at two distinct points $A, B$ satisfying $O A=4 O B$. Determine the sum of all elements of $S$.
A. $\frac{2}{9}$.
B. $\frac{-32}{9}$.
C. $2$.
D. $\frac{-16}{9}$.

Question 15. Which of the following two inequations are not equivalent?
A. $2 x-1>0$ and $2 x-1+\frac{1}{2 x^{2}+1}>\frac{1}{2 x^{2}+1}$.
B. $-2 x+1>0$ and $2 x-1<0$.
C. $3 x^{2}+1 \leq 2 x-1$ and $3 x^{2}-2 x+2 \leq 0$.
D. $5 x-1+\frac{1}{x-2}>\frac{1}{x-2}$ and $5 x-1>0$.

Question 16. Given an isosceles right triangle $A B C$ with sides $A B=A C=42 \mathrm{~cm} .$ Two medians $B E$ and $C F$ intersect at point $G$. The area of the triangle $G E C$ is
A. $7 \sqrt{21} \mathrm{~cm}^{2}$.
B. $21 \sqrt{7} \mathrm{~cm}^{2}$.
C. $147 \mathrm{~cm}^{2}$.
D. $174 \mathrm{~cm}^{2}$.

Question 17. In the $O x y$ coordinate plane, given two vectors $\vec{a}=(6 ;-4)$ and $\vec{b}=(-10,-2) .$ Compute the angle between two vectors $\vec{a}$ and $\vec{b}$.
A. $45^{\circ}$.
B. $60^{\circ}$.
C. $135^{\circ}$.
D. $120^{\circ}$.

Question 18. Given rectangle $A B C D$ with $A D=2$. Suppose that $E$ is the point which lies on the side $A B$ such that $A E=2 B E$ and $\sin \widehat{B D E}=\frac{1}{5}$. Compute the length of the segment $A B$.
A. $A B=2 \sqrt{2}$.
B. $A B=3 \sqrt{3}$.
C. $A B=\sqrt{3}$.
D. $A B=\sqrt{6}$.

Question 19. In the $O x y$ coordinate plane, given $A(2 ;-6)$. Let $B$ be the point which is symmetric to point $A$ with respect to the origin $O$. Find the coordinates of point $C$ satisfying that its horizontal coordinate equals $-4$ and $\triangle A B C$ has the right angle at $C$.
A. $C(2 \sqrt{6} ;-4)$ or $C(-2 \sqrt{6} ;-4)$.
B. $C(-4 ; 24)$ or $C(-4 ;-24)$.
C. $C(-4 ;-2 \sqrt{6})$ or $C(-4 ; 2 \sqrt{6})$.
D. $C(24 ;-4)$ or $C(-24 ;-4)$.

Question 20. In the $O x y$ coordinate plane, let $M$ be the vertex of Parabol $y=a x^{2}+b x+c(a \neq 0)$. The coordinates of $M$ are
A. $\left(\frac{b}{2 a} ; \frac{4 a c-b^{2}}{4 a}\right)$.
B. $\left(\frac{-b}{4 a} ; \frac{4 a c-b^{2}}{4 a}\right)$.
C. $\left(\frac{-b}{2 a} ; \frac{b^{2}-4 a c}{4 a}\right)$.
D. $\left(\frac{-b}{2 a} ; \frac{4 a c-b^{2}}{4 a}\right)$.

Question 21. In the $O x y$ coordinate plane, given $A(1 ;-3)$ and $B(-5 ; 4)$. The coordinates of vector $\overrightarrow{B A}$ are
A. $\overrightarrow{B A}=(6 ; 7)$.
B. $\overrightarrow{B A}=(6 ;-7)$.
C. $\overrightarrow{B A}=(-4 ; 1)$.
D. $\overrightarrow{B A}=(-6 ; 7)$.

Question 22. Among the following propostions, whose inverse proposition is true?
A. If a triangle is not regular then it has at least one interior angle less than 60 degrees.
B. If two triangles are congruent then their corresponding angles are equal.
C. If $n$ is a natural number then $n$ is a real number.
D. If a quadrilateral is an isosceles trapezoid then its two diagonals have the same length.

Question 23. Given $\triangle A B C$. Let $M$ and $N$ be the mid-points of sides $A B$ and $A C$, respectively. Find the scalars $m$ and $n$ such that $\overrightarrow{N M}=m \overrightarrow{A B}-n \overrightarrow{A C}$.

A. $m=-\frac{1}{2}, n=\frac{1}{2}$.
B. $m=-\frac{1}{2}, n=-\frac{1}{2}$.
C. $m=\frac{1}{2}, n=\frac{1}{2}$.
D. $m=\frac{1}{2}, n=-\frac{1}{2}$.

Question 24. Given two non-zero vectors $\vec{a}$ and $\vec{b}$. Which of the following statements is false?
A. Two vectors $\vec{a}$ and $\vec{b}$ with opposite direction to another nonzero vector are parallel.
B. Two vectors $\vec{a}$ and $k \vec{a}$ are parallel.
C. Two vectors $\vec{a}$ and $-3 \vec{a}$ have the same direction.
D. Two vectors $\vec{a}$ and $\vec{b}$ with the same direction are parallel.

Question 25. The domain of the function $y=\frac{2}{\sqrt{6-2 x}}$ is
A. $D=(-\infty ; 3]$.
B. $D=(-\infty ; 3)$.
C. $D=(3 ;+\infty)$.
D. $D=\mathbb{R} \backslash\{3\}$.

Question 26. In the $O x y$ coordinate plane, given $\triangle A B C$. Points $M(-2 ; 3), N(4 ;-1), P(1 ; 1)$ are the mid-points of sides $B C, C A$ and $A B$, respectively. The coordinates of vertex $A$ are
A. $A(-10 ; 0)$.
B. $A(7 ;-3)$.
C. $A(-7 ; 3)$.
D. $A(10 ; 0)$.

Question 27. Which of the following sentences is not a proposition?
A. Five divides twenty.
B. If “$3+x=4$” then “$x=1$”.
C. If “$1+2=7$” then “$7$ is an odd number.
D. What a nice day!

Question 28. In the $O x y$ coordinate plane, let $A(-3 ;-5) ; B(2 ; 5)$. Determine the slope of line $A B$.
A. $-5$.
B. $5$.
C. $2$.
D. $-3$.

Question 29. Given a right triangle $A B C$ at $B$ with $A B=2 a, A C=5 a$. Compute the dot product $\overrightarrow{A B} \cdot \overrightarrow{C A}$.
A. $-5 a^{2}$
B. $4 a^{2}$.
C. $-4 a^{2}$
D. $5 a^{2}$

Question 30. Given an isosceles triangle $A B C$ with the right angle $A$, inscribed in a circle with center $O$ and radius $R$. Let $r$ be the radius of the incircle of triangle $A B C$. The ratio of $R$ to $r$ is
A. $\frac{R}{r}=\frac{1+\sqrt{2}}{2}$.
B. $\frac{R}{r}=1+\sqrt{2}$.
C. $\frac{R}{r}=\frac{2+\sqrt{2}}{2}$
D. $\frac{R}{r}=\frac{\sqrt{2}-1}{2}$.

Question 31. A man travels from city $X$ to city $Y$ by train, then returns to city $Y$ by his car. Given that the distance between these two cities is $200 \mathrm{~km}$ and the average speed of his car is $10 \mathrm{~km} / \mathrm{h}$ faster than the train’s average speed. His journey takes 9 hours, find the sum of average speeds of the train and his car.
A. $90$.
B. $80$.
C. $60$.
D. $100$.

Question 32. Let $a, b, c$ be real numbers and $a+2021 c>b+2021 c$. Which of the following statements is true?
A. $a^{2}>b^{2}$.
B. $-2020 a>-2020 b$.
C. $2021 a>2021 b$
D. $\frac{1}{a}>\frac{1}{b}$.

Question 33. Given two sets $X=\{A ; 1 ; 2 ; 4 ; 6\}, Y=\{3 ; 7 ; 4 ; \varnothing\}$, the union of $X$ and $Y$ is
A. $\{1 ; 2 ; 3 ; 4 ; 6 ; 7\}$.
B. $\{A ; 1 ; 2 ; 3 ; 4 ; 6 ; 7 ; \varnothing\}$.
C. $\{1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7\}$.
D. $\{A ; 1 ; 2 ; 3 ; 4 ; 6 ; 7\}$.

Question 34. In the $O x y$ coordinate plane, let Parabol $(P): y=a x^{2}+b x+3$ and a point $M(-1 ; 9)$ belongs to the graph of $(P)$. The symmetric axis of $(P)$ has equation $x=-2$. Find the value of $S=a+b$.
A. $-6$.
B. $16$.
C. $6$.
D. $-10$.

Question 35. The negation of the proposition “Fourteen is a composite number” is
A. Fourteen has four positive factors.
B. Fourteen has only two factors 1 and 14 .
C. Fourteen is a prime number.
D. Fourteen is not a composite number.

PART 2. PROBLEMS SOLVING (3,0 points)

Write the solutions to the following problems in the provided space on your answer sheet.

Problem 1. (1,0 point)

To measure the height of the Cham temple tower Po Klong Garai in Ninh Thuan province (Figure 1), two points $A$ and $B$ which are chosen on the ground with the length $A B=16 \mathrm{~m}$ and the bottom $C$ of the tower are collinear (Figure 2). Two total stations whose tripods have a height $h=1,6 m$ are put at point $A$ and point $B$. Let $D$ be the top of the tower and two points $A_{1}, B_{1}$ be collinear to $C_{1}$ on height $C D$ of the tower. The measurements are $\widehat{D A_{1} C_{1}}=54^{0}$ and $\widehat{D B_{1} C_{1}}=32^{\circ} .$ Caculate the height $C D$ of the tower then round the result to 3 decimal places.

Problem 2 (1,0 point).

Let $f(x)=a x^{3}+b x^{2}+c x+d$ be a cubic function with $f(0)=k, f(1)=2 k, f(-1)=3 k$, where $k$ is a given constant. What is the value of $f(2)+f(-2)$?

Problem 3 (1,0 point).

The sum of 2025 consecutive positive integers is a perfect square. Find the minimum value of the largest of these integers?