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Bộ 40 đề thi vào lớp 10 môn Toán chọn lọc và hay nhất Đề thi tuyển sinh lớp 10 môn Toán (Có đáp án)

The set of 40 exam questions for 10th grade math is an extremely useful document that Pubokid.vn would like to introduce to teachers and 9th grade students for reference.

10th Math exam questions Below are issued by Ha Tinh Department of Education and Training, including 40 math entrance exam questions for 10th grade with detailed answers attached. The 10th grade math exam is compiled according to the central and scientific topics, suitable for all students with average, good to excellent academic ability. Thereby helping students consolidate and firmly grasp the foundational knowledge and apply it with basic exercises; Students with good academic performance improve their thinking and problem-solving skills with advanced application exercises. So here are 40 entrance exams for 10 math subjects, please read and download here.

Maths 10th grade entrance exam with answers

  • 10th Math Exam – Topic 1
  • 10th Math Exam – Question 2
  • Maths 10th grade entrance exam – Question 3
  • Maths 10th grade entrance exam – Question 4
  • Maths 10th grade entrance exam – Topic 5
  • Maths 10th grade entrance exam – Topic 6
  • Maths 10th grade entrance exam – Question 7

10th Math Exam – Topic 1

Question 1: a) Tell a =2+sqrt{3} and mathrm{b}=2-sqrt{3}. Calculate the expression value: mathrm{P}=mathrm{a}+mathrm{b}-mathrm{ab}.

b) Solve the system of equations: left{begin{array}{l}3 x+y=5 \ x-2 y=-3end{array}right..

Verse 2: For the expression mathrm{P}=left(frac{1}{mathrm{x}-sqrt{mathrm{x}}}+frac{1}{sqrt{mathrm{x}}-1} right): frac{sqrt{mathrm{x}}}{mathrm{x}-2 sqrt{mathrm{x}}+1}( with mathrm{x}>0, mathrm{x} neq 1)” width=”103″ height=”21″ data-latex=”mathrm{x}>0, mathrm{x} neq 1)” class=”lazy” data-src=”https://tex.vdoc.vn?tex=%5Cmathrm%7Bx%7D%3E0%2C%20%5Cmathrm%7Bx%7D%20%5Cneq%201)”></p>
<p>a) Simplify the expression P</p>
<p>b) Find the values ​​of x to <img src=Sentence 3: Given the equation: mathrm{x}^{2}-5 mathrm{x}+mathrm{m}=0 (m is the parameter).

a) Solve the above equation when mathrm{m}=6.

b) Find m where the above equation has two solutions mathrm{x}_{1}, mathrm{x}_{2} satisfy: left|mathrm{x}_{1}-mathrm{x}_{2}right|=3.

Question 4: Given a circle with center O and diameter AB. Draw the chord CD perpendicular to AB at I (I lies between A and ). mathrm{O}). Taking the point E on the minor arc BC E is different from B and C, AE cuts CD at F. Prove:

a) BEFI is a quadrilateral inscribed in the circle.

b)mathrm{AE} cdot mathrm{AF}=mathrm{AC}^{2}

c) When E runs on the minor arc BC, the circumcenter of the circle Delta CEF always on a fixed line.

Question 5: Given two positive numbers a, b satisfying: mathrm{a}+mathrm{b} leq 2 sqrt{2}. Find the minimum value of the expression: quad mathrm{P}=frac{1}{mathrm{a}}+frac{1}{mathrm{~b}}.

10th Math Exam – Question 2

Question 1: a) Simplify the expression: frac{1}{3-sqrt{7}}-frac{1}{3+sqrt{7}}.

b) Solve the equation: x^{2}-7 x+3=0.

Verse 2: a) Find the coordinates of the intersection of the line d: y=-x+2 and the Parabola (P): y =x^{2}.

b) For the system of equations: left{begin{array}{l}4 x+ay=b \ xb y=aend{array}right.. Find a and b of the given system that has a unique solution (mathrm{x} ; mathrm{y})=(2 ;-1) .

Question 3: A train needs to transport a quantity of goods. The driver calculated that if each carriage was loaded with 15 tons of cargo, there would be 5 tons left over, and if each carriage was packed with 16 tons, it could carry another 3 tons. Bald trains have how many cars and how many tons of goods must be carried.

Verse 4: From a point A outside the circle (O;R), we draw two tangents AB, AC to the circle (B, C are the tangent). On minor arc BC take a point M, draw mathrm{MI} perp mathrm{AB}, mathrm{MK} perp mathrm{AC}(mathrm{I} in mathrm{AB}, mathrm{K} in mathrm{AC })

a) Prove: AIMK is a quadrilateral inscribed in a circle.

b) operatorname{Ver} mathrm{MP} perp mathrm{BC}(mathrm{P} in mathrm{BC}). Prove: mathrm{MPK}=mathrm{MBC}.

c) Determine the position of the point M on the minor arc BC the product MI.MK.MP has the maximum value.

Question 5: Solve the equation: frac{sqrt{x-2009}-1}{x-2009}+frac{sqrt{y-2010}-1}{y-2010}+frac{sqrt{z-2011}-1 }{z-2011}=frac{3}{4}

Maths 10th grade entrance exam – Question 3

Question 1: Solve the following equation and system of equations:

a) x^{4}+3 x^{2}-4=0

b) left{begin{array}{l}2 x+y=1 \ 3 x+4 y=-1end{array}right.

Verse 2: Extract the expressions:

a) A=frac{sqrt{3}-sqrt{6}}{1-sqrt{2}}-frac{2+sqrt{8}}{1+sqrt{2}}

b) mathrm{B}=left(frac{1}{mathrm{x}-4}-frac{1}{mathrm{x}+4 sqrt{mathrm{x}}+4} right) cdot frac{mathrm{x}+2 sqrt{mathrm{x}}}{sqrt{mathrm{x}}} quad

Question 3:

a) Graph the functions y = – x2 and y = x – 2 on the same coordinate system.

b) Find the intersection coordinates of the graphs drawn above by calculation.

Question 4: Let ABC be a triangle with three acute angles inscribed in the circle (O; R). The altitudes BE and CF intersect at H.

a) Prove: AEHF and BCEF are quadrilaterals inscribed in a circle.

b) Let M and N, respectively, be the second intersection of the circle (O;R) with BE and CF. Proof: MN // EF.

c) Prove that OA is perpendicular to EF.

Question 5: Find the minimum value of the expression:

mathrm{P}=mathrm{x}^{2}-mathrm{x} sqrt{mathrm{y}}+mathrm{x}+mathrm{y}-sqrt{mathrm{y }}+1

Maths 10th grade entrance exam – Question 4

Question 1:

a) Radial axis in the form of the following expressions: frac{4}{sqrt{3}} ;  frac{sqrt{5}}{sqrt{5}-1}.

b) In the coordinate system mathrm{Oxygen}know the graph of the function mathrm{y}=mathrm{ax}^{2} pass through point mathrm{M}left(-2 ; frac{1}{4}right). Find the coefficient a.

Verse 2: Solve the following equation and system of equations:

a) sqrt{2 x+1}=7-x

b) left{begin{array}{l}2 x+3 y=2 \ xy=frac{1}{6}end{array}right.

Question 3: For the hidden equation mathrm{x}: mathrm{x}^{2}-2 mathrm{mx}+4=0 (1)

a) Solve the given equation when m = 3

b) Find the value of m so that equation (1) has two constraints mathrm{x}_{1}, mathrm{x}_{2} satisfy: left(mathrm{x}_{1}+1right)^{2}+left(mathrm{x}_{2}+1right)^{2}=2.

Question 4: Let ABCD be a square with two diagonals intersecting at E. Take I on side AB, M on side BC such that: mathrm{IEM}=90^{circ} (I and M do not coincide with the vertices of the square).

a) Prove that BIEM is a quadrilateral inscribed in a circle.

b) Calculate the measure of angle IME

c) Let N be the intersection of ray AM and ray DC; K is the intersection of BN and ray EM. Prove mathrm{CK} perp mathrm{BN}

Question 5: Let a, b, c be the lengths of the 3 sides of a triangle. Prove:

a b+b c+ca leq a^{2}+b^{2}+c^{2}<2(a b+b c+ca)

Maths 10th grade entrance exam – Topic 5

Question 1:

a) Perform the calculation:left(sqrt{frac{3}{2}}-sqrt{frac{2}{3}}right) cdot sqrt{6}

b) In the coordinate system Oxy, know the line mathrm{y}=mathrm{ax}+mathrm{b} passing through point A (2 ; 3 ) and point B (-2 ; 1) Find the coefficients a and b.

Verse 2: Solve the following equation:

a) x^{2}-3 x+1=0

b) frac{x}{x-1}+frac{-2}{x+1}=frac{4}{x^{2}-1}

Sentence 3: Two cars start at the same time on a distance of 120 km from A to B. The first car is 10 km faster than the second car every hour, so it will arrive at B 0.4 hours before the second car. Calculate the velocity of each car.

Question 4: Given circle (O, R) ; AB and CD are two different diameters of the circle. The tangent at B of the circle (O; R) intersects the lines mathrm{AC}, mathrm{AD} order at E and F.

a) Prove quadrilateral mathrm{ACBD} is a rectangle.

b) Prove triangle mathrm{ACD} sim triangle mathrm{CBE}

c) Prove that quadrilateral CDFE is inscribed in a circle.

d) Call mathrm{S}, mathrm{S}_{1}, mathrm{~S}_{2} order is the area of triangle mathrm{AEF}, triangle mathrm{BCE} and triangle mathrm{BDF}. Prove: sqrt{mathrm{S}_{1}}+sqrt{mathrm{S}_{2}}=sqrt{mathrm{S}}.

Question 5: Solve the equation:10 sqrt{mathrm{x}^{3}+1}=3left(mathrm{x}^{2}+2right)

Maths 10th grade entrance exam – Topic 6

Question 1: Simplify the following expressions:

a) mathrm{A}=left(2+frac{3+sqrt{3}}{sqrt{3}+1}right) cdotleft(2-frac{3-sqrt {3}}{sqrt{3}-1}right)

b) mathrm{B}=left(frac{sqrt{mathrm{b}}}{mathrm{a}-sqrt{mathrm{ab}}}-frac{sqrt{mathrm {a}}}{sqrt{mathrm{ab}}-mathrm{b}}right) cdot(mathrm{a} sqrt{mathrm{b}}-mathrm{b} sqrt {mathrm{a}}) quad( with mathrm{a}>0, mathrm{~b}>0, mathrm{a} neq mathrm{b})” width=”585″ height=”59″ data-type=”0″ data-latex=”b) mathrm{B}=left(frac{sqrt{mathrm{b}}}{mathrm{a}-sqrt{mathrm{ab}}}-frac{sqrt{mathrm{a}}}{sqrt{mathrm{ab}}-mathrm{b}}right) cdot(mathrm{a} sqrt{mathrm{b}}-mathrm{b} sqrt{mathrm{a}}) quad( với mathrm{a}>0, mathrm{~b}>0, mathrm{a} neq mathrm{b})” class=”lazy” data-src=”https://tex.vdoc.vn?tex=b)%20%5Cmathrm%7BB%7D%3D%5Cleft(%5Cfrac%7B%5Csqrt%7B%5Cmathrm%7Bb%7D%7D%7D%7B%5Cmathrm%7Ba%7D-%5Csqrt%7B%5Cmathrm%7Bab%7D%7D%7D-%5Cfrac%7B%5Csqrt%7B%5Cmathrm%7Ba%7D%7D%7D%7B%5Csqrt%7B%5Cmathrm%7Bab%7D%7D-%5Cmathrm%7Bb%7D%7D%5Cright)%20%5Ccdot(%5Cmathrm%7Ba%7D%20%5Csqrt%7B%5Cmathrm%7Bb%7D%7D-%5Cmathrm%7Bb%7D%20%5Csqrt%7B%5Cmathrm%7Ba%7D%7D)%20%5Cquad(%20v%E1%BB%9Bi%20%5Cmathrm%7Ba%7D%3E0%2C%20%5Cmathrm%7B~b%7D%3E0%2C%20%5Cmathrm%7Ba%7D%20%5Cneq%20%5Cmathrm%7Bb%7D)”></p>
<p></strong><span style=Verse 2:

a) Solve the system of equations: left{begin{array}{l}xy=-1 \ frac{2}{x}+frac{3}{y}=2end{array}right.  (2)

b) Call mathrm{x}_{1}, mathrm{x}_{2} are two solutions of the equation:mathrm{x}^{2}-mathrm{x}-3=0. Calculate the expression value: mathrm{P}=mathrm{x}_{1}^{2}+mathrm{x}_{2}^{2}.

Question 3:

a) Know the straight line mathrm{y}=mathrm{ax}+mathrm{b} pass through point mathrm{M}left(2 ; frac{1}{2}right) and parallel to the line 2 mathrm{x}+mathrm{y}=3. Find the coefficients a and b.

b) Calculate the dimensions of a rectangle whose area is 40 mathrm{~cm}^{2}knowing that if each dimension is increased by 3 cm, the area increases by 48 cm2

Question 4: For triangle mathrm{ABC} square at mathrm{A}, mathrm{M} is a point on side AC (M is different from A and C). The circle with diameter MC intersects BC at N and intersects ray BM at I. Prove that:

a) ABNM and ABCI are quadrilaterals inscribed in the circle.

b) NM is the bisector of angle widehat{mathrm{ANI}}.

c) mathrm{BM} .  mathrm{BI}+mathrm{CM} cdot mathrm{CA}=mathrm{AB}^{2}+mathrm{AC}^{2}.

Question 5: For the expression A=2 x-2 sqrt{xy}+y-2 sqrt{x}+3. Does A have a minimum value? Why?

Maths 10th grade entrance exam – Question 7

Question 1:

a) Find the condition of x the following expression means: mathrm{A}=sqrt{mathrm{x}-1}+sqrt{3-mathrm{x}}

b) Calculate:frac{1}{3-sqrt{5}}-frac{1}{sqrt{5}+1}

Verse 2: Solve the following equations and inequalities:

a) (x-3)^{2}=4

b) frac{x-1}{2 x+1}<frac{1}{2}

Question 3: Given the equation for the hidden x: x^{2}-2 m x-1=0 (1)

a) Prove that the given equation always has two distinct solutions x_{1} and x_{2}.

b) Find the values ​​of m so that: mathrm{x}^{2}+mathrm{x}^{2}{ }^{2}-mathrm{x}_{1} mathrm{X}_{2}=7.

Question 4: Given a circle (O ; R) with diameter AB. Draw chord CD perpendicular to AB (CD does not pass through center O). On the opposite ray of ray BA take the point S, SC intersects (O, R) at the second point M.

a) Prove triangle mathrm{SMA} similar to triangle mathrm{SBC}.

b) Let H be the intersection of MA and BC; K is the intersection of MD and AB. Prove that BMHK is a cyclic quadrilateral and mathrm{HK} / / mathrm{CD}.

c) Prove: mathrm{OK} .  mathrm{OS}=mathrm{R}^{2}.

Question 5: Solve the system of equations: left{begin{array}{l}x^{3}+1=2 y \ y^{3}+1=2 xend{array}right..

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