"Solving Triangle" Multiple-choice questions in previous college entrance examinations (self-test)
1. Multiple choice questions: (5 points for each small question, 40 points)
1. (Beijing 2008) It is known that in △ABC, a =, b =, b = 60, then the angle a is equal to ().
135 90(C)45(D)30
2.(2007 Chongqing Li) is in China, so BC = ()
A. 2 D BC.
3. (Shandong Arts and Sciences in 2006) In △ABC, if the opposite sides of angles A, B and C are A, B, C, A=, a=, b= 1, then c= ().
(A) 1 (B)2 (C)— 1 (D)
4. (Fujian 2008) In Chinese, the sides corresponding to angles A, B and C are A, B and C respectively. If so, the value of angle b is ().
ABC or d or
5. (Shanghai in 2005) is in △, if, then △ is ()
Right triangle. An equilateral triangle. Obtuse triangle. (d) an isosceles right triangle.
6. (A Volume of Literature and Theory in China in 2006) The opposite sides of inner angles A, B and C are A, B and C respectively. If a, b and c are in geometric series, then ().
A.B. C. D。
7.(2005 Beijing Spring Recruitment Literature and Theory) is in the middle, known, so it must be ()
A. right triangle B. isosceles triangle C. isosceles right triangle D. regular triangle
8. In △ABC, A, B and C are opposite sides of ∠A, ∠B and ∠C respectively. If a, b and c
Arithmetic progression, ∠ b= 30, and the area of △ABC is, then b= ().
A.B. C. D。
Fill in the blanks: (5 points for each small question, counting 30 points)
In △ABC, if AB = 1, BC = 2 and B = 60, then AC =.
10. (Hubei in 2008) In △ABC, A, B and C are opposite sides of angles A, B and C respectively, which is known.
Then a =
1 1.(2006 Beijing Science) Yes, if, the size is _ _ _ _.
12.(2007 Beijing Arts and Sciences) Chinese, if,,, then _ _ _ _.
13. (Hubei Theory in 2008) In △ABC, the opposite lengths of three angles A, B and C are A = 3, B = 4 and C = 6, respectively, so the values of bc cosA+ca cosB+ab cosC are.
14. (On Shanghai in 2005) In Chinese, if,,, then the area of S = _ _ _ _ _.
3. Problem-solving: (for each small question, 15, 16, 12 points, each is 14 points, accounting for 80 points).
15. (National Volume II in 2008) In China, …
(i) the value of; (II) Set and find the area.
16. (Shandong 2007) In Chinese, the opposite sides of the angle are.
( 1); (2) If, and ask.
17, (written in Ningxia, Hainan in 2008) as shown in the figure, △ACD is an equilateral triangle, △ABC is an isosceles right triangle, ∠ ACB = 90, BD intersects AC at point E, and AB=2. (1) Find the value of cos∠CBE; (2) looking for AE.
18. (National Volume 2, 2006), seeking
(1) (2) If the point
19.(2007 State Theory I) Let the opposite sides of the internal angles A, B and C of the acute triangle ABC be A, B, C and A = 2bsina respectively.
(i) Find out the size of b; (ii) The numerical range to be obtained.
O
20. (Guangdong, China, 2003) There was a typhoon near a coastal city. According to the monitoring, at present, the typhoon center is located on the sea surface 300km southeast of O city (as shown in the figure), and it moves to the northwest at the speed of 20 km/h. The typhoon attacks a circular area with the current radius of 60km and the speed of10 km/h.
"Solving Triangle" Multiple-choice questions in previous college entrance examinations (self-test)
Reference answer
1. Multiple choice questions: (5 points for each small question, 40 points)
Fill in the blanks: (5 points for each small question, counting 30 points)
9.； 10.30 ; . 1 1.__ 60O _。 12.； 13.； 14.
3. Problem-solving: (for each small question, 15, 16, 12 points, each is 14 points, accounting for 80 points).
15. Solution: (I) You, De, You, De.
So ...
(2) Derived from sine theorem.
So this area.
16. Solution: (1)
Also solved.
, is an acute angle.
(2)∫ that is, abcosC= and cosC=.
and .. / with ..
. .
17. Solution: (1) Because,, so.
So ...
(ii) In the middle,
According to the sine theorem.
therefore
18. Solution: (1) by
According to sine theorem
(2),
According to cosine theorem
19. Solution: (i) is derived from sine theorem, so,
This is an acute triangle.
(Ⅱ)
.
From an acute triangle.
So,
Therefore, from this point of view,
Therefore, the value range of is.
20. Solution: Let the typhoon center be located at point Q at time t, and at this time |OP|=300, |PQ|=20t.
The radius of the circular area of the typhoon attack range is r(t)= 10t+60,
O
It can be seen from that,
cos∠OPQ = cos(θ-45o)= cosθcos 45o+sinθsin 45o
=
At △OPQ, from cosine theorem, we get
=
=
If the city O encounters a typhoon, there is |OQ|≤r(t), that is
Sort out, get, solve 12≤t≤24,
A: 12 hours later, the city began to be hit by a typhoon.
20 10 Mathematics Target Training for College Entrance Examination (1) (Liberal Arts Edition)
Time: 60 minutes, full mark: 80 minutes, class: name: score:
Personal goal: □ Excellent (70'~80') □ Good (60' ~ 69') □ Qualified (50' ~ 59')
First, multiple-choice questions: this big question is ***5 small questions, with 5 points for each small question, out of 25 points.
1. If the complex number is purely imaginary, the value of the real number A is
A. 1 B.2 C. 1 or 2 D.- 1
2. Let the common ratio of geometric series q=2, and the sum of the first n terms is Sn, then = ().
A.B. C. D。
3. Let p be the point on curve C: y=x2+2x+3, and the range of tangent inclination of curve C at point P is
The value range of the abscissa of point P is
(A) (B) (C)[0， 1] (D)
4. In △ABC, the opposite sides of the angle ABC are A, B and C respectively. If so, the value of angle b is
ABC or d or
5. Cut the ball with a plane with a distance from the center of the ball, and the cross-sectional area obtained is, and the volume of the ball is.
A.B. C. D。
Fill-in-the-blank question: There are three small questions in this big question, each with 5 points, out of 15.
6, then the included angle is 0, then
7. If the constraint conditions are met, the maximum value is.
8. If the straight line and the circle (as parameters) have no common point,
The range of real number m is
Three. Solution: This big question * * * has 3 small questions, out of 40, the ninth small question 12, the first 10, the first1/each small question 14. The solution must be written in words, proof process or calculation steps.
9. Because of the snow and ice disaster, the fruit forest of a citrus base was seriously damaged. Therefore, experts put forward a plan to save fruit trees, which needs to be implemented within two years and independent of each other. It is estimated that the probability of citrus yield recovering to 1.0 times, 0.9 times and 0.8 times in the first year is 0.2, 0.4 and 0.4 respectively. The probability that the citrus yield in the second year is 1.5 times, 1.25 times and 1.0 times that in the first year is 0.3, 0.3 and 0.4 respectively.
(1) Find the probability that the citrus yield will just reach the pre-disaster yield after two years;
(2) Find the probability that the citrus yield will exceed the pre-disaster yield after two years.
10, set the plane rectangular coordinate system xoy, set the image of quadratic function and two coordinate axes with three intersections, and mark the circle passing through these three intersections as c, and ask:
(1) range of real number b
(2) find the equation of circle c
(3) Does circle C pass through a fixed point (its coordinates have nothing to do with B)? Please prove your conclusion.
1 1, in the series, …
(1) Hypothesis. Prove that the sequence is arithmetic progression;
(2) Find the sum of the previous paragraph of the series.
Detailed explanation of the answer
First, multiple-choice questions: this big question is ***5 small questions, with 5 points for each small question, out of 25 points.
1. If the complex number is purely imaginary, the value of the real number A is
A. 1 B.2 C. 1 or 2 D.- 1
Solution: yes, and (pure imaginary number must make the imaginary part not 0)
2. Let the common ratio of geometric series q=2, and the sum of the first n terms is Sn, then = ().
A.B. C. D。
Solution:
3. Let p be the point on curve C: y=x2+2x+3, and the range of tangent inclination of curve C at point P is
The value range of the abscissa of point P is
(A) (B) (C)[0， 1] (D)
Analysis: This small question mainly examines the problem of finding tangent slope by using the geometric meaning of derivative. Sets the abscissa of the tangent point according to the theme.
Is, and (is the inclination of the tangent at point P), and \
∴,∴
4. In △ABC, the opposite sides of the angle ABC are A, B and C respectively. If so, the value of angle b is
ABC or d or
Solution: What you get is what you get.
And in δ, so b is or.
5. Cut the ball with a plane with a distance from the center of the ball, and the cross-sectional area obtained is, and the volume of the ball is.
A.B. C. D。
Solution: The cross-sectional area is 1 and the radius of the sphere is 1.
So according to the volume formula of the ball, B is the correct answer.
Fill-in-the-blank question: There are three small questions in this big question, each with 5 points, out of 15.
6. If the included angle is 0, it is 7.
7. If the constraint conditions are met, the maximum value is 9.
8. If the straight line and the circle (as parameters) have no common point,
The range of real number m is
Solution: the center of the circle is, if there is no common point, it can be obtained according to the fact that the distance from the center of the circle to the straight line is greater than the radius.
, that is,
Three. Solution: This big question * * * has 3 small questions, out of 40, the ninth small question 12, the first 10, the first1/each small question 14. The solution must be written in words, proof process or calculation steps.
9. Because of the snow and ice disaster, the fruit forest of a citrus base was seriously damaged. Therefore, experts put forward a plan to save fruit trees, which needs to be implemented within two years and independent of each other. It is estimated that the probability of citrus yield recovering to 1.0 times, 0.9 times and 0.8 times in the first year is 0.2, 0.4 and 0.4 respectively. The probability that the citrus yield in the second year is 1.5 times, 1.25 times and 1.0 times that in the first year is 0.3, 0.3 and 0.4 respectively.
(1) Find the probability that the citrus yield will just reach the pre-disaster yield after two years;
(2) Find the probability that the citrus yield will exceed the pre-disaster yield after two years.
Solution: (1) Order A represents an event in which the citrus output just reached the pre-disaster output two years later.
(2) Order B indicates the event that the citrus output exceeds the pre-disaster output two years later.
10, set the plane rectangular coordinate system xoy, set the image of quadratic function and two coordinate axes with three intersections, and mark the circle passing through these three intersections as c, and ask:
(1) range of real number b
(2) find the equation of circle c
(3) Does circle C pass through a fixed point (its coordinates have nothing to do with B)? Please prove your conclusion.
Analysis: This small question examines the solution of the equation of quadratic function image in nature and circle.
(1) Let x=0, and the intersection of parabolas on the y axis is (0, b).
Let f(x)=0, and x2+2x+b=0, from b≠0 and △ >; 0 and b
(2) Let the general equation of a circle be x2+ y2+Dx+Ey+F=0.
Let y=0, x2+Dx+F=0, and x2+2x+b=0 are the same equation, so D=2 and f = b.
Let x=0, and you get y2+ Ey+b=0. This equation has a root of b, and you get E=-b- 1.
So the equation of circle C is x2+ y2+2x -(b+ 1)y+b=0.
(3) Circle C must pass through a fixed point (0, 1), (-2, 1).
The proof is as follows: If (0, 1) is substituted into the equation of circle C, the left side = 02+12+2× 0-(b+1)×1+b = 0, and the right side = 0.
So circle C must pass through a fixed point (0,1); It can also be proved that circle C must pass through a fixed point (-2, 1).
1 1, in the series, …
(1) Hypothesis. Prove that the sequence is arithmetic progression;
(2) Find the sum of the previous paragraph of the series.
Solution: (1),
It was arithmetic progression,
,.
(2)
Subtract two expressions to get.
.