Questions and Answers on Unit Training of Necessary and Sufficient Conditions in the Second Volume of Mathematics in Senior Two.
First, multiple-choice questions (6 points for each small question, ***42 points)
1. It is known that A and B are two propositions. If A is a sufficient but unnecessary condition of B, then A is B's ().
A. Sufficient but not necessary conditions B. Necessary but not sufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
Answer: b
Analysis:? A B B A？ ,? B A？ Equivalent to? A b? .
2.(20 10 Zhejiang Hangzhou No.2 middle school simulation, 4)? A>2 and b>2? what's up a+b & gt； 4 and ab>4? ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
A: A.
Analysis: the sufficiency is obvious, when A = 5 and B = 1, there is a+b >; 4. ab>4, but? A>2 and b>2? It's not true.
3.(20 10, Model 1, Xicheng District, Beijing 5) Set A, B? What about r? a & gtb？ what's up a & gt|b|？ ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition
Answer: b
Analysis: a>b didn't get a & gt|b|.
For example, 2 >; -5, but 2 < |-5|, and a>| b | a>b. So choose B.
4. Known condition p:|x|=x, condition q:x2? -x, then p is q's ().
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Necessary and sufficient conditions are neither sufficient nor necessary.
A: A.
Analysis: p:A={0, 1}, q:B={x|x? -1 or x? 0}.
∫A B，？ A necessary and sufficient condition for p to be q.
5. known real question:? Answer? Is b a necessary and sufficient condition for c>d? And then what? a
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. sufficient and necessary conditions are neither sufficient nor necessary.
A: A.
Analysis:? Answer? Is b a necessary and sufficient condition for c>d? Equivalent to? c？ District attorney
6.(20 10 National College Entrance Examination, 2) Inequality 10 holds ()
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. necessary and sufficient conditions D. insufficient and unnecessary conditions.
A: A.
Analysis: when 10, tanx & gt0,? That is tan (x-1) tanx >; 0, but when x=, (x- 1)tanx=(-1)? 1 & gt； 0, and (1,), so choose a.
7. Known parabola y = AX2+BX+C (a >; 0，b，c？ R) then what? On the inequality ax2+bx+c of x
A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions
C. Sufficient and necessary conditions D. Conditions that are neither sufficient nor necessary
Answer: b
Analysis: ax2+bx+c0, the vertex (-) is in the straight line Y = X-(B-1) 2 > 4ac+ 1, so choose B.
Two. Fill in the blanks (5 points for each small question, *** 15 points)
8. Equation 3x2- 10x+k=0 has two real roots with the same sign but not equal if and only if it is _ _ _ _ _ _ _ _.
Answer: 0
Analysis: Its necessary and sufficient condition is 0.
9. Known pests: |x+ 1| >2 and q: > 0, then P is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (Fill in? Sufficient, unnecessary, unnecessary, insufficient, sufficient and necessary conditions are neither sufficient nor necessary? Conditions? )
Answer: It is totally unnecessary.
Analysis: p: x < -3 or x> 1,
Q: x & lt-4 or x> 1,
? p:-3？ x？ 1，q:-4？ x？ 1.
? A necessary and sufficient condition for p to be q.
10. Give the following p and q groups:
( 1)p:x2+x-2=0，q:x =-2；
(2)p:x=5，q:x & gt； -3;
(3)p: internal dislocation angles are equal, and q: two straight lines are parallel to each other;
(4)p: two angles are equal, q: two angles are diagonal;
(5)p:x？ M and x? p，q:x？ m？ P(P，M？ ).
The serial number of the group where P is the necessary and sufficient condition and the unnecessary condition of Q is _ _ _ _ _ _ _ _ _ _ _ _.
Answer: (2)(5)
Analysis: (1) (4) Necessary and sufficient conditions for P to be Q; ? (3)P is a necessary and sufficient condition of Q; (2)(5) Satisfy the meaning of the question.
Third, answer questions (1 1? 13 is 10, 14 is 13, ***43)
1 1. let x, y? R, verification: | x+y | = | x |+y | is a necessary and sufficient condition for xy? 0.
Proof: sufficiency: if xy=0, then ①x=0, y? 0; ②y=0，x？ 0; ③x=0，y=0。 So | x+y | = | x |+y |.
If xy>0 is x>0, y>0 or X.
When x>0, y>0, |x+y|=x+y=? |x|+|y|？ ;
When x
Necessity: solution 1: from | x+y | = | x |+y | and x, y? R, get (x+y) 2 = (| x |+y|) 2, that is, x2+2xy+y2=x2+2|xy|+y2, |xy|=xy,? xy？ 0.
Solution 2: | x+y | = | x |+y| (x+y) 2 = (| x |+y |) 2x2+y2+2xy = x2+y2+2 | xy | xy = | xy | xy? 0.
12. It is known that A and B are real numbers. It is proved that the sufficient condition that a4-b4= 1+2b2 is a2-b2= 1. Is this a necessary condition? Prove your conclusion.
Prove that this condition is necessary.
When a2-b2= 1, that is, a2=b2+ 1,
a4-B4 =(B2+ 1)2-B4 = 2 B2+ 1。
? A4-b4= 1+2b2 holds if a2-b2= 1, a4-b4= 1+2b2, then a4=(b2+ 1)2.
? A2=b2+ 1, that is, a2-b2= 1, so this condition is necessary.
13. The equation about x is known: (a-6)x2-(a+2)x- 1=0. (a？ R), and find the necessary and sufficient condition that the equation has at least one negative root.
Analysis: ∵ When a=6, the original equation is 8x=- 1 and the negative root x=-.
When a. At 6 o'clock, the equation has positive roots and negative roots if and only if: x 1x2 =-
A necessary and sufficient condition for an equation to have two negative roots is:
That's two? a & lt6.
? The necessary and sufficient conditions for an equation to have at least one negative root are: 2? A<6 or a=6 or a>6, that is, a? 2.
14.( 1) Is there a real number p? 4x+p & lt； 0? what's up x2-x-2 & gt； 0? Sufficient conditions? If it exists, find out the range of p;
(2) Is there a real number p that makes? 4x+p & lt； 0? what's up x2-x-2 & gt； 0? Necessary conditions? If it exists, find the range of p.
Analysis: (1) When x>2 or X; 0,
Increase by more than 4 times
? x & lt-x & lt； - 1 x2-x-2 & gt； 0? .
? p？ At 4 o'clock, 4x+p <; 0? what's up x2-x-2 & gt； 0? Sufficient conditions.