Find out when the expression is exactly divisible by
Let where A and B are to be determined in terms of p, q, r, a and b. We can assume so because we know from the fundamental theorem of algebra that the if the LHS has to be of degree three in x, the remaining factor in RHS has to be linear in x.
So, expanding out the RHS of above, we get:
We are saying that the above is true for all values of x: hence, coefficients of like powers of x on LHS and RHS are same; we equate them and get a system of equations:
Hence, we get and or that
Also, so that which means
but and hence,
So, the required conditions are and .
2) Find the condition that may be a perfect square.
Let where A and B are to be determined in terms of p and q; finally, we obtain the relationship required between p and q for the above requirement.
which is true for all real values of x;
Hence, so or
Also, and hence, or
Also, so that so , which is the required condition.
3) To prove that is a perfect square if and .
PS: Note in the method of undetermined coefficients, we create an identity expression which is true for all real values of x.
Discuss the following “proof” of the (false) theorem:
If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:
PROOF BY INDUCTION:
If , the result is evident.
Step 2: By the induction hypothesis the result is true when ; we must prove that it is correct when . Let S be any set containing exactly real numbers and denote these real numbers by . If we omit from this list, we obtain exactly k numbers ; by induction hypothesis these numbers are all equal:
If we omit from the list of numbers in S, we again obtain exactly k numbers ; by the induction hypothesis these numbers are all equal:
It follows easily that all numbers in S are equal.
For the following tutorial problems, it helps to know/remember/understand/apply the following identities (in addition to all other standard/famous identities you learn in high school maths):
By the way, I hope you also know how to derive the above.Let me mention two methods to derive the above :
Method I: Using polynomial division in three variable, divide the dividend by the divisor .
Method II: Assume that is a polynomial with roots a, b and c. So, we know by the fundamental theorem of algebra that . Now, we also know that a, b and c satisfy P(X). Now, proceed further and complete the proof.
Let us now work on the tutorial problems below:
1) If , prove that
2) If , prove that .
Prove the following identities:
11) Prove that
12) If3 , prove that
13) If , prove that
14) If , prove that
15) If , then prove that
16) If , then prove that
17) If , then prove that
18) Prove that
19) If prove that
20) If , , , find the value of
21) Prove that
22) Prove that
23) if , prove that .
24) If , , , prove that
25) If , prove that
26) If , simplify:
27) Prove that the equation is equivalent to the equation , hence show that the only possible values of x and y are: ,
28) If , prove that and therefore that and are the only possible solutions.
1) Show that quadrilateral ABCD can be inscribed in a circle iff and are supplementary.
2) Prove that a parallelogram having perpendicular diagonals is a rhombus.
3) Prove that a parallelogram with equal diagonals is a rectangle.
4) Show that the diagonals of an isosceles trapezoid are equal.
5) A straight line cuts two concentric circles in points A, B, C and D in that order. AE and BF are parallel chords, one in each circle. If CG is perpendicular to BF and DH is perpendicular to AE, prove that .
6) Construct triangle ABC, given angle A, side AC and the radius r of the inscribed circle. Justify your construction.
7) Let a triangle ABC be right angled at C. The internal bisectors of angle A and angle B meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.
8) Three circles with radii , with . They are placed such that lies to the right of and touches it externally; lies to the right of and touches it externally. Further, there exist two straight lines each of which is a direct common tangent simultaneously to all the three circles. Find in terms of and .