Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Let a be any positive integer and b = 3. Then, by euclid’s algorithm
a = 9q + r, where q ≥ 0 and r = 0, 1, 2 [ 0 ≤ r ≤ b ]
For r = 0, x = 3q, or
For r = 1, x = 3q +1
For r = 2, x = 3q + 2
Now by taking the cube of all the three above terms, we get,
Case (i) : when r = 0, then,
x3 = (3q)3 = 27q3 = 9 (3q3) = 9m; where m = 3q
Case (ii) : when r = 1, then,
x3 = (3q +1)3 = (3q)3 + 13 +3 × 3q × 1 (3q + 2 ) = 27q3 + 1 +27q2 + 9q
Taking 9 as common factor, we get,
x3 = 9 (3q3 +3q2 + q) +1
Putting (3q3 + 3q2 + q) = m, we get,
x3 = 9m + 1
Case (iii) : when r = 2, then,
x3 = (3q + 2)3 = (3q)3 + 23 + 3 × 3q × 2 (3q + 2) = 27q3 + 54q2 + 36q + 8
Taking 9 as common factor , we get
x = 9 (3q + 6q + 4q) + 8
Putting (3q + 6q + 4q) = m, we get,
x = 9m + 8,
Therefore, it is proved that the cube of any positive integer is of the form 9m, 9m + 1, 9m + 8.
Prove that 3 + 2√5 is irrational.
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Check whether 6n can end with the digit 0 for any natural number n.
Prove that √5 is irrational.
Given that HCF (306, 657) = 9, find LCM (306, 657).
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p/q what can you say about the prime factors of q?
Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.
Complete the following statements:
(i) Probability of an event E + Probability of the event ‘not E’ = .
(ii) The probability of an event that cannot happen is . Such an event is called .
(iii) The probability of an event that is certain to happen is . Such an event is called .
(iv) The sum of the probabilities of all the elementary events of an experiment is .
(v) The probability of an event is greater than or equal to and less than or equal to .
Check whether the following are quadratic equations :
(i) (x + 1)2 = 2(x – 3) (ii) x2 – 2x = (–2) (3 – x) (iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x2+ 3x + 1 = (x – 2)2 (vii) (x + 2)3 = 2x (x2 – 1) (viii) x3 – 4x2 – x + 1 = (x – 2)3
How many tangents can a circle have?
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically.
Which of the following experiments have equally likely outcomes? Explain.
(i) A driver attempts to start a car. The car starts or does not start.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.
Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3.
Find the number of blue balls in the jar.
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Refer to Example 13. (i) Complete the following table:
(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability
Do you agree with this argument? Justify your answer.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.
Find the roots of the following quadratic equations, if they exist, by the method of
completing the square:
(i) 2x2 – 7x + 3 = 0 (ii) 2x2 + x – 4 = 0 (iv) 2x2 + x + 4 = 0
Suppose you drop a die at random on the rectangular region shown in Fig. 15.6. What is the probability that it will land inside the circle with diameter 1m?