Welcome to the NCERT Solutions for Class 10th Mathematics - Chapter Real Numbers. This page offers a step-by-step solution to the specific question from Exercise 1, Question 2: **show that any positive odd integer is of the form...**.

Question 2

Show that any positive odd integer is of the form 6*q* + 1, or 6*q* + 3, or 6*q* + 5, where *q* is some integer.

Answer

Let *a* be any positive integer b = 6. Then by Euclid’s algorithm,

a = 6*q *+ r for some integer *q* ≥ 0

So, values of *r* we get, *r* = 0, 1, 2, 3, 4, 5

When, *r* = 0, then, a = 6*q* + 0,

similarly for r = 1, 2, 3, 4, 5 the value of a is 6*q*+1, 6*q*+2, 6*q*+3, 6*q*+4 and 6*q*+5 respectively.

If a = 6*q*, 6*q*+2, 6*q*+4, then a is even and divisible by 2.

Therefore, any positive odd integer is in the form of 6*q*+1, 6*q*+3, 6*q*+5

Where *q* is some integer.

- Q:-
Prove that 3 + 2√5 is irrational.

- Q:-
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 - Q:-
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:

- Q:-
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

- Q:-
Check whether 6

^{n}can end with the digit 0 for any natural number n. - Q:-
Prove that √5 is irrational.

- Q:-
Given that HCF (306, 657) = 9, find LCM (306, 657).

- Q:-
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?

- Q:-
Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 - Q:-
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?

- Q:-
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.

- Q:-
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).

- Q:-
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.

- Q:-
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

(iii) The probability of an event that is certain to happen is

(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

- Q:-
Check whether the following are quadratic equations :

(i) (x + 1)^{2}= 2(x – 3) (ii) x^{2}– 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) x

^{2}+ 3x + 1 = (x – 2)^{2}(vii) (x + 2)^{3}= 2x (x2 – 1) (viii) x^{3}– 4x^{2}– x + 1 = (x – 2)^{3} - Q:-
How many tangents can a circle have?

- Q:-
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.

- Q:-
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.

- Q:-
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. - Q:-
Represent the following situations in the form of quadratic equations :

(i) The area of a rectangular plot is 528 m^{2}. 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.

- Q:-
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

- Q:-
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).

- Q:-
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

- Q:-
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

- Q:-
Find the roots of the following quadratic equations, if they exist, by the method of

completing the square:

(i) 2x^{2 }– 7x + 3 = 0 (ii) 2x^{2 }+ x – 4 = 0 (iv) 2x^{2}+ x + 4 = 0 - Q:-
In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

- Q:-
Find two consecutive positive integers, sum of whose squares is 365.

- Q:-
(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?

(ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective ? - Q:-
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.

- Q:-
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to

(A) 60° (B) 70°

(C) 80° (D) 90°

- All Chapters Of Class 10 Mathematics

- All Subjects Of Class 10