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.
(i) Total no. Of outcomes = 36
• (1, 2) and (2, 1) are events for getting a sum as 3
P (E) = 2/36 = 1/18
• (1, 3), (2, 2) and (3, 1) are the events of getting the Sum 4
P(E) = 3/36 = 1/12
• (1, 4), (2, 3), (3, 2) and (4, 1) are the events of getting the sum 5
P(E) = 4/36 = 1/9
• (1, 5), (2, 4), (3, 3), (4, 2) and (5, 1) are the events of Getting a sum 6
P(E) = 5/36
• (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1) are the event of getting a sum 7
P(E) = 6/36 = 1/6
• (3, 6), (4, 5), (5, 4) and (6, 3) are the events of getting a sum 9
P(E) = 4/36 = 1/9
• (4, 6), (5, 5) and (6, 4) are the events of getting a sum 10
P(E) = 3/36 = 1/12
• (5, 6), (6, 5) are the events of getting a sum 11
P(E) = 2/36 =1/18
(ii) No, the eleven sum is not equally likely.
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8 (ii) 4s2 – 4s + 1 (iii) 6x2 – 3 – 7x (iv) 4u2 + 8u (v) t2 – 15 (vi) 3x2 – x – 4
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Find two consecutive positive integers, sum of whose squares is 365.
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.
Prove that 3 + 2√5 is irrational.
Find two numbers whose sum is 27 and product is 182.
Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
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
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t2 – 3, 2t4 + 3t3 – 2t2 – 9t – 12
(ii) x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2
(iii) x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1
Find two numbers whose sum is 27 and product is 182.
Solve the following pair of linear equations by the substitution method.
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.
Prove that the parallelogram circumscribing a circle is a rhombus.
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
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.
Prove that 3 + 2√5 is irrational.
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