Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(i) Area : 25a 2 – 35a + 12
(ii) Area : 35y 2 + 13y –12
Area = Length × Breadth
The expression given for the area of the rectangle has to be factorised. One of its factors will be its length and the other will be its breadth.
(i) 25a 2 – 35a + 12 = 25a2 - 15a - 20a + 12
= 5a(5a - 3) - 4 (5a - 3)
= (5a - 3) (5a - 4)
Therefore, possible length = 5a - 3
And, possible length = 5a - 4
(ii) 35y 2 + 13y –12 = 35y2 + 28y - 15y - 12
= 7y(5y + 4) - 3 (5y + 4)
= (5y + 4)(7y - 3)
Therefore, possible length = 5y + 4
And, possible breadth = 7y - 3
Write the coefficients of x 2 in each of the following:
(i) 2 + x 2 + x
(ii) 2 – x 2 + x 3
(iii) \(\frac{π }{ 2}\) x2 + x
(iv) √2 x -1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x 2 – 3x + 7
(ii) y 2 + √2
(iii) 3 √t + t√2
(iv) y +\(\frac{ 2 }{ y} \)
(v) x 10 + y 3 + t 50
What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
(i) Volume : 3x 2 – 12x
(ii) Volume : 12ky2 + 8ky – 20k
Without actually calculating the cubes, find the value of each of the following:
(i) (–12)3 + (7)3 + (5)3
(ii) (28)3 + (–15)3 + (–13)3
Use these adverbs to fill in the blanks in the sentences below.
awfully sorrowfully completely loftily carefully differently quickly nonchalantly
(i) The report must be read ________ so that performance can be improved.
(ii) At the interview, Sameer answered our questions _________, shrugging his shoulders.
(iii) We all behave _________ when we are tired or hungry.
(iv) The teacher shook her head ________ when Ravi lied to her.
(v) I ________ forgot about it.
(vi) When I complimented Revathi on her success, she just smiled ________ and turned away.
(vii) The President of the Company is ________ busy and will not be able to meet you.
(viii) I finished my work ________ so that I could go out to play