How will you describe the position of a table lamp on your study table to another person?
Consider the lamp as a point and table as a plane. Choose any two perpendicular edges of the table. Measure the distance of the lamp from the longer edge, suppose it is 25 cm. Again, measure the distance of the lamp from the shorter edge, and suppose it is 30 cm. You can write the position of the lamp as (30, 25) or (25, 30), depending on the order you fix.
(Street Plan): A city has two main roads which cross each other at the centre—one running North–South and the other East–West. All other streets run parallel to these and are 200 m apart, with 5 streets in each direction. Using the convention that the cross-street formed by the 2nd North–South street and 5th East–West street is written as (2, 5), find:
(i) how many cross-streets can be referred to as (4, 3)
(ii) how many cross-streets can be referred to as (3, 4)
The Street plan is shown in the figure given below.
Both the cross-streets are marked in the figure above. They are uniquely found because of the two reference lines we have used for locating them.
Write the answer of each of the following questions:
(i) What is the name of the horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?
(ii) What is the name of each part of the plane formed by these two lines?
(iii) Write the name of the point where these two lines intersect.
(i) The x-axis and the y-axis
(ii) Quadrants
(iii) The origin
See Fig. 3.14, and write the following:
(i) The coordinates of B.
(ii) The coordinates of C.
(iii) The point identified by the coordinates (–3, –5).
(iv) The point identified by the coordinates (2, –4).
(v) The abscissa of the point D.
(vi) The ordinate of the point H.
(vii) The coordinates of the point L.
(viii) The coordinates of the point M.
(i) (–5, 2)
(ii) (5, –5)
(iii) E
(iv) G
(v) 6
(vi) –3
(vii) (0, 5)
(viii) (–3, 0)
Determine which of the following polynomials has \((x + 1)\) as a factor:
(i) \(x^3 + x^2 + x + 1\)
(ii) \(x^4 + x^3 + x^2 + x + 1\)
(iii) \(x^4 + 3x^3 + 3x^2 + x + 1\)
(iv) \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
(x + 1) is a factor of (i), but not a factor of (ii), (iii), and (iv).
Use the Factor Theorem to determine whether \(g(x)\) is a factor of \(p(x)\) in each case:
(i) \(p(x) = 2x^3 + x^2 - 2x - 1\), \(g(x) = x + 1\)
(ii) \(p(x) = x^3 + 3x^2 + 3x + 1\), \(g(x) = x + 2\)
(iii) \(p(x) = x^3 - 4x^2 + x + 6\), \(g(x) = x - 3\)
(i) Yes
(ii) No
(iii) Yes
Find the value of \(k\), if \(x - 1\) is a factor of \(p(x)\) in each of the following cases:
(i) \(p(x) = x^2 + x + k\)
(ii) \(p(x) = 2x^2 + kx + \sqrt{2}\)
(iii) \(p(x) = kx^2 - \sqrt{2}x + 1\)
(iv) \(p(x) = kx^2 - 3x + k\)
(i) -2
(ii) \(-(2 + \sqrt{2})\)
(iii) \(\sqrt{2} - 1\)
(iv) \(\dfrac{3}{2}\)
Factorise the following:
(i) \(12x^2 - 7x + 1\)
(ii) \(2x^2 + 7x + 3\)
(iii) \(6x^2 + 5x - 6\)
(iv) \(3x^2 - x - 4\)
(i) \((3x - 1)(4x - 1)\)
(ii) \((x + 3)(2x + 1)\)
(iii) \((2x + 3)(3x - 2)\)
(iv) \((x + 1)(3x - 4)\)
Factorise the following:
(i) \(x^3 - 2x^2 - x + 2\)
(ii) \(x^3 - 3x^2 - 9x - 5\)
(iii) \(x^3 + 13x^2 + 32x + 20\)
(iv) \(2y^3 + y^2 - 2y - 1\)
(i) \((x - 2)(x - 1)(x + 1)\)
(ii) \((x + 1)(x + 1)(x - 5)\)
(iii) \((x + 1)(x + 2)(x + 10)\)
(iv) \((y - 1)(y + 1)(2y + 1)\)
Use suitable identities to find the following products:
(i) \((x + 4)(x + 10)\)
(ii) \((x + 8)(x - 10)\)
(iii) \((3x + 4)(3x - 5)\)
(iv) \((y^2 + \dfrac{3}{2})(y^2 - \dfrac{3}{2})\)
(v) \((3 - 2x)(3 + 2x)\)
(i) \(x^2 + 14x + 40\)
(ii) \(x^2 - 2x - 80\)
(iii) \(9x^2 - 3x - 20\)
(iv) \(y^4 - \dfrac{9}{4}\)
(v) 9 - 4x^2
Evaluate the following products without multiplying directly:
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
(i) 11021
(ii) 9120
(iii) 9984
Factorise the following using appropriate identities:
(i) \(9x^2 + 6xy + y^2\)
(ii) \(4y^2 - 4y + 1\)
(iii) \(x^2 - \dfrac{y^2}{100}\)
(i) \((3x + y)(3x + y)\)
(ii) \((2y - 1)(2y - 1)\)
(iii) \((x + \dfrac{y}{10})(x - \dfrac{y}{10})\)
Expand each of the following using suitable identities:
(i) \((x + 2y + 4z)^2\)
(ii) \((2x - y + z)^2\)
(iii) \((-2x + 3y + 2z)^2\)
(iv) \((3a - 7b - c)^2\)
(v) \((-2x + 5y - 3z)^2\)
(vi) \([\dfrac{1}{4}a - \dfrac{1}{2}b + 1]^2\)
(i) \(x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8xz\)
(ii) \(4x^2 + y^2 + z^2 - 4xy - 2yz + 4xz\)
(iii) \(4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8xz\)
(iv) \(9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ac\)
(v) \(4x^2 + 25y^2 + 9z^2 - 20xy - 30yz + 12xz\)
(vi) \(\dfrac{a^2}{16} + \dfrac{b^2}{4} + 1 - \dfrac{ab}{4} - b + \dfrac{a}{2}\)
Factorise:
(i) \((2x + 3y - 4z)(2x + 3y - 4z)\)
(ii) \(( -\sqrt{2}x + y + 2\sqrt{2}z)( -\sqrt{2}x + y + 2\sqrt{2}z)\)
(i) \(8x^3 + 12x^2y + 6x + 1\)
(ii) \(8a^3 - 27b^3 - 36a^2b + 54ab^2\)
Write the following cubes in expanded form:
(i) \((2x + 1)^3\)
(ii) \((2a - 3b)^3\)
(iii) \([ \dfrac{3}{2}x + 1]^3\)
(iv) \([x - \dfrac{2}{3}y]^3\)
(i) \(8x^3 + 27/8 x^3 + 27/4 x^2 + 9/2 x + 1\)
(ii) \(x^3 - \dfrac{8}{27}y^3 - 2x^2y + \dfrac{4xy^2}{3}\)
Evaluate the following using suitable identities:
(i) \((99)^3\)
(ii) \((102)^3\)
(iii) \((998)^3\)
(i) 970299
(ii) 1061208
(iii) 994011992
Factorise each of the following:
(i) \(8a^3 + b^3 + 12a^2b + 6ab^2\)
(ii) \(8a^3 - b^3 - 12a^2b + 6ab^2\)
(iii) \(27 - 125a^3 - 135a + 225a^2\)
(iv) \(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
(v) \(27p^3 - \dfrac{1}{216} - \dfrac{9}{2}p^2 + \dfrac{1}{4}p\)
(i) (2a + b)(2a + b)(2a + b)
(ii) (2a - b)(2a - b)(2a - b)
(iii) (3 - 5a)(3 - 5a)(3 - 5a)
(iv) (4a - 3b)(4a - 3b)(4a - 3b)
(v) (3p - 1/6)(3p - 1/6)(3p - 1/6)
Verify:
(i) \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
(ii) \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)
Simplify RHS.
Factorise the following:
(i) \(27y^3 + 125z^3\)
(ii) \(64m^3 - 343n^3\)
(i) \((3y + 5z)(9y^2 - 15yz + 25z^2)\)
(ii) \((4m - 7n)(16m^2 + 49n^2 + 28mn)\)
Factorise: \(27x^3 + y^3 + z^3 - 9xyz\)
(3x + y + z)(9x^2 + y^2 + z^2 - 3xy - yz - 3xz)
Verify that \(x^3 + y^3 + z^3 - 3xyz = \dfrac{1}{2}(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]\)
Simplify RHS.
If \(x + y + z = 0\), show that \(x^3 + y^3 + z^3 = 3xyz\)
Put \(x + y + z = 0\) in the identity in Q12.
Without calculating cubes, find the value of each:
(i) \((-12)^3 + (7)^3 + (5)^3\)
(ii) \((28)^3 + (-15)^3 + (-13)^3\)
(i) -1260
(ii) 16380
Give possible expressions for the length and breadth of rectangles whose areas are:
(i) \(25a^2 - 35a + 12\)
(ii) \(35y^2 + 13y - 12\)
(i) One possible answer: Length = \(5a - 3\), Breadth = \(5a - 4\)
(ii) One possible answer: Length = \(7y - 3\), Breadth = \(5y + 4\)
Find possible expressions for the dimensions of cuboids whose volumes are:
(i) \(3x^2 - 12x\)
(ii) \(12ky^2 + 8ky - 20k\)
(i) One possible answer: \(3x, x, x - 4\)
(ii) One possible answer: \(4k, 3y + 5, y - 1\)