Are the following pair of linear equations consistent? Justify your answer. (i) -3x- 4y = 12 4y + 3x = 12 (ii) (3/5)x – y = ½ (1/5)x – 3y= 1/6 (iii) 2ax + by = a ax + 2by – 2a = 0; a, b ≠0 (iv) x + 3y = 11 2 (2x + 6y) = 22 Solution: We know that Conditions for pair of linear equations to be consistent are: a1/a2 ≠ b1/b2. [unique solution]... View Article
Do the following equations represent a pair of coincident lines? Justify your answer. (i) 3x + 1/7y = 3 7x + 3y = 7 (ii) -2x – 3y = 1 6y + 4x = – 2 (iii) x/2 + y + 2/5 = 0 4x + 8y + 5/16 = 0 Solution We know that the condition for coincident lines, a1/a2 = b1/b2 = c1/c2; (i) 3x + 1/7y = 3 7x + 3y = 7 No. The given pair... View Article
Do the following pair of linear equations have no solution? Justify your answer. (i) 2x + 4y = 3 12y + 6x = 6 (ii) x = 2y y = 2x (iii) 3x + y – 3 = 0 2x + 2/3y = 2 Solution We know that the condition for no solution is a1/a2 = b1/b2 ≠ c1/c2 (parallel lines) (i) 2x + 4y = 3... View Article
The pair of equations x = a and y = b graphically represents lines which are (A) parallel (B) intersecting at (b, a) (C) coincident (D) intersecting at (a, b) Answer A correct answer is an option (D) intersecting at (a, b) Solution Graphically in every condition, we know that a, b>>0 a, b<... View Article
The pair of equations y = 0 and y = -7 has (A) one solution (B) two solutions (C) infinitely many solutions (D) no solution Answer The correct answer is an option (D) no solution Solution The given pair of equations are y = 0 and y = – 7. Graphically, both... View Article
If a pair of linear equations is consistent, then the lines will be (A) parallel (B) always coincident (C) intersecting or coincident (D) always intersecting Answer The correct answer is an option (C) intersecting or coincident Solution Condition for a pair of linear equations to be consistent are:... View Article
The pair of equations x + 2y + 5 = 0 and -3x – 6y + 1 = 0 have (A) a unique solution (B) exactly two solutions (C) infinitely many solutions (D) no solution Answer The correct answer is an option (D)No solution Solution The given equations are: x + 2y + 5 = 0 –3x – 6y... View Article
Graphically, the pair of equations 6x – 3y + 10 = 0 2x – y + 9 = 0 represents two lines which are (A) Intersecting at exactly one point. (B) Intersecting at exactly two points. (C) Coincident (D) parallel. Answer The correct answer is option (D) Parallel Solution The given equations are, 6x-3y+10 = 0 On dividing by 3, we get ⇒ 2x-y+ 10/3=... View Article
Given that √2 is a zero of the cubic polynomial 6x3 + √2 x2 – 10x – 4√2 , find its other two zeroes. Given √2 is a zero of the cubic polynomial 6x3 + √2 x2 – 10x – 4√2 Find out We... View Article
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial. Given a, a+b, a+2b are roots of given polynomial x³-6x²+3x+10 Find out We have to determine the values of a and b as well as... View Article
For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation. (i) (-8/3), 4/3 (ii) 21/8, 5/16 (iii) -2√3, -9 (iv) (-3/(2√5)), -½ Solution: (i) (–8/3), 4/3 Sum of the zeroes = – 8/3 Product of the zeroes = 4/3 We know that, P(x) = x2 – (sum... View Article
Find the zeroes of the following polynomial by factorisation method: 2x2 +(7/2)x +3/4 Solution: 2x2 +(7/2)x +3/4 The equation can also be expressed as, 8x2+14x+3 By splitting the middle term, we obtain, 8x2+12x+2x+3... View Article
Find the zeroes of the following polynomial by factorisation method: t3 – 2t2 – 15t Solution: t3 – 2t2 – 15t On taking out t common, we obtain, t ( t2 -2t -15) By splitting the middle term of the... View Article
Find the zeroes of the following polynomial by factorisation method: 5t2 + 12t + 7 Solution 5t2 + 12t + 7 By splitting the middle term, we obtain, 5t2 +5t + 7t + 7 On taking the common factors out, we obtain, 5t... View Article
Find the zeroes of the following polynomial by factorisation method: 3x2+ 4x – 4 Solution 3x2 + 4x – 4 By splitting the middle term, we obtain, 3x2 + 6x – 2x – 4 On taking out the common factors... View Article
Find the zeroes of the following polynomial by factorisation method: 4x2 – 3x – 1 Solution 4x2 – 3x – 1 On splitting the middle term, we obtain, 4x2-4x+1x-1 Ontaking out the common factors out, we... View Article
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1? Solution A Quadratic Equation will have equal roots if it satisfies the condition given below: b² – 4ac = 0 Given equation is... View Article
If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)? Solution In order to divide p(x) by g(x) We know that, Degree of p(x) > degree of g(x) or Degree of p(x)= degree of g(x) Hence, we can... View Article
If on division of a polynomial p (x) by a polynomial g (x), the quotient is zero, what is the relation between the degrees of p (x) and g (x)? Solution We know that, p(x)= g(x) × q(x)+r(x) According to the given details q(x) =0 When q(x)=0, then r(x) is also = 0 So, now when... View Article
What will the quotient and remainder be on division of ax2 + bx + c by px3 + qx2 + rx + s, p ≠0? Solution Degree of the polynomial px3 + qx2 + rx + s is 3 Degree of the polynomial ax2 + bx + c is 2 Here, we observe that... View Article
