Tiêu đề 03-Linear equation in two variables-(Part-2)(1).pdf ✅

EXERCISE # 1 A. Very Short Answer Type Questions In each of the following verify whether the given value of the x is a solution or not : Q.1 4 x 3 x + = 8, x = 12 Q.2 (4x + 7) – 2 = 3x + 1, x = – 4 Q.3 4 5x + 4 – 2 3x − 2 = 5, x = 2 1 Q.4 2x – 4 + 1 = 3x – 6, x = 3 Q.5 Solve : x 6 + 11 = x 3 + 12 Q.6 If 2x – 8 = 8, then find the value of x 2 + x – 70. Q.7 For each of the following, state the quadrant in which the point lies. (i) (3, 3) (ii) (–3, 2) (iii) (2, –4) (iv) (–1, –2) (v) (–5, –5) (vi) (5, 3). Q.8 Draw the graph of y = x. Show that point (4, 4) is on the graph. Q.9 Express x in terms of y, given that 3x + 4y = 6. Check whether the point (3, 2) is on the given line. Q.10 Draw the graph of y = – 2x. Show that the point (2, –5) is not on the graph. B. B. Short answer type Questions Q.11 Indicate the quadrants in which the following points lie and plot them on a graph paper. (i) (–2, 0) (ii) (0, 1) (iii) (–2, –3) Q.12 Draw the graph of (i) x = 3 (ii) y = –2. Q.13 Find the value of k, if line represented by the equation 2x – ky = 9 passes through the point (–1, –1). Q.14 Express x in terms of y, it is being given that 7x – 3y = 15. Check if the line represented by the given equation intersects the y-axis at y = – 5 Q.15 Draw the graph of 6 – 1.5x = 0. Q.16 The following observed values of x and y are thought to fulfil the law y = ax + b. Find the values of a and b. x 1 2 –3 0 5 y 12 19 –16 5 –30 Q.17 Show that the points A (1, 2), B (–1, –16), C(0, –7) are on the graph y = 9x – 7. Q.18 Find the point of intersection of the line represented by the equation 7x + y = –2 with x-axis. Check whether the point (2, 1) is a solution set of the given equation. Q.19 Express y in terms of x, given that 2x – 5y = 7. Check whether the point (–3, –2) is on the given line. Q.20 Verify whether x = 2, y = 1 and x = 1 and y = 2 are the solutions of the linear equation 2x + y = 5. Find two more solutions. Q.21 Draw the graph of the equation 4x – 5y = 20 and check whether the points (3, 1) and and (5, 0) lie on the graph. Q.22 Draw the graph of the equation 3x + 4y = 14 and check whether x = 1 and y = 2 is a solution or not. Q.23 Draw the graph of the equation 2y + x = 7 and determine from the graph whether x = 3 and y = 2 is a solution Q.24 Solve the following system of equations graphically. Also, find out the points, where these lines meet the x-axis. x – 2y = 1 2x + y = 7 Q.25 Solve the following system of equations graphically. Also, find out the points, where these lines meet the y-axis. (i) x + 2y – 7 = 0 (ii) 2x + y = 8 2x – y + 1 = 0 x + 1 = 2y (iii) 2x + 3y = 12 2y – 1 = x Q.26 Draw the graphs of the following systems of equations, state whether it is consistent (dependent), consistent (independent) or inconsistent : (i) x + y = 7 (ii) 2x + 4y = 7 2x – 3y = 9 3x + 6y = 10 (iii) 2x + 3y – 12 = 0 (iv) 3x – 5y + 4 = 0 2x + 3y – 6 = 0 9x = 15y – 12 (v) x + 3y = 1 (vi) x + 4y = 7 2x + 6y = 2 2x – y = 5 Q.27 Solve the following pair of linear equations by the substitution method : (i) 7x – 15y = 2 (ii) 2x + 3y = 9 x + 2y = 3 4x + 6y = 18 (iii) x + 2y = 5 (iv) 0.2x+0.3y = 1.3 2x + 3y = 8 0.4x+0.5y =2.3
(v) x + 2y = – 1 (vi) 3x – 5y + 1 = 0 2x – 3y = 12 x – y + 1 = 0 Q.28 Solve the following equations by the method of elimination by equating the coefficients. (i) 12x + 5y = 17; 7x – y = 6 (ii) 17x + 12y = – 2; 15x + 8y = 6 (iii) 23x + 17y = 6; 39x – 19y = 58 (iv) 43x – 37y = 31; 13x + 23y = – 59 (v) 0.4x + 3y = 1.2, 7x – 2y = 6 17 (vi) (a + 2b) x + (2a – b) y = 2, (a – 2b) x + (2a + b) y = 3 (vii) a(x + y) + b(x – y) = a2 – ab + b2 , a(x + y) – b(x – y) = a2 + ab + b2 Q.29 Solve the following system of equations by cross-multiplication method : (i) 3x – 4y = 7 (ii) 3x – 5y = 1 5x + 2y = 3 7x + 2y = 16 (iii) 2x + 3y = 8 (iv) 3x – 4y = 1 3x + 2y = 7 4x – 3y = 6 (v) 3x – 4y = 10 (vi) 2x – 6y + 10 = 0 4x + 3y = 5 3x – 9y + 15 = 0 (vii) x 1 2 − + y 1 3 + = 2 x 1 3 − + y 1 2 + = 6 13 , x  1, y  – 1 (viii) x y 5 + – x y 2 − = – 1 x y 15 + + x y 7 − = 10; x + y  0, x – y  0 Q.30 For what value of k will the following system of equations have a unique solution. (i) 2x + ky = 1 and 3x – 5y = 7 (ii) x – 2y = 3 and 3x + ky = 1 (iii) 2x + 5y = 7 and 3x – ky = 5 Q.31 For what value of k will the following system of equations have infinitely many solutions. (i) 7x – y = 5 and 21x – 3y = k (ii) 5x + 2y = k and 10x + 4y = 3 (iii) kx + 4y = k – 4 and 16x + ky = k Q.32 Find the conditions so that the following systems of equations have infinitely many solutions. (i) 3x – (a + 1) y = 2b – 1 and 5x + (1 – 2a) y = 3b, find a and b. (ii) 2x + 3y = 7 and (p + q) x + (2p – q) y = 3(p + q + 1), find p and q. (iii) 2x – (2a + 5) y = 5 and (2b + 1) x – 9y = 15, find a and b. Q.33 Show that the following systems of equation are inconsistent. (i) x – 2y = 6 (ii) 2y – x = 9 3x – 6y = 0 6y – 3x = 21 (iii) 2x – y = 9 4x – 2y = 15 Q.34 For what value of k the following systems of equations have no solution. (i) 8x + 5y = 9 and kx + 10y = 8 (ii) x – 4y = 6 and 3x + ky = 5 (iii) kx – 5y = 2 and 6x + 2y = 7 (iv) 4x + 6y = 11 and 2x + ky = 7 (v) 2x + ky = 11 and 5x – 7y = 5 Q.35 Solve the following pair of linear equations (i) 2x 1 – y 1 = – 1. x 1 + 2y 1 = 8, x  0, y  0 (ii) x 2 + 3y 2 = 6 1 , x 3 + y 2 = 0; x  0 y  0 and hence, find a for which y = ax – 4. (iii) 7x 1 + 6y 1 = 3, 2x 1 – 3y 1 = 5; x  0 y  0 (iv) x m – y n = a, px – qy = 0; x  0 y  0 (v) y 2 + x 3 = xy 7 , y 1 + x 9 = xy 11 ; x  0, y  0 (vi) x y xy + = 5 6 , y x xy − = 6; xy  0, y  0 (vii) x + y = 5xy 3x + 2y = 13 xy Q.36 Solve the following pair of linear equations. (i) 3(a + 3b) = 11 ab, 3(2a + b) = 7ab (ii) 5x + y 4 = 9,
7x – y 2 = 5; y  0 (iii) 3/x + 4y = 7, x −2 + 7y = 3 19 ; x  0 (iv) x 1 5 + – y 1 2 − = 2 1 (x 1) 10 + + (y 1) 2 − = 2 5 , x  –1, y  1 (v) x y 6 + = x y 7 − + 3, 2(x y) 1 + = 3(x y) 1 − , x + y  0 x – y  0 (vi) ax + by = c, bx + ay = 1 + c (vii) ax + by = 1, bx + ay = 2 2 2 a b (a b) + + – 1 (viii) x 148 + y 231 = xy 527 ; x 231 + y 148 = xy 610 ; x  0, y  0 Q.37 2 tables and 3 chairs together cost −j 2000 whereas 3 tables and 2 chairs together cost −j 2500. Find the total cost of 1 table and 5 chairs. Q.38 3 bags and 4 pens together cost −j 257 whereas 4 bags and 3 pens together cost −j 324. Find the total cost of 1 bag and 10 pens. Q.39 Two numbers differ by 4 and their product is 192. Find the numbers. Q.40 Five years hence, father’s age will be three times the age of his son. Five years ago, father was seven times as old as his son Find their present ages. Q.41 The age of father is 4 times the age of his son. 5 years hence, the age of father will be three times the age of his son. Find their present ages. Q.42 The sum of a two-digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of the digits in the first number. Find the first number. Q.43 The sum of a two-digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number. Q.44 If 2 be added to the numerator of a fraction, it reduces to 1/2 and if 1 be subtracted from the denominator, it reduces to 1/3. Find the fraction. Q.45 The sum of the numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction. Q.46 The length of a rectangle exceeds its width by 8 cm and the area of the rectangle is 240 sq. cm. Find the dimensions of the rectangle. Q.47 The side of a square exceeds the side of another square by 4 cm and the sum of the area of the two squares is 400 sq. cm. Find the dimensions of the squares. Q.48 The area of a rectangle gets reduced by 8 sq. metres, if its length is reduced by 5 metres and width is increased by 3 metres. If we increase the length by 3 metres and breadth by 2 metres, the area is increased by 74 sq. metres. Find the length and breadth of the rectangle. Q.49 In a triangle, the sum of two angles is equal to the third. If the difference between them is 50o, find the angles. Q.50 Find the four angles of the following cyclic quadrilateral ABCD in which (i) A = 5xo, B = 9xo + 2yo, C = xo + 8yo and D = xo + 4yo. (ii) A=(2x+y)o,B = 2(x + y)o, C = (3x + 2y)o, D = (4x – 2y)o. C. Long answer type Questions Q.51 The ages of Ram and Mohan are in ratio 2 : 3. If sum of their ages is 65, find the difference of their ages. Q.52 The difference between two numbers is 1365. When larger is divided by the smaller one, the quotient is 6 and remainder is 15. Find the numbers.
Q.53 The denominator of a fraction is 1 more than its numerator. If 1 is subtracted from both the numerator and denominator, the fraction becomes 1/2. Find the fraction. Q.54 The measures of angles of a triangle in degrees are x, x + 12 and x + 27. Find the measure of angles. Q.55 Solve for x 3 x 17x 32 13x 2 18 4x 17 + − − − + = 36 x 16 12 7x + − Q.56 The coach of a cricket team buys 3 bats and 6 balls for −j 3900. Later, she buys another bat and 2 balls of the same kind for −j 1300. Represent this situation algebraically and geometrically. Q.57 Gloria is walking along the path joining (–2, 3) and (2, –2) while Suresh is walking along the path joining (0, 5) and (4, 0). Represent this situation graphically. Q.58 Solve the following system of equations by cross-multiplication method : (i) ax + by = a2 bx + ay = b2 (ii) a 2x + b y = 2. a x – b y = 4; a  0, b  0 (iii) x – y = a + b ax + by = a2 – b 2 (iv) a x + b y = 2, ax – by = a2 – b 2 ; a  0, b  0 (v) x + y = a + b ax – by = a2 – b 2 Q.59 Two numbers differ by 4 and their product is 96. Find the numbers. Q.60 Two numbers are in the ratio of 3 : 5, If 5 is subtracted from each of the number, they become in ratio of 1 : 2. Find the numbers. Q.61 Two numbers are in the ratio of 3 : 4. If 8 is added to each number, they become in the ratio of 4 : 5. Find the numbers. ANSWER KEY A. VERY SHORT ANSWER TYPE QUESTIONS: 1. No 2. Yes 3. No 4. Yes 5. 3 6. 2 7. (i) Ist (ii) IInd (iii) IVth (iv) IIIrd (v) IIIrd (vi) Ist 9. (i) x = 3 6 − 4y , (ii) No B. SHORT ANSWER TYPE QUESTIONS : 11. (i) lies on x-axis on negative side (ii) lies on y-axis on + ve side. (iii) IIIrd quadrant 12. (i) The graph of x = 3 is a straight line parallel to y-axis . (ii) The graph of y = –2 is a straight line below x-axis. 13. k = 11 14. (i) x = 7 15 + 3y , (ii) Yes 16. a = 7, b = 5 18. (i) (–2/7, 0) (ii) No 19. (i) y = 5 2x − 7 (ii) No 20. x = 2, y = 1 is the solution but x = 1 and y = 2 is not the solution. Other solutions are x = 3, y = – 1 and x = 1, y = 3. 21. Point (3, 1) does not lie on the lines and the point (5, 0) lies on the line. 22. Not 23. Yes 24. x = 3, y = 1, (1, 0),       , 0 2 7