1Solve graphically and check: \( x + y = 5 \) and \( x - y = 3 \).
2Solve graphically: \( 3x + y = 7 \) and \( x = 2y \).
3Solve graphically: \( x + y = 13 \) and \( 2x = y + 8 \).
4Solve graphically: \( 4x + y = 2 \) and \( 3x - 2y = 7 \).
5Solve each pair graphically and check: (a) \( x + y = 6 \), \( x - y = 2 \) (b) \( x + y = 8 \), \( x - y = 4 \) (c) \( x + 2y = 6 \), \( 2y - x = 2 \) (d) \( 3x + 2y = 4 \), \( x - 3y = 5 \) (e) \( 2x = 5 + 3y \), \( 5y = 2x - 3 \) (f) \( 2x - 1 = y \), \( 3x - 2y = 0 \) (g) \( x + 3 = 2y \), \( 2x + y = 14 \) (h) \( x - 2y = 5 \), \( 2x + 3y = 10 \).
6The sum of two numbers is 15 and their difference is 5. Find the numbers.
7The sum of two numbers is 12 and the larger is three times the smaller. Find the numbers.
8The difference of two numbers is 5, and 5 times the smaller equals 4 times the larger. Find the numbers.
9Three exercise books and four pens cost Rs 200, while five exercise books and two pens cost Rs 240. Find the price of one exercise book and one pen.
10A father's age is 3 years less than three times his daughter's age. The difference between their ages is 37 years. Find their ages.
11Kamala is 5 years older than Vimala. Five years from now Kamala's age will be double Vimala's present age. How old is each of them now?
12Bipin is 4 years older than Bipana. Two years ago Bipin's age was twice Bipana's. Find their ages.
13Kusum and her father's ages differ by 20 years. The father's age is 4 more than twice Kusum's age. Find their ages.
14In an examination Ram scored 20 marks more than Shyam. Ram's marks were double Shyam's. How many marks did each get?
15Solve: (a) \( x^2 - 3x = 0 \) (b) \( 2x^2 - x = 0 \) (c) \( 9x^2 + 3x = 0 \) (d) \( 9y^2 - 4 = 0 \) (e) \( 5x + 9x^2 = 0 \) (f) \( 4y^2 - 7y = 0 \) (g) \( x^2 - 49 = 0 \) (h) \( \frac{x^2}{4} - 36 = 0 \) (i) \( 5x^2 - 125 = 0 \) (j) \( x^2 - 7 = 29 \) (k) \( x^2 - 4x = 0 \).
16Solve: \( x^2 - 11x + 30 = 0 \).
17Solve: \( 2x^2 - x - 6 = 0 \).
18Solve: (a) \( x^2 + 2x + 1 = 0 \) (b) \( y^2 - y - 2 = 0 \) (c) \( x^2 + x - 2 = 0 \) (d) \( x^2 - 10x - 24 = 0 \) (e) \( x^2 - 9x + 18 = 0 \) (f) \( x^2 + 2x - 3 = 0 \) (g) \( x^2 + 8x + 16 = 0 \) (h) \( x^2 - 8x + 15 = 0 \) (i) \( y^2 + 7y + 12 = 0 \) (j) \( 7x^2 + 13x - 2 = 0 \) (k) \( x^2 - 18x + 77 = 0 \) (l) \( 3x^2 - 11x - 20 = 0 \) (m) \( 12x^2 - 11x + 2 = 0 \) (n) \( (x + 6)^2 - 36 = 0 \) (o) \( 100 - (x - 5)^2 = 0 \).
19(a) Find the quadratic equation whose values of \( x \) are 1 and 2. (b) Find the quadratic equation whose values of \( x \) are 3 and \( -2 \). (c) Find the quadratic equation whose values of \( y \) are 2 and \( \frac{1}{2} \).
20Solve using a graph: \( 2x - y = 5 \) and \( x - y = 1 \).
21If \( 3x^2 - 8x - 16 = 0 \), find the values of \( x \) and verify that both values satisfy the given equation.
22The sum of two numbers is 12 and their difference is 4. (a) Taking the larger as \( x \) and the smaller as \( y \), form the equations. (b) Solve them using a graph.
23(a) Solve \( x + 2y = 8 \) and \( x + y = 5 \) using a graph. (b) What kind of equations are these called?
24(a) Solve \( x + 2y = 6 \) and \( 2y - x = 2 \) using a graph. (b) What kind of equations are these called?
25For \( x^2 - 7x + 12 = 0 \): (a) What is the degree of the equation? (b) What kind of equation is it called? (c) Find its roots.
26There are some cats and some hens in a yard. Counting them gives 10 heads and 26 legs in all. How many cats and how many hens are there?
27Without drawing anything, check whether \( (3, 1) \) satisfies both \( 2x + y = 7 \) and \( x - y = 2 \).
28Build a quadratic equation in which both values of \( x \) are 5, and say what its factorization looks like.
29A student writes \( x = 6 \) and \( x = 4 \) from \( (x - 6)(x + 4) = 0 \). Where is the mistake, and what is the correct answer?
30On a bus, 2 adult tickets and 3 child tickets cost Rs 700, while 1 adult ticket and 4 child tickets cost Rs 600. Find the price of one adult ticket and one child ticket.