4. QUADRATIC EQUATIONS

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1 . QUADRATIC EQUATIONS Important Terms, Definitions and Results The roots of a quadratic equation ax + bx + c = 0 An equation of the form ax + bx + c = 0, where a, b, c are real numbers and a 0, is called a quadratic equation in x. A real number a is called a root of the quardratic equation ax + bx + c = 0, a 0, if aa + ba + c = 0. Any quardratic equation can have at most two roots. Note : If a is a root of ax + bx + c = 0, then we say that (i) x = a satisfies the equation ax + bx + c =0 or (ii) x = a is a solution of the equation ax + bx + c = 0 are called the zeros of the polynomial ax + bx + c. Solving a quadratic equation means finding its roots. If ax + bx + c can be factorised as (x a) (x b), then ax + bx + c = 0 is equivalent to (x a) (x b) = 0 Thus, (x a) (x b) = 0 x a = 0 or x b = 0 i.e., x = a or x = b. Here a and b are called the roots of the equation ax + bx + c = 0. To solve a quadratic equation by factorisation : (a) Clear fractions and brackets, if necessary. (b) Transfer all the terms to L.H.S. and combine like terms. (c) Write the equation in the standard form, i.e., ax + bx + c = 0. (d) Factorise the L.H.S. (e) Put each factor equal to zero and solve. (f) Check each value by substituting it in the given equation. The roots of a quadratic equation can also be found by using the method of completing the square. The roots of the quadratic equation ax + bx + c = 0, a, b, c R and a 0 are given by x = b b ac (Shridharacharya s formula) a The expression b ac is called the discriminant of the quadratic equation ax + bx + c = 0. The discriminant, usually denoted by D, decides the nature of roots of a quadratic equation. (i) If D > 0, the equation has real roots and roots are unequal, i.e., unequal-real roots. If D is a perfect square, the equation has unequal-rational roots. (ii) If D = 0, the equation has real and equal roots and each root is b a (iii) If D < 0, the equation has no real roots. (i) If p 5, then p 5 (ii) If p 5, then p 5 (iii) If p, then either p or p 5 (Important) (iv) If p, then p lies between and, i.e., p (Important) Quadratic equations can be applied to solve word problems involving various situations. To solve problems leading to quadratic equations, following steps may be used :. Represent the unknown quantity in the problem by a variable (letter).. Translate the problem into an equation involving this variable.. Solve the equation for the variable.. Check the result by satisfying the conditions of the original problem. 5. A root of the quadratic equation, which does not satisfy the conditions of the problem, must be rejected. Multiple choice Questions[ Mark] Summative Assessment A. Important Questions. If p(x) = 0 is a quadratic equation, then p(x) is a polynomial of degree : (a) one (c) three (b) two (d) four. Which of the following is a quadratic equation in x? (a) x + = 0 (b) 7x = x (c) x + = (d) 6 x(x + ) = 0 x

2 . Which of the following is a root of the equation x 5x = 0? (a) x = (b) x = (c) x = (d) x =. Which of the following is a root of the equation x x = 0? (a) x = (b) x = (c) x = (d) x = 5. x = is a solution of the equation : (a) x + x = 0 (b) x x = 0 (c) x + 5x + = 0 (d) (a) and (b) both 6. Which of the following is a quadratic equation in x? (a) x + = x (b) x = 5 x x (c) x x x + = 0 (d) x + 7x + 5 = 0 7. For what value of k, x = is a solution of kx + x = 0? (a) k = (b) k = (c) k = (d) k = 8. Which of the following is a root of the equation x x +? (a) (b) (c) (d) 9. If x = is a root of the equation x kx + 5 = 0, then k is equal to : (a) (b) 7 (c) (d) Which of the following is a quadratic equation in x? x + = (a) x (b) x + x = (c) x + = (d) x + x = 0 x. Which of the following is a quadratic equation? (a) x + x + = ( x) + (b) x = (5 x) x (c) (k + )x + x = 7, where k = (d) x x = (x ). Which of the following is not a quadratic equation? 5 (a) (x ) = x x + (b) x x = x + 5 (c) x + + x = x 5x (d) (x + x) = x + + x. Which of the following equations has as a root? (a) x x + 5 = 0(b) x + x = 0 (c) x 7x + 6 = 0 (d) x 6x = 0. If is a root of the equation x + kx 5 = 0, then the value of k is : (a) (b) (c) (d) 5. Which of the following equations has the sum of its roots as? (a) x x + 6 = 0 (b) x + x = 0 (c) x x + = 0 (d) x x + = 0 6. The roots of x + x + = 0 are : (a) real and equal (b) real and unequal (c) not real 7. Discriminant of x + px + q = 0 is : (a) p 8q (b) p + 8q (c) p 8q (d) q 8p 8. If the equation x + x + k = 0 has real and distinct roots, then : (a) k < (b) k > (c) k > (d) k < 9. The quadratic equation ax + bx + c = 0, a 0 has two distinct real roots, if D is equal to : (a) b ac > 0 (b) b ac = 0 (c) b ac < 0 0. If ax + bx + c = 0 has equal roots, then c is equal to : b (a) a (c) b a (b) (d) b a. If the equation 9x + 6kx + = 0 has equal roots, then the roots are : (a) ± (b) ± (c) 0 (d) ± b a

3 . The roots of the equation x + 5x + 5 = 0 are: (a) real and distinct (c) real and equal (b) not real (d) real and unequal. If the equation (a + b )x (ac + bd)x + c + d = 0 has equal roots, then : (a) ad = cd (b) ad = bc (c) ad = bc (d) ab = cd`. If the roots of the equation (a + b )x b (a + c) x + (b + c ) = 0 are equal, then : (a) b = a + c (b) b = ac (c) b = ac a + c (d) b = ac 5. If x = is a common root of the equations ax + ax + = 0 and x + x + b = 0, then ab is equal to : (a) (b).5 (c) 6 (d) 6. Values of k for which the quadratic equation x kx + k = 0 has equal roots is : (a) 0 only (b) (c) 8 only (d) 0, 8 7. Which constant must be added and subtracted to solve the quadratic equation 9x + x = 0 by the method of completing the square? (a) 9 (b) (c) (d) The quadratic equation x + x + = 0 has : (a) two distinct real roots (b) two equal real roots (c) no real roots (d) more than real roots 9. Which of the following equations has two distinct real roots? 9 (a) x x + = 0 (b) x + x 5 = 0 (c) x + x + = 0 (d) 5x x + = 0 0. Which of the following equations has no real roots? (a) x + x + = 0 (b) x + x = 0 (c) x x = 0 (d) x + x + = 0. (x + ) x = 0 has : (a) four real roots (b) two real roots (c) no real roots (d) one real root. If kx 5x + = 0 and x kx + = 0 have equal discriminants, then the value of k is : (a) (b) (c) (d). The sum and product of the roots of x 5x + are in the ratio : (a) 5 : (b) 5 : (c) 0 :. The quadratic equation (p + q) x + (p + q) x + = 0 has : (a) equal roots (b) no real roots (c) real but not equal roots 5. If p x + (p + q )x + q = 0 has equal roots, then p q is equal to : (a) q (b) q (c) 0 (d) 6. If x = 7 and x = 5 are roots of x + mx + n = 0, then the values of m and n are : (a) and 6 (b) and 5 (c) and 5 7. A boy said, the product of my age 5 years before and after 5 years is 75, then the present age of the boy is : (a) 8 years (c) years (b) 0 years (d) 5 years 8. If the price of a pen is increased by Rs, a person can buy pen less for Rs 0, then the original price of one pen is : (a) Rs 0 (b) Rs 8 (c) Rs 6 (d) Rs 9. If the sum of the squares of three consecutive integers is 9, then one of the integer is : (a) (b) (c) 5 (d) 6 B. Questions From CBSE Examination Papers. The roots of the equation x x x + = 0 are : (a), (b), (c), (d),. The roots of the quadratic equation x x = 0 are : (a), (b),

4 (c), (d),. Which of the following is not a quadratic equation : (a) (x ) + = x (b) x(x + ) + 8= (x + ) (x ) (c) x(x + ) = x + (d) (x + ) = x. The roots of the quadratic equation x + 7x + = 0 are : (a), (b), (c), (d), 5. The quadratic equation whose roots are real and equal is : (a) x x + = 0 (b) x x + = 0 (c) x 5x + = 0 (d) x x 6 = 0 6. Which of the following equations has two distinct real roots? (a) x x + 9 / = 0 (b) x + x 5 = 0 (c) x + x + = 0 (d) 5x x + = 0 7. If 8 is a root of the equation x 0x + k = 0, then the value of k is : (a) (b) 8 (c) 8 (d) 6 8. If the discriminant of x + x + a = 0, is double the discriminant of x x + = 0, then the value of a is: (a) (b) (c) (d) 9. The positive root of x + 6 = 9 is : (a) (b) (c) 5 (d) 7 0. One of the roots of the quadratic equation 6x x = 0 is : (a) (b) (c) (d) 7. For what value of k will be a root of x x k = 0? 7 (a) (b) 7 (c) (d). If x + kx + = 0 has a root x =, then the value of k is? (a) (b) (c) (d). The value of k for which the equation x (k ) x + 8 = 0 will have real and equal roots are : (a) 9 and 7 (b) only 9 (c) only 7 (d) 9 and 7. Which of the following is a solution of the quadratic equation x b = a (x a)? a (a) a + b (b) b a (c) ab (d) b 5. If one root of the equation x 0x + p = 0 is, then the value of p is : (a) (b) 6 (c) 9 (d) 6. Which of the following is a root of the equation x 5x = 0? (a) x = (b) x = (c) x = (d) x = 7. If is the root of the equation x + kx 5 = 0, then the value of k is : (a) (b) (c) (d) 8. The roots of the equation ax + x + b = 0 are equal if : (a) b = a (b) b < a (c) b > a (d) ab = 9. If the equation 9x + 6kx + = 0 has equal roots, then the value of k is : (a) (b) ± (c) 0 (d) 0. The roots of the equation x x + = 0 are : (a) real and unequal (c) imaginary (b) real and equal. If the equation kx + x + = 0 has real and distinct roots, then : (a) k < (b) k > (c) k (d) k. Value of k for which quadratic equation x kx + k = 0 has equal roots is : (a) (b) (c) 8 (d) 8. If r = is a root of quadratic equation kr kr = 0, value of k is : (a) (b) (c) (d)

5 . The value of k, for which the quadratic equation x + x + k = 0 has equal roots is : (a) k = (b) k = (c) k = (d) k = 5. For what value of k the equation kx 6x = 0 has equal roots? (a) 7 (b) 9 (c) (d) 7 6. The value of p for which the quadratic equation x(x ) + p = 0 has real roots, is : (a) p < (b) p (c) p = 7. The value of k for which x + x + k = 0 has real roots is : (a) k > (b) k (c) k (d) k < 8. For the quadratic equation x x + = 0 the value of x + is : x (a) (b) (c) (d) SHORT ANSWER TYPE QUESTIONS [ Marks] A. Important Questions. Find the roots of the quadratic equation x x = 0.. Are x = 0, x = the solution of the equation x + x + = 0?. Find the discriminant of the quadratic equation x x + = 0.. If a x abx + k = 0 has equal roots, then find the value of k. 5. Find the discriminant of the quadratic equation bx acx + b = 0 6. If kx 5x + = 0 has real and equal roots, then find the value of k. 7. If is a root of x ( + )x + k = 0 and x + kx + a = 0 has equal roots, then find the value of a. 8. If is a root of the equation x + bx + = 0 and the equation has equal roots, find the value of b. 9. If one of the roots of the equation a x + 8abx + b = 0 is b, then find the other root. a 0. If x = b c is a root of abx + (b ac)x k = 0, then find the value of k.. State whether the quadratic equation (x ) (x + ) = 0 has two distinct real roots. Justify your answer.. Write whether the following statement is true or false. Justify your answer. If the coefficient of x and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.. If b = 0, c < 0, is it true that the roots of the quadratic equation x + bx + c = 0 are numerically equal and opposite in sign? Justify. [HOTS]. Does a quadratic equation with integral coefficient has always integral roots. Justify your answer. [HOTS] 5. Is the following statement true or false? Justify your answer. If in a quadratic equation the coefficient of x is zero, then the quadratic equation has no real roots. [HOTS] 6. Does (x ) + (x + ) = 0 have a real root? Justify your answer. [HOTS] Solve the following quadratic equations by factorisation (7-6): 7. x + x + = 0 8. x + x + = 9. x + bx (a b ) = 0 0. ax + (a b)x ab = 0. a x + (a + b )x + b = 0, a 0. m n x + m n = x. (a + b) x abx (a b) = 0. a(x + ) x(a + ) = 0 5. x x a(a + ) = 0 6. (x )(x ) = () 5

6 Solve the following quadratic equations by completing the squares (7-): 7. x + x + = 0 8. x x = 0 9. x + 0x + 7 = 0 0. x - x + = 0. a x abx +b = 0. y + y = 0 Solve the following equations by using the formula (- 8):. x ax + x 6a = 0. (p + q)x = x + pq 5. a b x + b x a x = 0 6. a b = ax x 7. (a b)x + (b c)x + (c a) = 0 8. x (m + n)x + (m + n) = Find the values of k for which the roots are real and equal in each of the following equations (9-): 9. (k + )x + (k + )x + (k + 8) = 0 0. x kx + 7k = 0. 5x x + + k(x x ) = 0. ( k)x + (k + )x + (8k + ) = 0. Show that the equation x + ax = 0 has real and distinct roots for all real values of a.. For what value of k, ( k)x + (k + ) x + (8k + ) = 0, is a perfect square. 5. If the roots of the equations ax + bx + c = 0 and bx acx + b = 0 are simultaneously real, then prove that b = ac. B. Questions From CBSE Examination Papers. Find the values of k for which roots of the equation x 8kx + k = 0 are equal.. Find the values of p such that the quadratic equation (p )x (p )x + = 0 has equal roots.. Find the roots of the quadratic equation x x + 8 = 0.. Find the value(s) of k for which the equation x x + k = 0 has equal roots. 5. For what value(s) of k, the equation x kx k = 0 will have equal roots? 6. Solve the equation : 0ax + 5ax 6x 9 = 0, a Write all the values of k for which the quadratic equation x + kx + 8 = 0, has equal roots. Also, find the roots. 8. Find the value of k for which the equation : kx (x ) + 6 = 0 has equal roots. 9. Solve : x = x 0. Find the roots of the following quadratic equation by factorisation method. x + 7x + 5 = 0. One root of the equation x 8x m = 0 is 5. Find the other root and the value of m.. For what value of k the equation x (k + ) x + (k + ) = 0 has real and equal roots?. Using quadratic formula, determine the roots of following equation : x `x =. Find the values of k for which the following quadratic equation has two equal roots. x + kx + = 0 5. For what value of k, the quadratic equation 9x + 8kx + 6 = 0 has equal roots? 6. Find the roots of the quadratic equation x 6 x + = 0 7. Find the roots of the following quadratic equation: (x + )(x ) = x 8. Find the value of k such that the quadratic equation x(x k) + 6 = 0 has real and equal roots. 9. Find the value of k such that the quadratic equation x kx + (7k ) = 0 has real and equal roots. 6

7 0. Find the nature of roots of the quadratic equation x x + = 0. Find the roots of 6x x = 0. Find the roots of the quadratic equation x 5x + = 0. Find the roots of the quadratic equation 6x + 5x 6 = 0. If is a root of the quadratic equation x + px = 0 and the equation x + px + k = 0 has equal roots, find the value of k. 5. If a x abx + k = 0 has equal roots of x, then find the value of k. 6. For what value of k does (k ) x + (k ) x + = 0 have equal roots? [008C] SHORT ANSWER TYPE QUESTIONS 7. Solve for x : x (a + b )x + a b = 0. [00] 8. Solve for x : x a x + (a b ) = 0. [00] 9. Using quadratic formula, solve the following quadratic equation for x : p x + (p q )x q = 0. [00] 0. Using quadratic formula, solve the following quadratic equation for x : x ax + (a b ) = 0. [00]. Using quadratic formula, solve the following quadratic equation for x : x ax + a b = 0 [00]. Solve for x : 6x 8a x + (a b ) = 0 [00]. Solve for x : 6x ax + (a b ) = 0. [00C] [ Marks] A. Important Questions Solve the following quadratic equations by factorisation ( 6) : 6. + =, (x 0) x x x. [HOTS] a. ax + bx = a + b. x a + x b = a + b x b x a b a 5. b a b c + = x a x b x c x x = 0,(x 0) x + x + 7. If a, b, c, are real numbers such that ac 0 then show that at least one of the equations ax + bx + c = 0 and ax + bx + c = 0 has a real root. 8. If the equation ( + m )x + mcx + (c a ) = 0 has equal roots, prove that c = a ( + m ). 9. If p, q, r and s are real numbers such that pr = (q + s), then show that at least one of the equations x + px + q = 0 and x + rx + s = 0 has real roots. 0. Prove that the equation x (a + b ) + x(ac + bd) + (c + d ) = 0 has no real root, if ad bc. If the roots of the equation x + cx + ab = 0 are real and unequal, prove that the equation x (a + b)x + a + b + c = 0 has no real roots.. The sum of two numbers is 8 and 5 times the sum of their reciprocals is also 8. Find the numbers.. The product of two successive integral multiples of 5 is 00. Determine the multiples.. The sum of the squares of two numbers is and one of the numbers is less than twice the other number. Find the numbers. 5. Find two consecutive positive integers, the sum of whose squares is The difference of two numbers is 8 and the sum of their squares is 7. Find the numbers. 7. Find three consecutive positive numbers such that the square of the middle number exceeds the difference of the squares of the other two by If a number is added to three times its reciprocal, the result is 5 5. Find the number. 9. The length of a rectangle is cm more than its width and its area is 0 cm. Find the dimensions of the rectangle. 7

8 0. The denominator of a fraction is more than its numerator. The sum of the fraction and its reciprocal is 0 9 Find the fraction.. A two-digit number is times the sum of its digits and also equal to twice the product of its digits. Find the number. B. Questions From CBSE Examination Papers. The sum of the squares of two consecutive natural numbers is. Find the numbers.. Solve for x by using quadratic formula 6x ax + (a b ) = 0. The sum of a number and its reciprocal is 0. Find the number. x + x = 0; x, x x + (x )(x + ) 5. Solve the equation : [0 (T-II] = x,7 x + x 7 0 [0 (T-II] 6. If one root of the quadratic equation x 5x + 6k = 0 is reciprocal of other, find the value of k. Also find the roots. 7. Solve the given equation for x : + + = a b x a + b + x 8. Solve : = x x 9. Find the roots of the following quadratic equation by the factorisation method. x + 5x = 0 0. Find the roots of the following quadratic equation. x + 7x 0 = 0. Find two natural numbers, which differ by and whose squares have the sum 9.. Solve for x : 5 x = x +, x 0,. The sum of two natural numbers is 8. Determine the numbers, if the sum of their reciprocals is 8/5.. An express train takes hour less than a passenger train to travel km between stations A and B (without taking into consideration the time they 8 stop at intermediate stations). If the average speed of the express train is km/hour more than that of the passenger train, find the average speed of the two trains. 5. If two pipes function simultaneously, a reservoir will be filled in twelve hours. First pipe fills the reservoir 0 hours faster than the second pipe. How many hours will the second pipe take to fill the reservoir? 6. Solve the equation for x : x + x = 7 x x 7. Solve for x : a(a + b )x + b x a = 0 8. Solve for x : = x x Solve for x : x + + x = ; x, x x + 0. If the equations 5x + (9 + p)x + p = 0 and 5x + 9 = 0 are satisfied by the same value of x, find the value of p.. Solve the following quadratic equation : x x 0 = 0. Solve for x : x x + x + + x = 5. Find the value of k for which the quadratic equation (k + ) x + (k + ) x + = 0 has equal roots.. The sum of the reciprocals of Rehman s ages (in years) years ago and 5 years from now is. Find his present age. 5. Solve for x : + = x + x + x + 6. Solve for x : 9x (a + b)x + ab = 0 ; x,,

9 7. Find two positive numbers whose squares have the difference 8 and the sum of the numbers is. 8. Solve for x : x 0 x + =, x, x + x 9. Find the roots of the quadratic equation : a b x + b x a x = 0 0. The length of a rectangular plot is greater than LONG ANSWER TYPE QUESTIONS thrice its breadth by m. The area of the plot is 0 sq. m. Find the length and breadth of the plot.. Solve for x : 7x x 7 = 0 [0 (T- II)]. Solve the following quadratic equation for x : p x + ( p q )x q = 0. Solve for x : [00] x x + = 5; given that x, x. x + x [ Marks]. The difference of the squares of two numbers is 5. The square of the smaller number is times the larger number. Find the numbers.. Out of a number of Saras birds, one fourth the number are moving about in lotus plants, th 9 coupled (along) with as well as 7 times the square root of the number move on a hill, 56 birds remain in Vakula trees. What is the total number of birds? (Mahavira around 850 A.D.). A train, travelling at a uniform speed for 60 km, would have taken 8 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.. If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been more than five times her actual age. What is her age now? 5. At present Asha s age (in years) is more than the square of her daughter Nisha s age. When Nisha grows to her mother s present age, Asha s age would be one year less than 0 times the present age of Nisha. Find the present ages of both Asha and Nisha. 6. In the centre of a rectangular lawn of dimensions 50 A. Important Questions B. Questions From CBSE Examination Papers. Three consecutive positive integers are taken such that the sum of the square of the first and the product of the other two is 5. Find the integers.. The speed of a boat in still water is km/hr. It can go km upstream and return downstream to the original point in hrs 5 min. Find the speed of the stream. 9 m 0 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 8 m. Find the length and breadth of the pond. 7. A sailor can row a boat 8 km downstream and return back to the starting point in hour 0 minutes. If the speed of the stream is km/hr, find the speed of the boat in still water. 8. The length of a rectangle is thrice as long as the side of a square. The side of the square is cm more than the width of the rectangle. If their areas being equal, find their dimensions. 9. Out of a group of swans, 7/ times the square root of the total number are playing on the shore of a pond. The two remaining ones are swinging in water. Find the total number of swans. 0. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m? If so, find its length and breadth.. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 0 years. Four years ago, the product of their ages in years was 8.. Is it possible to design a rectangular park of perimeter 80 m and area 00 m? If so, find its length and breadth.. The product of the digits of a two digit positive number is. If 8 is added to the number, then the digits of the number are interchanged. Find the number.. Two taps together can fill a tank in 9 8 hours. The tap of larger diameter takes 0 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

10 5. A person on tour has Rs 60 for his daily expenses. If he extends his tour for four days, he has to cut down his daily expenses by Rs. Find the original duration of the tour. 6. A train travels at a uniform speed for a distance of 6 km and then travels a distance of 7 km at an aveage speed of 6 km/h more than its original speed. If it takes hours to complete the total journey, what is the original speed of the train? 7. Sum of the areas of two squares is 68 m. If the difference of their perimeters is m, find the sides of the two squares. 8. The difference of squares of two numbers is 80. The square of the smaller number is 8 times the larger number. Find the two numbers. 9. Two pipes running together can fill a tank in 6 minutes. If one pipe takes 5 minutes more than the other to fill the tank, find the time in which each pipe would fill the tank separately. 0. A motorboat whose speed is 8 km/hr in still water takes hr more to go km upstream than to return downstream to the same spot. Find the speed of stream.. Some students planned a picnic.the budget for food was Rs 80. But 8 of them failed to go, the cost of food for each member increased by Rs 0. How many students attended the picnic?. A fast train takes hours less than a slow train for a journey of 600 km. If the speed of the slow train was 0 km/hr less than that of the fast train, find the speeds of the trains.. A person has a rectangular garden whose area is 00 sq m. He fences three sides of the garden with 0 m barbed wire. On the fourth side, the wall of his house is constructed; find the dimensions of the garden.. Two pipes can together fill a tank in minutes. If one pipe takes minutes more than the other to fill it, find the time in which each pipe can fill the tank. 5. By increasing the speed of a bus by 0 km/hr, it takes one and half hours less to cover a journey of 50 km. Find the original speed of the bus. 6. A factory produces certain pieces of pottery in a day. It was observed on a particular day that the cost of production of each piece (in rupees) was more than twice the number of articles produced in the day. If the total cost of production on that day was Rs 90, find the number of pieces produced and cost of each piece. 7. The hypotenuse of a right triangle is 5 cm. If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 5 cm. Find the length of each side. [0 (T-II] 8. A man bought a certain number of toys for Rs 80. He kept one for his own use and sold the rest for one rupee each more than he gave for them. Besides getting his own toy for nothing, he made a profit of Rs 0. Find the number of toys, he initially bought. 9. The product of Tanay s age (in years) five years ago and his age ten years later is 6. Determine Tanay s present age. 0. A plane left 0 minutes later than the schedule time and in order to reach its destination 500 km away in time, it has to increase its speed by 50 km/hr from its usual speed. Find its usual speed.. A train travels 00 km at a uniform speed. If the speed of the train had been 5 km/hour more, it would have taken hours less for the same journey. Find the usual speed of the train.. The sum of two natural numbers is 8. Determine the numbers, if the sum of their reciprocals is A two digit number is four times the sum of its digits. It is also equal to three times the product of its digits. Find the number.. The speed of a boat in still water is 5 km/hr. It can go 0 km upstream and return downstream to the original point in hours 0 minutes. Find the speed of the stream. 0

11 5. The sum of ages of father and his son is 5 years. 5 years ago, the product of their ages was. Determine their present ages. [0 ( T-II)] 6. The numerator of a fraction is less than its denominator. If is addd to both numerator as well as denominator, the sum of new and original fraction is, find the frction. 7. The hypotenuse of a right angled triangle is 6 cm more than twice its shortest side. If third side is cm less than the hypotenuse, find the sides of this triangle. 8. A takes 6 days less tahn the time taken by B to finish a piece of work. If both A and B together can finish it in days, find the time taken by B to finish the work. 9. The differene of squares of two natural numbers is 5. The square of the smaller number is four times the larger number. Find the numbers. 0. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 6, find the fraction.. In a class test, the sum of the marks obtained by P in Mathematics and Science is 8. Had he got more marks in Maths and marks less in Science, the product of marks obtained in the two subjects would have been 80. Find the marks obtained in the two subjects separately. [008]. A peacock is sitting on the top of a pillar which is 9 m high. From a point 7 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal, at what distane from the hole is the snake caught? [008]. In a class test, the sum of Kamal s marks in Maths and English is 0. Had he got marks more in Maths and marks less in English, the product of their marks would have been 60. Find his marks in two subjects. [008C]. A person on tour has Rs 00 for his expenses. If he extends his tour for days, he has to cut down his daily expenses by Rs 70. Find the duration of the tour. [008C] 5. In a class test the sum of Gagan s, marks in Mathematics and English is 5. If he had more mark in Maths and less in English, the product of marks would have been 500. Find the original marks obtained by Gagan in Maths and English separately. [008C] 6. Places A and B are 00 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in hour. What are the speeds of the two cars? [009] 7. If 5 is a root of the quadratic equation x + px 5 = 0 and the quadratic equation p(x + x) + k = 0 has equal roots, then find the values of p and k. [009] 8. A trader bought a number of articles for Rs 900, five articles were found damaged. He sold each of the remaining articles at Rs more than what he paid for it. He got a profit of Rs 80 on the whole transaction. Find the number of articles he bought. [009] 9. Two years ago a man s age was three times the square of his son s age. Three years hence his age will be four times his son s age. Find their present ages. [009]