CBSE Class 10 Mathematics Quadratic Equation Worksheet Set Q

VERY SHORT ANSWER TYPE QUESTIONS

Question. If 1/2 is a root of x2 + px – 5/4 = 0 then value of p is
(a) 2
(b) –2
(c) 1/4
(d) 1/2
Answer. (a) 2

Question. Roots of Quadratic equation x² – 7x = 0 will be
(a) 7
(b) 0, –7
(c) 0, 5
(d) 0, 7
Answer. (d) 0, 7

Question. Which of the following is not a Quadratic Equation?
(a) 2(x – 1)² = 4x² – 2x + 1
(b) 3x – x² = x² + 6
(c) (√3x+√2)² = 2x² – 5x
(d) (x² + 2x)² = x⁴ + 3 + 4x²
Answer. (d) (x² + 2x)² = x⁴ + 3 + 4x²

Question. 0.3 is a root of x² – 0.9 = 0. True or False
Answer. False ((0.3)2 – 0.9 = 0.09 – 0.9 ≠ 0)

Question. Every quadratic equation has atleast one real roots. True or False
Answer. False (A quadratic equation has atmost two real root).

Question. If px² + qx + r = 0 has equal roots then value of r will be ______ .
Answer. [r = q²/4p (D = 0 ⇒ q² - 4pr = 0)]

Question. If in a quadratic equation ax² + bx + c = 0, value of a is zero then it become a _____ equation.
Answer. Linear equation (x = 0 ⇒ ax² + bx + c = 0 reduces to bx + c = 0)

Question. The value of k for which the roots of qaudratic equations x² + 4x + k = 0 are real is ______ .
Answer. k ≤ 4 [D ≥ 0 ⇒ 16 - 4 k ≥ 0 ⇒ 16 ≥ 4k ⇒ 4 ≥ k]

SHORT ANSWER TYPE QUESTIONS-I

Question. Solve for x by factorisation 2x² + ax – a² = 0
Answer. x = a/2, x = -a

Question. Solve for x by factorisation 3√5x² + 25x + 10√5 = 0
Answer. x = -√5, x = (-2√5)/3

Question. If – 5 is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots find the value of k.
Answer. 2(– 5)2 + p(– 5) – 15 = 0 ⇒ p = 7
∴ 7x² + 7x + k = 0, D = 49 – 28 k = 0
⇒ k = 49/28 = 7/4

Question. Solve for x by factorisation 3x² – 2√6 x + 2 = 0
Answer. x = √2/3, x = √2/3

Question. The sides of two squares are x cm and (x + 4) cm. The sum of their areas is 656 cm² Find the sides of these two squares.
Answer. x² + (x + 4)² = 656
x² + 4x – 320 = 0
D = 1296 x = (-4±√1296)/2 = (-14+36)/2, (-4-36)/2
x = 32/2 = 16, (rejecting –ve value)
Sides are 16 cm, 20 cm

Question. If the Quadratic equation px2 – 2 5 px + 15 = 0 (p ≠ 0) has two equal roots then find the value of p.
Answer. D = 0
20p² – 60p = 0, p ≠ 0
20p (p – 3) = 0
p = 3

SHORT ANSWER TYPE QUESTIONS-II

Question. Solve for x
2x/(x-3) + 1/(2x+3) + (3x+9)/((x-3)(2x+3)) = 0, x ≠ 3, -3/2
Answer. Take LCM to get 2x² + 5x + 3 = 0, x = – 1, x ≠ -3/2. (given)

Question. Solve for x
1/(x+4) - 1/(x-7) = 11/30, x ≠ -4, 7
Answer. Take LCM to get x² – 3x + 2 = 0
Solve to get x = 1, x = 2

Question. If the roots of the quadratic equation (p + 1)x² – 6(p + 1) x + 3(p + 9) = 0 are equal find p and then find the roots of this quadratic equation.
Answer. D = 0
∴ p² - 2p = 0 ; p = -1, 3
rejecting p = –1,
∴ Answer is p = 3

LONG ANSWER TYPE QUESTIONS

Question. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2(16/21). Find the fraction.
Answer. Fraction is x/(2x+1)
ATQ x/(2x+1) + (2x+1)/x = 2(16/21) = 58/21
Solve to get x = 3, x = -7/11 (rejected)
∴ Answer is Fraction = 3/7.

Question. Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Answer. Time taken by top of smaller diameter = x hrs
Time taken by larger tap = (x – 9) hrs
ATQ 1/x + 1/(x-9) = 1/6 and get x² – 21x + 54 = 0
x = 3, x = 18
x = 3 rejeced as x – 9 = – 6 < 0
∴ x = 18 hrs x – 9 = 18 – 9 = 9 hrs

Question. Sum of areas of two squares is 400 cm². If the difference of their perimeter is 16 cm. Find the side of each square.
Answer. x² + y² = 400 ...(1)
4x – 4y = 16 ⇒ x – y = 4 ...(2)
y – x = 4 ...(3)
Solve (1) and (2) to get x = 16, x = –12 (rejected)
Solve (1) and (3) to get x = 12, x = –16 (rejected)
x = 16 m, y = 12 m from (1) and (2)
x = 12 m, y = 16 m from (1) and (3)

Question. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
Answer. Let the natural number be x.
ATQ, x + 12 = 160/x to get x² + 12x – 160 = 0
(x + 20) (x – 8) = 0
x = 8, x = – 20 (rejected)

Question. If the price of a book is reduced by ₹ 5, a person can buy 5 more books for ₹ 300. Find the original list price of the book.
Answer. Let original list price = ₹ x
ATQ 300/(x-5) - 300/x = 5
Solve and get x = 20, x = –15 → rejected
∴ ₹ 20 is Answer

Question. If the roots of the quadratic equation (b – c)x² + (c – a)x + (a – b) = 0 are equal, prove 2b = a + c.
Answer. Find D and let D = 0
(c – a)² – 4(b – c) (a – b) = 0
Solve to get (a + c – 2b)² = 0
∴ a + c = 2b

Question. A motor boat whose speed is 24 km/hr in still water takes 1 hour more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream
Answer. Let speed of boat = x
ATQ 32/(24-x) - 32(24+x) = 1
x² + 64x – 576 = 0
(x + 72) (x – 8) = 0
x = 8 km/hr
x = –72 km/hr (rejected)

Question. A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hr less than the fast train, find the speed of the two trains.
Answer. ATQ 600/x - 600/(x+10) = 3 (Speed of slow train x km/hr)
Solve to get x = 40, x = – 50 (rejected).
∴ Answer is 40 km/hr, 50 km/hr.

Question. A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less, then it would have taken 3 hours more to cover the same distance. Find the original speed of the train.
Answer. Let the speed of the train = x km/hr
ATQ, 480/(x-8) - 480/x = 3
x² – 8x – 1280 = 0
x = 40, –32 (rejected)
x = 40 km/hr

Question. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?
Answer. Equation 54/x + 63/(x+6) = 3, x → speed of train at first, x + 6 → Increased speed.
x = 36, x ≠ – 3.

Question. ₹ 6500 were divided equally among a certain number of persons. If there been 15 more persons, each would have got ₹ 30 less. Find the original number of persons.
Answer. Let original number of persons be x
ATQ 6500/x - 6500/(x+15) = 30
Solve and get x = 50, x = – 65 (rejected).

I. Water Distribution System: Delhi Jal Board (DJB) is the main body of the Delhi Government which supplies drinking water in the National Capital Territory of Delhi. Distribution system is well knit and properly planned. Maintenance of underground pipe and hose system is also performed at regular interval of time. Many rivers and canals are inter-connected in order to ensure un-interrupted water supply. It has been meeting the needs of potable water for more than 16 million people. It ensures availability of 50 gallons per capita per day of pure and filtered water with the help of efficient network of water treatment plants and pumping stations.
In our locality, DJB constructed two big reservoir labelled as Reservoir–A and Reservoir–B.
Reservoir–A: In order to fill it, department uses two pipes of different diameter.
Reservoir–B: Department uses two taps to store water in this reservoir.
Refer to Reservoir-A

Question. Two taps running together can fill the reservoir in 3 , 1/13 minutes. If one tap takes 3 minutes more than the other to fill it, how many minutes each tap would take to fill the reservoir?         
(a) 12 min, 15 min
(b) 6 min, 9 min
(c) 18 min, 14 min
(d) 5 min, 8 min

Answer: D

Question. Two pipes running together can fill a reservoir in 6 minutes. If one pipe takes 5 minutes more than the other to fill the reservoir, the time in which each pipe would fill the reservoir separately is         
(a) 8 min, 6 min
(b) 10 min, 15 min
(c) 12 min, 16 min
(d) 16 min, 18 min

Answer: B

Question. If two tapes function simultaneously, reservoir will be filled in 12 hours. One tap fills the reservoir 10 hours faster than the other. The time that the second tap takes to fill the reservoir is given by     
(a) 25 hrs
(b) 28 hrs
(c) 30 hrs
(d) 32 hrs

Answer: C

Question. Two water taps together can fill a reservoir in 9,3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the reservoir separately. The time in which each tap can separately fill the reservoir will be         
(a) 15 hrs, 25 hrs
(b) 20 hrs, 22 hrs
(c) 14 hrs, 18 hrs
(d) 18 hrs, 16 hrs

Answer: 4

Question. Two pipes running together can fill the reservoir in 11, 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the reservoir, the time in which each pipe alone would fill the reservoir is     
(a) 10 min, 12 min
(b) 25 min, 20 min
(c) 15 min, 18 min
(d) 22 min, 28 min

Answer: B

II. A Hill Station: In the last summer, I enjoyed a tour to a hill station at Shimla. I was accompanied by my five friends and enjoyed the natural beauties of mountains, rivers, streams, forests etc. The beginning of the tour was the most adventurous itself! How amazingly my group win the bet! Actually, the story is that my two friends along with me prefered train to go to Shimla, but other three were forcing for a car or a bus. At last the consensus was reached and we were divided ourselves in two groups of 3 each and started for Shimla at the same time. It was decided that the group who reach the destination first, would be declared as the winner, and runner up the group have to bear the expanses of the tour. I named my group, ‘Group A’ while the second group was named as ‘Group B’. Luckily we reached Shimla 1 hour before the Group-B and enjoyed the trip for absolutely FREE!! How thrilling it was the tour!

Question. An express train makes a run of 240 km at a certain speed. Another train whose speed is 12 km/hr less takes an hour longer to make the same trip. The speed of the express train will be             
(a) 60 km/h
(b) 50 km/h
(c) 65 km/h
(d) 48 km/h

Answer: A

Question. A bus travels a distance of 300 km at a uniform speed. If the speed of the bus is increased by 5 km an hour, the journey would have taken two hours less. The original speed of the bus will be           
(a) 20 km/h
(b) 15 km/h
(c) 22 km/h
(d) 25 km/h

Answer: D

Question. An express train takes 1 hour less than a passenger train to travel 132 km between Delhi and Shimla (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/hr more than that of the passenger train, the average speeds of the two trains will be     
(a) 33 km/h, 44 km/hr
(b) 40 km/h, 45 km/h
(c) 30 km/h, 38 km/h
(d) 42 km/h, 62 km/h

Answer: A

Question. A deluxe bus takes 3 hours less than a ordinary bus for a journey of 600 km. If the speed of the ordinary bus is 10 km/hr less than that of the deluxe bus, the speeds of the two buses will be             
(a) 35 km/h, 42 km/h
(b) 42 km/h, 52 km/h
(c) 40 km/h, 50 km/h
(d) 30 km/h, 58 km/h

Answer: C

Question. A journey of 192 km from Delhi to Shimla takes 2 hours less by a super fast train than that by an ordinary passenger train. If the average speed of the slower train is 16 km/hr less than that of the faster train, average speed of super fast train is
(a) 50 km/h
(b) 48 km/h
(c) 55 km/h
(d) 60 km/h

Answer: B