Quadratic Equations Chapter Wise Important Questions Class 10 Mathematics

Quadratic Equations Chapter Wise Important Questions Class 10 Mathematics

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2016

Short Answer Type Questions I [2 Marks]

Question 1.
If x= 2/3 and x = – 3 are roots of the quadratic equations ax² + lx + b = 0, find the values of a and b.
Solution :

Question 2.
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.
Solution :


Question 3.
Solve for x: + x = 13.
Solution :

Question 4.
Solve for x :√3x²-2√2x-2√3=0 = 0.
Solution :

Question 5.
A two digit number is four times the sum of the digits. It is also equal to 3 times the product of digits. Find the number.
Solution :

Question 6.
Solve for x:√3x² – 2√2x-2√3=0

Solution :

Question 7.

Solve for x:
Solution :

Short Answer Type Questions II [3 Marks]

Question 8.

Solve for x:
Solution :

Question 9.

Solve for x:
Solution :

Question 10.
Solve the following quadratic equation for x:

Solution :

Question 11.

Solve for x:
Solution :

Question 12.
If the roots of the quadratic equation (a – b)x² + (b – c)x + (c – a) = 0 are equal, prove that 2a = b + c.
Solution :

Question 13.
Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60. Find the numbers
Solution :

Question 14.
Two water taps together can fill a tank in 9 hours 36 minutes. The tap of larger diameter takes 8 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Solution :
Let x be the time taken by larger diameter tap.
x + 8 be the time taken by smaller diameter tap.

Question 15.
Solve the given quadratic equation for x: 9x² – 9(a + b)x + (2a² + 5ab + 2b² ) =0.
Solution :

Question 16.

Solve for x:
Solution :

Question 17.

Solve for x (in terms of a and b):
Solution :

Long Answer Type Questions [4 Marks]

Question 18.
A passenger, while boarding the plane, slipped from the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane. What value is depicted in this question?
Solution :

Question 19.
Find x in terms of a, b and c:

Solution :

Question 20.
The time taken by a person to cover 150 km was 2. 1/2 hours more than the time taken in the return journey. If he returned at a speed of 10 km/hour more than the speed while going, find the speed per hour in each direction.
Solution :

Let t₁ and t₂ be the time taken in going and returning respectively.
Also, v and v + 10 be the speed in going and returning respectively.

Question 21.
Solve for x:

Solution :

Question 22.
A motor boat whose speed is 24 km/h 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.
Solution :

Question 23.
A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find the length and breadth of the rectangular park.
Solution :

Question 24.
Two pipes running together can fill a tank in 11. 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.
Solution :

Question 25.
A pole has to be erected at a point on the boundary of a circular park of diameter 17 m in such a way that the differences of its distances from two diametrically opposite fixed gates where the pole is to be erected.
Solution :

Question 26.
Find the positive value(s) of k for which quadratic equations x² + kx + 64 = 0 and x² -8X+ k = 0 both will have real roots.
Solution :

Question 27.
If roots of the quadratic equation x² + 2px + mn = 0 are real and equal, show that the roots of the quadratic equation x² – 2(m + n)x + (m² + n² + 2p² ) =0 are also equal.
Solution :

Question 28.
The denominator of a fraction is one more than twice its numerator. If the sum of the fraction and its reciprocal 2. 16/21 find the fraction
Solution :

2015

Very Short Answer Type Questions [1 Mark]

Question 29.
If x =-1/2 , is a solution of the quadratic equation 3x² + 2kx -3 = 0, find the value of K .
Solution :

Question 30.
If the quadratic equation px² – 2√5px + 15 = 0 has two equal roots, then find the value of p.
Solution :

Short Answer Type Questions I [2 Marks]

Question 31.
Solve the following quadratic equation for x: 4x² – 4a² x + (a⁴ – b⁴ ) = 0
Solution :

Question 32.
Solve the following quadratic equation for x: 9x² – 6b² x – (a⁴ – b⁴ ) = 0
Solution :

Question 33.
Solve the following quadratic equation for x: 4x² + 4bx – (a² – b²) = 0
Solution :

Question 34.
Solve the following quadratic equation for x: x² – 2ax – (4b² -a²) = 0
Solution :

Question 35.
Solve for x  .x² -(√3+1)x+√3=0
Solution :

Short Answer Type Questions II [3 Marks]

Question 36.
Find the value ofp for which the quadratic equation (p + l)x² – 6(p + 1)x + 3(p + 9) = 0,p ≠ 1 has equal roots. Hence, find the roots of the equation.
Solution :

Question 37.
Find that non-zero value of k, for which the quadratic equation kx² + 1 – 2 (k – 1)x + x²= 0 has equal roots. Hence, find the roots of the equation.
Solution :

Question 38.
Solve for x: √3x² – 2√2x-2√3=0
Solution :

Question 39.
Solve for x: 2x² + 6√3-60 = 0
Solution :

Question 40.
Solve for x: x² + 5x – (a² + a – 6) =0
Solution :

Question 41.
Solve for x:
x² -(2b- l)x + (b² -b-20) = 0
Solution :

Question 42.
Solve for a:
x² + 6x – (a² + 2a – 8) = 0
Solution :

Long Answer Type Questions [4 Marks]

Question 43.

Solve for x:
Solution :

Question 44.

Solve for x
Solution :

Question 45.
The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original  fraction is 29/20. Find the original fraction.
Solution :

Question 46.
larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.
Solution :

Question 47.
The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field.
Solution :

Question 48.
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/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?
Solution :

Question 49.
A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/h more than the first speed. If it takes 3 hours to complete the total journey, find its first speed.
Solution :

Question 50.
A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the toal distance in 5 hours, find the first speed of the truck
Solution :

Question 51.
Ifx = – 2 is a root of the equation 3x² + 7x + p = 0, find the values of k so that the roots of the equation x² + k(4x + k-1) + p = 0 are equal.
Solution :

Question 52.
The total cost of a certain length of a piece of cloth is rs 200. If the piece was 5 cm longer and each metre of cloth costs rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre?
Solution :

Question 53.
Ifx = 3 is root of the equation x² -x + k = 0, find the value of p so that the roots of the equation x² + k(2x + k +2)+ p = 0 are equal.
Solution :

2014

Short Answer Type Questions I [2 Marks]

Question 54.
Solve the quadratic equation 2x² + ax- a²= 0 for x.
Solution :

Question 55.
Find the values of p for which the quadratic equation 4x² + px + 3 = 0 has equal roots.
Solution :

Question 56.
Find the values of k for which the quadratic equation 9x² – 3kx + k = 0 has equal roots.
Solution :

Question 57.
Find the value of p so that the quadratic equation px(x – 3) + 9 = 0 has equal roots.
Solution :

Question 58.
Solve for x: √3x²-2√2x-2√3=0
Solution :

Short Answer Type Questions lI [3 Marks]

Question 59.
Solve the equation:

Solution :

Question 60.
Solve the equation:

Solution :

Question 61.
Solve the equation:

Solution :

Question 62.
Solve for x:

Solution :

Question 63.
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.
Solution :

Question 64.
If 2 is a root of the quadratic equation 3x² + px – 8 = 0 and the quadratic equation 4x²– 2px + k = 0 has equal roots, find the value of k.
Solution :

Question 65.
If 1 is a root of the quadratic equation 3x² + ax – 2 = 0 and the quadratic equation a(x² + 6x) – b = 0 has equal roots, find the value of b.
Solution :

Long Answer Type Questions [4 Marks]

Question 66.
Find the value of p for which the quadratic equation (2p + l)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also find these roots.
Solution :

Question 67.
Find the values of k for which the quadratic equation (3k + 1)x² + 2(k + l)x + 1 = 0 has equal roots. Also find these roots.
Solution :

Question 68.
Find the values of k for which the quadratic equation (k + 4)x² + (k + l)x + 1 = 0 has equal roots. Also find these roots.
Solution :

Question 69.
The difference of two natural numbers is 5 and the difference of their reciprocals is 1/10. find the numbers.
Solution :

Question 70.
The difference of two natural numbers is 3 and the difference of their reciprocal is 3/28. find the numbers.
Solution :

Question 71.
The difference of two natural numbers is 5 and the difference of their reciprocal is 5/14. find the numbers.
Solution :

Question 72.
Solve for x

Solution :

Question 73.
A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Solution :

Question 74.

Solve for x
Solution :


Question 75.

Solve for x
Solution :

Question 76.
The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
Solution :

Question 77.

Solve for x:
Solution :


Question 78.
The sum of the squares of two consecutive even numbers is 340. Find the numbers.
Solution :

Question 79.
Solve for x
Solution :

Question 80.
The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples.
Solution :

Question 81.
Solve for x :

Solution :

2013

Short Answer Type Questions I [2 Marks]

Question 82.
Solve the following quadratic equation for x; 4√3x² + 5x -2√3 = 0
Solution :

Question 83.
Solve the following forjr. √2x² + 7x + 5√2= 0
Solution :

Question 84.
Solve for x: x² – (√2 + 1) x + √2 = 0.
Solution :

Short Answer Type Questions II [3 Marks]

Question 85.
For what value of k, are the roots of the quadratic equation kx(x – 2) + 6 = 0 equal?
Solution :

Question 86.
For what value of k, are the roots of the quadratic equation kx(x – 2√5) + 10 = 0, equal?
Solution :

Question 87.
For what value of k, are the roots of the quadratic equation(k + 4)x² + (k + l)x + 1 = 0, equal?
Solution :

Question 88.
For what value of k, are the roots of the quadratic equation (k – 12)x² + 2(k – 12)x + 2 = 0 equal ?
Solution :

Question 89.
For what value of A, are the roots of the quadratic equation y² + k² = 2(k + l)y equal?
Solution :

Question 90.
For what value of k, are the roots of the quadratic equation (k – 4)x² + (k – 4) x + 4 = 0 equal ?
Solution :

Question 91.
For what value of k, are the roots of the quadratic equation x² – (3k – l)x + 2k² + 2k -11 = 0 equal?
Solution :

Question 92.
For what value of m, are the roots of the quadratic equation x² – 2x (1 + 3m) + 7 (3 + 2m) = 0 equal?
Solution :

Long Answer Type Questions [4 Marks]

Question 93.
Solve for x:

Solution :

Question 94.
Sum of the areas of two squares is 400 cm2. If the difference of their perimeters is 16 cm, find the sides of the two squares.
Solution :

Question 95.
Solve for x

Solution :

Question 96.
Solve for x

Solution :

Question 97.
Solve for x

Solution :

Question 98.

Solve for x:
Solution :

Question 99.
The present age of a father is equal to the square of the present age of his son. One year ago, the age of the father was 8 times the age of his son. Find their present ages.
Solution :

Question 100.
Solve the following quadratic equation in variable x : abx² = (a + b)² (x -1).
Solution :

Short Answer Type Questions I [2 Marks]

Question 101.
Find the value(s) of k so that the quadratic equation 2x² + kx + 3 = 0 has equal roots.
Solution :

Question 102.
Find the value(s) of so that the quadratic equation x² – 4kx +k = 0 has equal roots.
Solution :

Question 103.
Find the value(s) of k so that the quadratic equation 3x² – 2kx + 12 = 0 has equal roots
Solution :

Question 104.
Find the value of m for which the roots of the equation mx(6x + 10) + 25 = 0, are equal.
Solution :

Question 105.
Find the value of k for which the roots of the equation kx(3x – 4) + 4 = 0, are equal.
Solution :

Question 106.
Find the value of ip’ for which the roots of the equation px(x – 2) + 6 = 0, are equal.
Solution :

Question 107.
Find the value ofp for which the roots of the quadratic equation (p + 3) x² + 2(p+3)x+ 4 = 0 are equal.
Solution :

Question 108.
Find the value of* for which the roots of the quadratic equation (k – 4)x² + 2(k – 4) x + 2 = 0 are equal.
Solution :

Question 109.
Find the value of ‘a’ for which the roots of the quadratic equation 2(a + 5)x² – (a + 5)x + 1 = 0 are equal.
Solution :

Short Answer Type Questions II [3 Marks]

Question 110.
Solve for x: 4x² – 4ax + (a² – b²) = 0
Solution :

Question 111.
Solve for x: 3x² – 2-√6x + 2 = 0
Solution :

Question 112.
If the sum of two natural numbers is 8 and their product is 15, find the numbers.
Solution :

Question 113.
Solve for x: x²– 4ax – b + 4a²= 0.
Solution :

Question 114.
Solve for x: x²– 5√5x + 30 = 0.
Solution :

Question 115.
Solve for x: x² + √5 x – 60 = 0
Solution :

Question 116.
Solve for x: 4√3x² + 5x- 2√3 = 0
Solution :

long Answer Type Questions [4 Marks]

Question 117.
A shopkeeper buys some books for rs 80. If he had bought 4 more books for the same amount, each book would have cost rs 1 less. Find the number of books he bought.
Solution :

Question 118.
The sum of two numbers is 9 and the sum of their reciprocals is 1/2. Find the numbers.
Solution :

Question 119.
In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.
Solution :

Question 120.
The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by 1/15. Find the fraction.
Solution :

Question 121.
A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits interchange their places. Find the number.
Solution :


Question 122.
Find two consecutive natural numbers, the sum of whose squares is 145
Solution :

2011

Short Answer Type Questions I [2 Marks]

Question 123.
Find the value of p so that the quadratic equation px(x – 3) + 9 = 0 has two equal roots.
Solution :

Question 124.
Find the value of & so that the quadratic equation kx(3x -10) + 25 = 0, has two equal roots.
Solution :

Question 125.
Find the value of m so that the quadratic equation mx(5x – 6) + 9 = 0 has two equal roots.
Solution :

Question 126.
Find the value of m so that the quadratic equation mx(x – 7) + 49 = 0 has two equal roots.
Solution :

Question 127.
For what value of k does the quadratic equation (k – 5)x²+ 2(k – 5)x + 2 = 0 have equal roots?
Solution :

Short Answer Type Questions II [3 Marks]

Question 128.
Find the roots of the following quadratic equation. 2√3x²-5x + √3 = 0
Solution :

Question 129.
Find the roots of the following quadratic equation: x²– 3√5x + 10 = 0
Solution :

Question 130.
Find the roots of the following quadratic equation:√3x²-2√2x-2√3=0
Solution :


Question 131.
Find the roots of the following quadratic equation: 3x² + 2 √5x — 5 = 0
Solution :

Long Answer Type Questions [4 Marks]

Question 132.
A motor boat whose speed is 20 km/h in still water, takes 1 hour more to go 48 km upstream than to return downstream to the same spot. Find the speed of the stream.
Solution :
Let the speed of the stream = km/h
Speed of the boat in still water = 20 km/h
Now, speed of boat during downstream = (20 + x) km/h

Question 133.
Find the roots of the equation:

Solution :

Question 134.
Find the roots of the equation:

Solution :

Question 135.
A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Solution :

Question 136.
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.
Solution :

Question 137.
Solve the following equation for x:

Solution :

2010

Very Short Answer Type Questions [1 Mark].

Question 138.

Solution :


Question 139.
If one zero of the polynomial x² – 4x + 1 is 2 + √3, write the other zero.
Solution :

Short Answer Type Questions I [2 Marks]

Question 140.
For what value of k, is -2 a zero of the polynomial 3x² + 4x + 2k?
Solution :

Question 141.
For what value of k, is 3 a zero of the polynomial 2x² + x + k?
Solution :

Question 142.
For what value of It, is 3 a zero of the polynomial x² + 11x + k?
Solution :

Long Answer Type Questions [4 Marks]

Question 143.
The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.
Solution :

Question 144.
Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the integers.
Solution :

Question 145.
A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.
Solution :

Question 146.
Solve the following equation for x.

Solution :


Question 147.
Some students planned a picnic. The total budget for food was rs 2,000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by rs 20. How many students attended the picnic and how much did each student pay for the food.
Solution :