CBSE Class 10 Quadratic Equations Sure Shot Questions Set P

Question. Find the value(s) of k for which \( 2x^2 + kx + 3 = 0 \) has equal roots.
Answer: \( \pm 2\sqrt{6} \)

Question. Find the value(s) of k for which \( (k + 4)x^2 + (k + 1)x + 1 = 0 \) has equal roots.
Answer: 5, –3

Question. Find the value(s) of k for which \( (k – 4)x^2 + 2(k – 4)x + 4 = 0 \) has equal roots.
Answer: 8

Question. Find the value(s) of k for which \( kx(x – 2) + 6 = 0 \) has equal roots.
Answer: 6

Question. Find the value(s) of k for which \( px^2 + 4x + 1 = 0 \) has real roots.
Answer: \( p \leq 4 \)

Question. Find the value(s) of k for which \( 4x^2 + 8x – p = 0 \) has real roots.
Answer: \( p \geq – 4 \)

Question. Find the value(s) of k for which \( 2x^2 + px + 8 = 0 \) has real roots.
Answer: \( p \geq 8 \) or \( p \leq –8 \)

Question. Find the value(s) of k for which the simultaneous linear equation have a unique solution : \( (k – 1)x – 3y = 4, 3x – (4k + 1)y = 5 \)
Answer: All real nos except 2 and \( – \frac{5}{4} \)

Question. Find two consecutive natural numbers such that the sum of their squares is 365.
Answer: 13, 14

Question. Find two consecutive integers such that the sum of their squares is 365.
Answer: 13, 14 or –13, –14

Question. If the product of two consecutive positive integers is 306, find the integers.
Answer: 17, 18

Question. If the product of two positive consecutive odd integers is 255, find the integers.
Answer: 15, 17

Question. If the product of two consecutive odd integers is 255, find the integers.
Answer: 15, 17 or –15, –17

Question. The sum of the squares of two consecutive odd natural numbers is 290, find the numbers.
Answer: 11, 13

Question. The sum of the squares of two natural numbers is 116. If the square of the larger is 25 times the smaller, find the numbers.
Answer: 4, 10

Question. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the large number. Find the two numbers.
Answer: 18, 12 or 18, –12

Question. The sum of the square of three consecutive natural numbers is 110, find the numbers.
Answer: 5, 6, 7

Question. Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the square of the other two by 60, find the numbers.
Answer: 9, 10, 11

Question. Divide 14 into two parts such that the sum of their reciprocals is \( \frac{7}{20} \).
Answer: 4, 10

Question. The difference of two natural number is 4 and the difference of their reciprocals is \( \frac{1}{8} \), find the numbers.
Answer: 8, 4

Question. The sum of the numerator and denominator of a certain fraction is 11. If 1 is added to both numerator and denominator, the fraction increases by \( \frac{3}{56} \). Find the fraction.
Answer: \( \frac{4}{7} \)

Question. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is \( 2 \frac{16}{21} \), find the fraction.
Answer: \( \frac{3}{7} \)

Question. A two digit number contains the bigger at ten's place. The product of the digit is 27 and the difference between the two digits is 6. find the number.
Answer: 93

Question. A two digit number is such that the product of its digits is 24. When 18 is subtracted from this number, the digits interchange their places. Find the number.
Answer: 64

Question. Mukesh and Suresh together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. How many marbles they had to start with.
Answer: 9, 36

Question. A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produces in a day. On a particular day, the total cost was Rs. 750. Find the number of toys produced on that day.
Answer: 25 or 30

Question. The area of rectangular plot is \( 528\text{ m}^2 \). The length of the plot (in meters) is one more than twice its breadth. find the length and breadth of the plot.
Answer: 33 m, 16 m

Question. A rectangle has an area of \( 24\text{ cm}^2 \). If the perimeter is 20 cm, find its length.
Answer: 6 cm

Question. A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. If the area of the walk is \( 120\text{ m}^2 \), find the width of the walk.
Answer: 2 m.

Question. Harish made a rectangular garden with its length 5 meters more than its breadth. Next year, he increased the length by 3 metres and decreased the width by 2 meters. If the area of the garden is 119 sq. m, was this garden larger or smaller ?
Answer: smaller by 7 sq. m.

Question. The sum of the areas of two squares is \( 468\text{ m}^2 \). If the difference of their perimeters is 24 m, find the sides of the two squares.
Answer: 18 m, 12 m

Question. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Answer: 12 cm, 5 cm

Question. A wire, 112 cm long is bent to form a right angled triangle. If the hypotenuse is 50 cm long, find the area of the triangle.
Answer: \( 336\text{ cm}^2 \)

Question. A rectangular park is to be designed whose length is 3m more than its breadth. Its area is 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 dimensions of the rectangular park.
Answer: 7 m \( \times \) 4 m

Question. A school bus transported an excursion party to a picnic spot 150 km away. While returning it was raining and the bus had to reduce its speed by 5 km / hr, and it took one hour longer to make the return trip. Find the time taken in return trip.
Answer: 6 hours

Question. An aeroplane flying with a wind of 30 km / hr takes 40 minutes less of fly 3600 km, than what it would have taken to fly the same wind. Find the plane's speed in still air.
Answer: 570 km/hr

Question. A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24 km upstream than to return to the same spot. Find the speed of the stream.
Answer: 6 km/ hr

Question. When the price of an article is reduced by Rs 2, 5 more articles could be bought for Rs 120. Find the original price of each article.
Answer: Rs. 8

Question. A trader bought a number of articles for Rs 900, five were damaged and he sold each of the rest at Rs 2 more than what he paid for it thus getting a profit of Rs 80 on the whole transaction. Find the number of articles he bought.
Answer: 75

Question. Rahul sold an article for Rs 56 which cost him Rs x. He finds that he has gained x% on his outlay. Find x.
Answer: 40

Question. Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. What is Rohan's present age ?
Answer: 7 years

Question. Forty years hence Mr. Pratap's age will be square of what it was 32 years ago. Find his present age.
Answer: 41 years

Question. The sum of the ages of father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Find the present ages.
Answer: 36 years, 9 years

Question. The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.
Answer: 32 years, 4 years

Question. B takes 16 days less than A to do a piece of work. If both working together can do it in 15 days, in how many days will be alone complete the work ?
Answer: 24 days