CBSE Class 10 Quadratic Equations Sure Shot Questions Set D

CBSE Class 10 Quadratic Equations Sure Shot Questions Set D. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

1. Find the value of k for which the quadratic equation 2x²+ kx + 3 = 0 has two real equal roots.

2. Find the value of k for which the quadratic equation kx(x – 3) + 9 = 0 has two real equal roots.

3. Find the value of k for which the quadratic equation 4x² – 3kx + 1 = 0 has two real equalroots..

4. If –4 is a root of the equation x²+ px – 4 = 0 and the equation x²+ px +q = 0 has equal roots, find the value of p and q.

5. If –5 is a root of the equation 2x²+ px – 15 = 0 and the equation p(x²+ x) +k = 0 has equal roots, find the value of k.

6. Find the value of k for which the quadratic equation (k – 12)x²+ 2(k – 12)x + 2 = 0 has two real equal roots..

7. Find the value of k for which the quadratic equation k²x² – 2(k – 1)x + 4 = 0 has two real equal roots..

8. If the roots of the equation (a – b)x²+ (b – c)x + (c – a) = 0 are equal, prove that b + c = 2a.

9. Prove that both the roots of the equation (x – a)(x – b) + (x – b)(x – c)+ (x – c)(x – a) = 0 arereal but they are equal only when a = b = c.

10. Find the positive value of k for which the equation x²+ kx +64 = 0 and x²– 8x +k = 0 will have real roots.

11. Find the value of k for which the quadratic equation kx²– 6x – 2 = 0 has two real roots.

12. Find the value of k for which the quadratic equation 3x² + 2x + k= 0 has two real roots.

13. Find the value of k for which the quadratic equation 2x²+ kx + 2 = 0 has two real roots.

14. Show that the equation 3x²+ 7x + 8 = 0 is not true for any real value of x.

15. Show that the equation 2(a²+ b²)x² + 2(a + b)x + 1 = 0 has no real roots, when a ¹ b.

16. Find the value of k for which the quadratic equation kx²+ 2x + 1 = 0 has two real and distinct roots.

17. Find the value of p for which the quadratic equation 2x²+ px + 8 = 0 has two real and distinct roots.

18. If the equation (1 + m²)x²+ 2mcx + (c²– a²) = 0 has equal roots, prove that c² = a²(1 + m²). 

Question. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number. 
Answer: 8

Question. By increasing the list price of a book by ₹ 10, a person can buy 10 books less for ₹ 1200. Find the original list price of the book. 
Answer: ₹ 30

Question. The hypotenuse of a right-angled triangle is \( 1\text{ cm} \) more than twice the shortest side. If the third side is \( 2\text{ cm} \) less than the hypotenuse, find the sides of the triangle. 
Answer: \( 8\text{ cm} \), \( 15\text{ cm} \), \( 17\text{ cm} \)

Question. A passenger train takes 2 hours less for a journey of \( 300\text{ km} \), if its speed is increased by \( 5\text{ km/hr} \) from its usual speed. Find its usual speed. 
Answer: \( 25\text{ km/hr} \)

Question. The numerator of a fraction is one less than its denominator. If three is added to each of the numerator and denominator, the fraction is increased by \( \frac{3}{28} \). Find the fraction.
Answer: \( \frac{3}{4} \)

Question. The difference of squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers. 
Answer: 9 and 6

Question. Solve: \( x^2 + 5\sqrt{5}x - 70 = 0 \). 
Answer: \( 2\sqrt{5} \); \( -7\sqrt{5} \)

Question. A train travels a distance of \( 300\text{ km} \) at a uniform speed. If the speed of the train is increased by \( 5\text{ km} \) an hour, the journey would have taken two hours less. Find the original speed of the train. 
Answer: \( 25\text{ km/hr} \)

Question. The speed of a boat in still water is \( 11\text{ km/hr} \). It can go \( 12\text{ km} \) upstream and returns downstream to the original point in 2 hours 45 minutes. Find the speed of the stream. 
Answer: \( 5\text{ km/hr} \)

Question. Determine the value of \( k \) for which the quadratic equation \( 4x^2 - 3kx + 1 = 0 \) has equal roots. 
Answer: \( k \leq -\frac{4}{3} \) or \( k \geq \frac{4}{3} \)

Question. Using quadratic formula, solve the following equation for ‘\( x \)’: \( abx^2 + (b^2 - ac)x - bc = 0 \) 
Answer: \( x = \frac{c}{b} \), \( x = -\frac{b}{a} \)

Question. The sum of the numerator and the denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes \( \frac{1}{2} \). Find the fraction. 
Answer: \( \frac{5}{7} \)

Question. Rewrite the following as a quadratic equation in \( x \) and then solve for \( x \): \( \frac{4}{x} - 3 = \frac{5}{2x + 3}; x \neq 0, -\frac{3}{2} \) 
Answer: \( -2, 1 \)

Question. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number. 
Answer: 92

Question. A train covers a distance of \( 90\text{ km} \) at a uniform speed. Had the speed been \( 15\text{ km/hr} \) more, it would have taken 30 minutes less for the journey. Find the original speed of the train.  
Answer: \( 45\text{ km/hr} \)

Question. Solve for \( x \): \( \frac{x+1}{x-1} + \frac{x-2}{x+2} = 3; (x \neq 1, -2) \) 
Answer: \( x = -5, 2 \)

Question. Using quadratic formula, solve the following for \( x \): \( 9x^2 - 3(a^2 + b^2)x + a^2b^2 = 0 \) 
Answer: \( \frac{a^2}{3}, \frac{b^2}{3} \)

Question. Find the equation whose roots are reciprocal of the roots of \( 3x^2 - 5x + 7 = 0 \)
Answer: \( 7x^2 - 5x + 3 = 0 \)

Question. A number consists of two digits whose product is 18. When 27 is subtracted from the number, the digits change their places. Find the number. 
Answer: 63

Question. A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number. 
Answer: 63

Question. The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
Answer: 13 and 15

Question. An aeroplane left 30 minutes later than its scheduled time and in order to reach its destination \( 1500\text{ km} \) away in time, it has to increase its speed by \( 250\text{ km/hr} \) from its usual speed. Determine its usual speed.
Answer: \( 750\text{ km/h} \)

Question. Using quadratic formula, solve for \( x \): \( 9x^2 - 3(a + b)x + ab = 0 \) 
Answer: \( \frac{a}{3}, \frac{b}{3} \)

Question. Find the number which exceeds its positive square root by 20.
Answer: \( x = 25 \) (Rejecting \( x = 16 \))

Question. The sum of two numbers is 15 and the sum of their reciprocals is \( \frac{1}{3} \). Find the numbers. 
Answer: 5, 10

Question. A two digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number. 
Answer: 27

Question. A two digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number. 
Answer: 45

Question. A two digit number is such that the product of its digits is 15. If 8 is added to the number, the digits interchange their places. Find the number. 
Answer: 35

Question. Solve for \( x \): \( \frac{x-1}{x-2} + \frac{x-3}{x-4} = 3\frac{1}{3}; (x \neq 2, 4) \) 
Answer: \( \frac{5}{2}, 5 \)

Question. Solve for \( x \): \( abx^2 + (b^2 - ac)x - bc = 0 \) 
Answer: \( x = -\frac{b}{a}, \frac{c}{b} \)

Question. The sum of two numbers is 18. The sum of their reciprocals is \( \frac{1}{4} \). Find the numbers. 
Answer: 6, 12

Question. The sum of two numbers is 16, and the sum of their reciprocals is \( \frac{1}{3} \). Find the numbers. 
Answer: 12, 4

Question. The sum of two numbers ‘\( a \)’ and ‘\( b \)’ is 15, and sum of their reciprocals \( \frac{1}{a} \) and \( \frac{1}{b} \) is \( \frac{3}{10} \). Find the numbers ‘\( a \)’ and ‘\( b \)’. 
Answer: \( a = 5 \) or 10, \( b = 10 \) or 5

Question. Solve for \( x \): \( \frac{1}{a+b+x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}; a \neq 0, b \neq 0, x \neq 0 \) 
Answer: \( x = -a, -b \)

Question. Find the roots of the following quadratic equation: \( \frac{2}{5}x^2 - x - \frac{3}{5} = 0 \) 
Answer: \( -\frac{1}{2} \) and \( x = 3 \)

Question. Find the roots of the equation: \( \frac{1}{2x-3} + \frac{1}{x-5} = 1; x \neq \frac{3}{2}, 5 \) 
Answer: \( 4, \frac{3\sqrt{2}}{2} \)

Question. A natural number when subtracted from 28, becomes equal to 160 times its reciprocal. Find the number. 
Answer: 8

Question. Find two consecutive odd positive integers, sum of whose squares is 290.
Answer: 11 and 13

Question. Find the values of \( k \) for which the quadratic equation \( (k + 4)x^2 + (k + 1)x + 1 = 0 \) has equal roots. Also find these roots.
Answer: \( k = 5 \) or \( k = -3 \). For \( k = 5 \), roots are \( -\frac{1}{3}, -\frac{1}{3} \); for \( k = -3 \), roots are 1, 1.

Question. Solve for \( x \): \( \frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, -1. \) 
Answer: \( x = \pm 4 \)

Question. Solve for \( x \): \( \frac{x-2}{x-3} + \frac{x-4}{x-5} = \frac{10}{3}; x \neq 3, 5 \) 
Answer: \( x = \frac{7}{2} \)

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