1. Which of the following is a quadratic equation?
(a) \(x^2 + 3x + 1 = (x-2)^2\)
(b) \(x(x+1) + 8 = (x+2)(x-2)\)
(c) \((x+2)^3 = x^3 - 4\)
(d) \(x(2x+3) = x^2 + 1\)
2. The standard form of a quadratic equation is:
(a) \(ax^2 + bx + c = 0\), \(a \neq 0\)
(b) \(ax^2 + bx + c = 0\), \(a = 0\)
(c) \(ax + b = 0\)
(d) \(ax^3 + bx^2 + cx + d = 0\)
3. If one root of \(x^2 - 5x + 6 = 0\) is 2, the other root is:
(a) 3 (b) 4 (c) -3 (d) 6
4. The discriminant of \(2x^2 - 4x + 3 = 0\) is:
(a) 16 (b) -8 (c) 8 (d) 0
5. The roots of \(x^2 - 45x + 324 = 0\) represent:
(a) number of marbles John and Jivanti had
(b) speed of a train
(c) age of Rohan
(d) number of toys produced
6. For the equation \(2x^2 + x - 300 = 0\), the positive root gives:
(a) breadth of the prayer hall
(b) length of the prayer hall
(c) area of the hall
(d) perimeter of the hall
7. Assertion (A): The equation \((x-2)^2 + 1 = 2x-3\) is quadratic.
Reason (R): After simplification it becomes \(x^2 - 6x + 8 = 0\).
(a) Both A and R true, R explains A
(b) Both A and R true, R does not explain A
(c) A true, R false
(d) A false, R true
8. Assertion (A): \(b^2 - 4ac > 0\) implies two distinct real roots.
Reason (R): The quadratic formula gives two different real values when discriminant is positive.
(a) Both A and R true, R explains A
(b) Both A and R true, R does not explain A
(c) A true, R false
(d) A false, R true
9. The equation \(x^2 - 55x + 750 = 0\) is obtained from:
(a) cost of toy production
(b) area of rectangular plot
(c) ages of two friends
(d) consecutive integers product
10. If \(b^2 - 4ac = 0\), the roots are:
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) imaginary roots
11. Check whether \(x^2 - 2x = (-2)(3-x)\) is a quadratic equation.
12. Represent the situation: “The product of two consecutive positive integers is 306” as a quadratic equation.
13. Find the roots of \(x^2 - 3x - 10 = 0\) by factorisation.
14. Find the discriminant of \(3x^2 - 4\sqrt{3}x + 4 = 0\) and state the nature of roots.
15. State the condition for a quadratic equation to have no real roots.
16. Write the quadratic equation whose roots are \(\frac{2}{3}\) (repeated).
17. Find the roots of \(6x^2 - x - 2 = 0\) by factorisation.
18. Find two consecutive positive integers whose sum of squares is 365.
19. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
20. Find the values of \(k\) so that \(2x^2 + kx + 3 = 0\) has two equal roots.
21. A cottage industry produces pottery articles. The cost of each article is ₹3 more than twice the number produced. Total cost on a day was ₹90. Find the number of articles and cost of each.
22. A charity trust decides to build a prayer hall of carpet area 300 m² with length one metre more than twice the breadth. Find the length and breadth of the hall. Show all steps of factorisation.
23. A train travels 480 km at uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more. Find the speed of the train. (Multi-step problem)
24. Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360. Find Rohan’s present age. Verify that the obtained ages satisfy the given condition.
25. Case: Designing a rectangular mango grove
A farmer wants to design a rectangular mango grove whose length is twice its breadth and area is 800 m².
(i) Form the quadratic equation for the breadth. (1 mark)
(ii) Find the discriminant of the equation. (1 mark)
(iii) State the nature of roots. (1 mark)
(iv) Find the length and breadth of the grove. (1 mark)
26. Case: Erecting a pole in a circular park
A pole is to be erected on the boundary of a circular park of diameter 13 m such that the difference of its distances from two diametrically opposite gates A and B is 7 m.
(i) Let BP = x. Form the quadratic equation satisfied by x. (1 mark)
(ii) Calculate the discriminant of the equation. (1 mark)
(iii) Comment on whether it is possible to erect the pole. (1 mark)
(iv) Find the distances of the pole from gates A and B. (1 mark)
1. (a) — expands to \(x^2 - 6x + 8 = 0\) (correct form)
2. (a)
3. (a) — factorisation or sum-product
4. (b) — \((-4)^2 - 4(2)(3) = -8\)
5. (a) — marbles problem (NCERT Ex 4.1)
6. (a) — breadth = 12 m (positive)
7. (a) — both true, R explains A
8. (a) — both true, R explains A
9. (a) — toy production example
10. (b) — equal real roots when discriminant zero
11. Simplifies to \(x + 12 = 0\) (linear) → not quadratic (2 marks)
12. Let integers be \(x, x+1\); \(x(x+1) = 306\) → \(x^2 - x - 306 = 0\) (2 marks)
13. \((x-5)(x+2)=0\) → \(x=5, -2\) (2 marks)
14. Discriminant = 0 → two equal real roots (2 marks)
15. \(b^2 - 4ac < 0\) (2 marks)
16. \((x - \frac{2}{3})^2 = 0\) or \(9x^2 - 12x + 4 = 0\) (2 marks)
17. \((3x-2)(2x+1)=0\) → \(x=\frac{2}{3}, -\frac{1}{2}\) (correct splitting 1, factors 1, roots 1)
18. \(n^2 + (n+1)^2 = 365\) → \(n=12,13\) (3 marks)
19. Let base = \(x\), altitude = \(x-7\); \(x^2 + (x-7)^2 = 169\) → sides 12 cm, 5 cm (3 marks)
20. Discriminant zero → \(k^2 = 24\) → \(k = \pm 2\sqrt{6}\) (3 marks)
21. \(x(2x+3)=90\) → 6 articles, ₹15 each (3 marks)
22. \(2x^2 + x - 300 = 0\) → \((x-12)(2x+25)=0\) → breadth 12 m, length 25 m (formula 1, factorisation 2, positive root selection 1, final dimensions 1)
23. Let speed = \(x\); \(\frac{480}{x} - \frac{480}{x-8} = 3\) → \(x=40\) km/h (multi-step: equation 2, simplification 2, answer 1)
24. Let Rohan’s age = \(x\); \((x+3)(x+29)=360\) → \(x=7\) years (equation 2, solution 2, verification 1)
25. (i) \(x(2x)=800\) → \(2x^2-800=0\) (ii) 6400 (iii) two distinct real (iv) breadth 20 m, length 40 m (1 each)
26. (i) \(x^2 + 7x - 60 = 0\) (ii) 289 (iii) possible (two real roots) (iv) 5 m from B, 12 m from A (1 each)
All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.