1. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.
(a) -8 (b) -1 (c) 3 (d) 8
2. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).
(a) 1 (b) 2 (c) 3 (d) 4
3. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).
4. In the polynomial \(2x^3 - 3x^2 + 4x + 5 = 0\), the coefficient of \(x^0\) is—?
(a) 2 (b) 3 (c) 4 (d) 5
5. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).
6. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).
7. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).
8. The graph of the equation \(2x + 3 = 0\) is:
(a) Parallel to the x-axis. (b) Parallel to the y-axis. (c) Not parallel to any axis. (d) Passing through the origin.
9. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants and \(a \ne 0\), \(b \ne 0\)) is:
10. The graph of the equation \(2x + 3y = 0\) is:
(a) Parallel to the x-axis. (b) Parallel to the y-axis. (c) Passing through the origin. (d) Passing through the point (2, 0).
11. The graph of the equation \(cx + d = 0\) (where \(c\) and \(d\) are constants, and \(c \ne 0\)) will be the equation of the y-axis when:
(a) d =-c (b) d =c (c) d =0 (d) d =1
12. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants, and \(a \ne 0\)) will be the equation of the x-axis when:
(a) b = a (b) b = -a (c) b = 2 (d) b = 0
13. The solution of the equations \(4x + 3y = 25\) and \(5x - 2y = 14\) is:
(a) x = 4, y = 3 (b) x = 3, y = 4 (c) x = 3, y = 3 (d) x = 4, y =-3
14. If the roots of the quadratic equation \(5x^2+13x+k=0\) are reciprocals of each other, then the value of \(k\) is:
(a) 3 (b) 4 (c) 5 (d) -5
15. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).
16. The condition under which the roots of the equation \(ax^2 + bx + c = 0\) are reciprocals of each other is:
(a) \(a=c\) (b) \(b^2=4ac\) (c) \(c=0\) (d) \(b^2-4ab\gt 0\)
17. In the equation \(x^2+4x+3=(x+3)\), the coefficient of \(x°\) is ____.
18. The condition for the quadratic nature of the equation \(ax^2+bx+c=0\) is: \(a \neq 0\).
(a) \(a=0\) (b) \(a\ne 0\) (c) \(a=1\) (d) none of the above
19. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters. The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\). Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters. From \( \triangle \)ABO, we get: \(\cfrac{AB}{BO} = \tan \theta\) Or, \(\cfrac{x}{60} = \tan\theta\) --------(i) From \( \triangle \)COD, we get: \(\cfrac{CD}{OD} = \tan(90^o - \theta)\) Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii) Multiplying equations (i) and (ii), we get: \(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\) Or, \(\cfrac{2x^2}{60 \times 60} = 1\) Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\) Or, \(x = 30\sqrt2\) \(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters. And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters. (Proved).
20. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
21. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
22. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).
23. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).
24. \(\cfrac{x}{4 - x} = \cfrac{1}{3x},\ (x ≠ 0,\ x ≠ 4)\) — Let us express this in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and determine the coefficient of \(x\).
25. The length of a rectangular field is 36 meters more than its breadth. The area of the field is 460 square meters. Form a quadratic equation in one variable from this statement and determine the coefficients of \(x^2\), \(x\), and \(x^0\).
26. Express the equation \(x^3 - 4x^2 - x + 1 = (x + 2)^3\) in the standard form of a quadratic equation and write the coefficients of \(x^2\), \(x\), and \(x^0\).
27. Let us check whether the given values of \(x\), namely \(x = 10\) and \(x = -7\), satisfy the equation \(x^2 - 3x - 70 = 0\).
28. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).
29. Translate: The condition for the roots of the quadratic equation \[ ax^2 + bx - c = 0 \quad (a \ne 0) \] to be equal is:
30. Prove that:The squares of the roots of the equation \(x^2 + x + 1 = 0\) are also roots of the same equation \(x^2 + x + 1 = 0\). Would you like me to walk through the proof again in English, or are you asking for a translated version of your original Bengali explanation? I’ve got both options ready.
