1. Solve the quadratic equation $2x^2 - 5x + 3 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=2$, $b=-5$, $c=3$. Calculate the discriminant: $\Delta = (-5)^2 - 4 \times 2 \times 3 = 25 - 24 = 1$. Find roots: $x = \frac{5 \pm 1}{4}$, so $x=\frac{6}{4} = \frac{3}{2}$ or $x=\frac{4}{4} = 1$. Answer: (a) 1, 3/2. 2. Check if $(x-2)^2 + 1 = 2x - 3$ is quadratic. Expand left: $(x-2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5$. Right side: $2x - 3$. Bring all terms to one side: $x^2 - 4x + 5 - 2x + 3 = x^2 - 6x + 8 = 0$. This is quadratic. Answer: (c) yes. 3. Identify which is not quadratic. (a) $x^2 + 3x = (-1)(1 - 3x)^2$ expands to quadratic. (b) $(x+2)^2 = 2(x+3)$ is quadratic. (c) $(x+3)(x-1) = x^2 - 2x - 3$ is quadratic. (d) $x^3 - x^2 + 2x + 1 = (x+1)^3$ involves cubic terms. Answer: (d) $x^3 - x^2 + 2x + 1 = (x+1)^3$. 4. For $2x^2 - kx + k = 0$ to have equal roots, discriminant $\Delta = 0$. Calculate: $\Delta = k^2 - 4 \times 2 \times k = k^2 - 8k = 0$. Solve: $k(k - 8) = 0$ so $k=0$ or $k=8$. Answer: (a) 0, 8. 5. Solve $(x^2 + 1)^2 - x^2 = 0$. Expand: $x^4 + 2x^2 + 1 - x^2 = x^4 + x^2 + 1 = 0$. Check discriminant for $x^2$: $b^2 - 4ac = 1^2 - 4 imes 1 imes 1 = 1 - 4 = -3 < 0$. No real roots. Answer: (a) no real root. 6. In AP, $d = -4$, $n=7$, $a_n = 4$. Use formula: $a_n = a + (n-1)d$. $4 = a + 6(-4) = a - 24$. Solve: $a = 28$. Answer: (a) 28. 7. 10th term of AP: 6, 9, 12, 15,... Common difference $d=3$. $a_{10} = a + 9d = 6 + 9 imes 3 = 6 + 27 = 33$. Answer: (a) 33. 8. $a_{19} - a_{14}$ with $d=5$. $a_{19} = a + 18d$, $a_{14} = a + 13d$. Difference: $a_{19} - a_{14} = (a + 18d) - (a + 13d) = 5d = 5 imes 5 = 25$. Answer: (c) 25. 9. Sum of first 100 natural numbers. Formula: $S_n = \frac{n(n+1)}{2} = \frac{100 imes 101}{2} = 5050$. Answer: (d) 5050. 10. Sum of first 16 terms of AP: 10, 6, 2,... $d = 6 - 10 = -4$. $S_n = \frac{n}{2}[2a + (n-1)d] = \frac{16}{2}[2 imes 10 + 15 imes (-4)] = 8[20 - 60] = 8 imes (-40) = -320$. Answer: (c) -320. 11. $E' = U - E$. $U = \{1,2,3,4,5,6,7,8,9\}$, $E = \{1,2,5,7,8\}$. $E' = \{3,4,6,9\}$. Answer: (c) \{3,4,6,9\}. 12. If $X \subseteq Y$, then $X - Y = \emptyset$. Answer: (d) $\Phi$. 13. People liking both coffee and tea. Total = 70, coffee = 37, tea = 52. Use formula: $|C \cap T| = |C| + |T| - |C \cup T| = 37 + 52 - 70 = 19$. Answer: (c) 19. 14. People speaking at least one language. French = 50, Spanish = 20, both = 10. $|F \cup S| = |F| + |S| - |F \cap S| = 50 + 20 - 10 = 60$. Answer: (d) 60. 15. If $X \cup Y = X \cap Y$, then $X = Y$. Answer: (d) $X = Y$. 16. Area of circle = 616 cm$^2$. Area formula: $\pi r^2 = 616$. $r^2 = \frac{616}{\pi} \approx \frac{616}{3.14} = 196$. $r = \sqrt{196} = 14$ cm. Answer: (b) 14 cm. 17. Perimeter and area equal. Perimeter $= 2\pi r$, area $= \pi r^2$. Set equal: $2\pi r = \pi r^2$. Divide by $\pi r$: $2 = r$. Answer: (a) 2 units. 18. Perimeter of circle = perimeter of square. Circle perimeter: $2\pi r$, square perimeter: $4a$. Set equal: $2\pi r = 4a \Rightarrow a = \frac{\pi r}{2}$. Area ratio: $\frac{\text{circle area}}{\text{square area}} = \frac{\pi r^2}{a^2} = \frac{\pi r^2}{(\frac{\pi r}{2})^2} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \frac{4}{\pi}$. Ratio $= \frac{4}{\pi} \approx \frac{4}{3.14} = 1.27$. Closest ratio: 14:11. Answer: (c) 14 : 11. 19. Area of circle inscribed in semicircle of radius $r$. Radius of inscribed circle is $\frac{r}{2}$. Area: $\pi \left(\frac{r}{2}\right)^2 = \frac{\pi r^2}{4}$. Given options suggest $r=6$, area $= 9\pi$. Answer: (a) 9π. 20. Circumference = 123.2 cm. Formula: $2\pi r = 123.2$. $r = \frac{123.2}{2\pi} = \frac{123.2}{6.28} \approx 19.6$ cm. Answer: (c) 19.6 cm. 21. Volume of cuboid = $49 \times 33 \times 24 = 38808$ cm$^3$. Volume of sphere = $\frac{4}{3} \pi r^3$. Set equal: $\frac{4}{3} \pi r^3 = 38808$. $r^3 = \frac{38808 \times 3}{4 \pi} = \frac{116424}{12.56} \approx 9270$. $r = \sqrt[3]{9270} \approx 21$ cm. Answer: (b) 21 cm. 22. Volume of cylinder = $\pi r^2 h = \pi (1)^2 (16) = 16\pi$. Volume of 12 spheres = $12 \times \frac{4}{3} \pi r^3 = 16\pi$. Solve for $r$: $16\pi = 16\pi r^3$. $r^3 = \frac{16\pi}{16\pi} = 1$. $r = 1$ cm, diameter = 2 cm. Answer: (d) 2 cm. 23. Sphere diameter enclosed in cylinder with radius $r$ and height $h$ (where $h > 2r$) is $2r$. Answer: (c) 2r cm. 24. Curved surface area of bucket = $\pi (R + r) l$. $R=28$, $r=7$, $l=45$. Calculate: $\pi (28 + 7) 45 = \pi \times 35 \times 45 = 1575\pi \approx 4950$ cm$^2$. Answer: (a) 4950 cm$^2$.