1. Problem: Solve for $m$ in the equation $42m - 1 = 64$. Step 1: Add 1 to both sides: $42m = 65$. Step 2: Divide both sides by 42: $m = \frac{65}{42} \approx 1.55$. Answer choices do not include this value; likely a typo in options. 2. Problem: Find $a$ if $\sqrt{32} = 2$. Step 1: $\sqrt{32} = 4\sqrt{2} \approx 5.66$, not 2. Step 2: Possibly a typo; if $\sqrt{a} = 2$, then $a = 4$. Answer: (b) 4. 3. Problem: Solve $4m m = 1$ (assuming $4m \times m = 1$ or $4m^2=1$). Step 1: $4m^2 = 1$. Step 2: Divide both sides by 4: $m^2 = \frac{1}{4}$. Step 3: Take square root: $m = \pm \frac{1}{2}$. Answer choices: closest is (a) -1, (b) 0, (c) 1, (d) 4; none match exactly. 4. Problem: Evaluate statements about exponents: (i) $a^0 = 1$ (true for $a>0$), (ii) $x^0 = y$ when $ax = ax$ (ambiguous, likely false), (iii) $a^n = \frac{1}{a^n}$ (false, should be $a^{-n} = \frac{1}{a^n}$). Correct: (a) i and ii (if ii is true), (b) i and iii (iii false), (c) ii and iii (both false), (d) i, ii and iii (false). Answer: (a) i and ii. 5. Problem: Evaluate logarithm properties: (i) $\log_a(MN) = \log_a M + \log_a N$ (true), (ii) $\log_a 1 = 0$ when $a \neq 1$ (true), (iii) If $a^x = N$ then $x = \log_a N$ (true). Answer: (d) i, ii and iii. 6. Given $\log x = 3$, $\log y = 2$, find $\log_a (xy)$. Step 1: $\log_a (xy) = \log_a x + \log_a y = 3 + 2 = 5$. Answer: (c) 5. 7. Question incomplete; assuming find $\log_a (\frac{x}{y})$. Step 1: $\log_a (\frac{x}{y}) = \log_a x - \log_a y = 3 - 2 = 1$. Answer: (a) 1. 8. Problem: Distance from center to point on circle is called? Answer: (c) Radius. 9. In circle with center O, AB is chord, OM perpendicular to AB. (i) $AM = BM$ (true, perpendicular bisector), (ii) $\angle OMB = \angle OMA$ (true, right triangles), (iii) $OM = \frac{1}{2} AB$ (false). Answer: (a) i and ii. 10. If two chords bisect each other, intersection point is? Answer: (c) Inside the circle. 11. Angle subtended by diameter of circle is? Answer: (b) Right angle. 12. Angle inscribed in tangential arc of circle? Answer: (c) Right angle. 13. Max common tangents between two circles? Answer: (d) 4. 14. Two circles with radii 10 cm and 5 cm touch each other. Distance between centers = sum of radii = 15 cm. Options do not include 15 cm; possibly error. 15. Given angle between OA and OB is 35°, find $\angle AOB$. Answer: (b) 145° (assuming supplementary angle). 16. Given $PO=13$ cm, $OB=5$ cm, find $PA$. Using Pythagoras: $PA = \sqrt{PO^2 - OB^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm. Answer: (b) 12 cm. 17. Ratio 3:5, common denominator 4, find LCM. LCM of 3 and 5 is 15. Answer: (b) 15. 18. Given $x:y=7:5$, $y:z=5:7$, find $x:z$. Step 1: $x:z = 7:7 = 1:1$. Answer: (b) 7:7. 19. Problem unclear; insufficient data. 20. Side length reduced by 10%, new side = 0.9 original. Area reduced by $1 - 0.9^2 = 1 - 0.81 = 0.19 = 19\%$. Answer: (b) 19%. 21. Ratio of area of circle to inscribed square. Area circle = $\pi r^2$, area square = $(\sqrt{2}r)^2 = 2r^2$. Ratio = $\frac{\pi r^2}{2 r^2} = \frac{\pi}{2}$. Answer: (b) $\pi:2$. 22. Mangoes divided in ratio $1:1: \frac{1}{2} 5 9$ unclear; assuming ratio $1:1:\frac{1}{2}$. Sum parts = $1 + 1 + 0.5 = 2.5$. Mangoes = 146. First brother gets $\frac{1}{2.5} \times 146 = 58.4$ approx. Closest answer: (c) 45. 23. Proportion statements: (i) $a:b = b:a$ then $a=b$ (false), (ii) $a:b = c:d$ then $ad=bc$ (true), (iii) $a:b=5:3$ then $a:5 = b:3$ (true). Answer: (c) ii and iii. 24. Range 110, class number 10, class gap = $\frac{110}{10} = 11$. Answer: (b) 11. 25. Serial number required in: (i) median (true), (ii) drawing straight line (false), (iii) determining abundance (true). Answer: (b) i and iii.