13. A body of mass 42 kg increases its speed from \( 15 \mathrm{~m} / \mathrm{s} \) to \( 43 \mathrm{~m} / \mathrm{s} \) in 12 seconds. Find the force acting c the body. (A) 52 N (B) 98 N (C) 150 N (D) 203 N 14. Find the minimum value of \( =x^{2}+6 x-12 \). (A) -12 (B) -6 (C) -3 (D) -21 15. Given that \( y=2 x-1 \) and \( \Delta x=0.1 \), find \( \Delta y \). (A) 0.20 (B) 0.15 (C) 0.10 (D) 0.05 16. An exponential sequence (G.P.) is given by \( \frac{9}{2}, \frac{3}{4}, \frac{1}{8}, \ldots \ldots \). Find its sum to infinity. (A) \( 13 \frac{1}{2} \) (B) \( 4 \frac{1}{5} \) (C) \( € \) (D) \( 5 \frac{2}{5} \) 17. If \( r=i+2 j \) and \( =-I+3 j \), find \( |2 n-r| \). (A) 8.5 (B) 5.0 (C) 4.0 (D) 3.6 18. Find the area of a circle whose equation is \( { }^{2}+y^{2}-4 \cdot x+8 y+11=0 \). (A) \( 3 \pi \) (B) \( 9 \pi \) (C) \( \epsilon \) 19. Find the fourth term in the binomial expansion of \( (3 x-y)^{6} \). (A) \( -540 x^{3} y^{3} \) (B) \( -27 x^{3} \) (D) \( 12 \pi \) (C) \( 540 x^{4} y^{2} \) (D) \( -540 x^{4} y^{2} \) 20. Given that \( f(x)=x^{2}+3 x+1 \), find the value of \( x \) at the turning point. (A) 2 (B) \( 1 \frac{1}{2}(C \) (C) \( -1 \frac{1}{2} \) 21. In how many ways can 12 people be seated on a bench if only 5 spaces are available? (A) 950 2 (B) 11880 (C) 792 (D) 495 22. If \( K x^{2}-12 x+K=0 \) has equal roots, find the values of \( K \). \( (A)+2(B)+3 \) \( (C)+.6 \) (D) +9

Sure, let's go through each of the questions you've provided and identify the correct answers along with brief explanations. --- ### **Question 13** **A body of mass 42 kg increases its speed from \( 15 \mathrm{~m/s} \) to \( 43 \mathrm{~m/s} \) in 12 seconds. Find the force acting on the body.** **Options:** - (A) 52 N - (B) 98 N - (C) 150 N - (D) 203 N **Solution:** - **Acceleration (a)** = \( \frac{\Delta v}{\Delta t} = \frac{43 - 15}{12} = \frac{28}{12} = \frac{7}{3} \approx 2.333 \, \mathrm{m/s^2} \) - **Force (F)** = mass (m) × acceleration (a) = \( 42 \times \frac{7}{3} = 42 \times 2.333 \approx 98 \, \mathrm{N} \) **Answer:** **(B) 98 N** --- ### **Question 14** **Find the minimum value of \( f(x) = x^{2} + 6x - 12 \).** **Options:** - (A) -12 - (B) -6 - (C) -3 - (D) -21 **Solution:** - The function is a quadratic equation of the form \( ax^2 + bx + c \). - The vertex (minimum point) occurs at \( x = -\frac{b}{2a} = -\frac{6}{2 \times 1} = -3 \). - Minimum value \( f(-3) = (-3)^2 + 6(-3) - 12 = 9 - 18 - 12 = -21 \). **Answer:** **(D) -21** --- ### **Question 15** **Given that \( y = 2x - 1 \) and \( \Delta x = 0.1 \), find \( \Delta y \).** **Options:** - (A) 0.20 - (B) 0.15 - (C) 0.10 - (D) 0.05 **Solution:** - \( \Delta y = 2 \times \Delta x = 2 \times 0.1 = 0.2 \) **Answer:** **(A) 0.20** --- ### **Question 16** **An exponential sequence (G.P.) is given by \( \frac{9}{2}, \frac{3}{4}, \frac{1}{8}, \ldots \). Find its sum to infinity.** **Options:** - (A) \( 13 \frac{1}{2} \) - (B) \( 4 \frac{1}{5} \) - (C) \( € \) - (D) \( 5 \frac{2}{5} \) **Solution:** - **First term (a)** = \( \frac{9}{2} \) - **Common ratio (r)** = \( \frac{3/4}{9/2} = \frac{3}{4} \times \frac{2}{9} = \frac{1}{6} \) - **Sum to infinity (S)** = \( \frac{a}{1 - r} = \frac{9/2}{1 - 1/6} = \frac{9/2}{5/6} = \frac{9 \times 6}{2 \times 5} = \frac{54}{10} = 5.4 = 5 \frac{2}{5} \) **Answer:** **(D) \( 5 \frac{2}{5} \)** --- ### **Question 17** **If \( \mathbf{r} = \mathbf{i} + 2\mathbf{j} \) and \( \mathbf{s} = -\mathbf{i} + 3\mathbf{j} \), find \( |2\mathbf{s} - \mathbf{r}| \).** **Options:** - (A) 8.5 - (B) 5.0 - (C) 4.0 - (D) 3.6 **Solution:** - Calculate \( 2\mathbf{s} - \mathbf{r} \): \[ 2(-\mathbf{i} + 3\mathbf{j}) - (\mathbf{i} + 2\mathbf{j}) = -2\mathbf{i} + 6\mathbf{j} - \mathbf{i} - 2\mathbf{j} = -3\mathbf{i} + 4\mathbf{j} \] - Magnitude: \[ \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0 \] **Answer:** **(B) 5.0** --- ### **Question 18** **Find the area of a circle whose equation is \( x^{2} + y^{2} - 4x + 8y + 11 = 0 \).** **Options:** - (A) \( 3\pi \) - (B) \( 9\pi \) - (C) \( \epsilon \) - (D) Not listed correctly **Solution:** - Rewrite the equation in standard form by completing the squares: \[ x^2 - 4x + y^2 + 8y = -11 \] \[ (x^2 - 4x + 4) + (y^2 + 8y + 16) = -11 + 4 + 16 \] \[ (x - 2)^2 + (y + 4)^2 = 9 \] - **Radius (r)** = √9 = 3 - **Area** = \( \pi r^2 = \pi \times 9 = 9\pi \) **Answer:** **(B) \( 9\pi \)** --- ### **Question 19** **Find the fourth term in the binomial expansion of \( (3x - y)^{6} \).** **Options:** - (A) \( -540x^{3}y^{3} \) - (B) \( -27x^{3} \) - (C) \( 540x^{4}y^{2} \) - (D) \( -540x^{4}y^{2} \) **Solution:** - The general term for the binomial expansion is: \[ T_{k+1} = C(6, k) \times (3x)^{6-k} \times (-y)^k \] - For the fourth term, \( k = 3 \): \[ T_4 = C(6, 3) \times (3x)^3 \times (-y)^3 = 20 \times 27x^3 \times (-y)^3 = 20 \times 27 \times (-1) x^3 y^3 = -540x^3y^3 \] **Answer:** **(A) \( -540x^{3}y^{3} \)** --- ### **Question 20** **Given that \( f(x) = x^{2} + 3x + 1 \), find the value of \( x \) at the turning point.** **Options:** - (A) 2 - (B) \( 1\frac{1}{2} \) - (C) \( -1\frac{1}{2} \) - (D) Not listed correctly **Solution:** - The x-coordinate of the turning point for a quadratic \( ax^2 + bx + c \) is \( -\frac{b}{2a} \): \[ x = -\frac{3}{2 \times 1} = -\frac{3}{2} = -1.5 \] **Answer:** **(C) \( -1\frac{1}{2} \)** --- ### **Question 21** **In how many ways can 12 people be seated on a bench if only 5 spaces are available?** **Options:** - (A) 95,040 - (B) 11,880 - (C) 792 - (D) 495 **Solution:** - This is a permutation problem where we select and arrange 5 people out of 12: \[ P(12,5) = 12 \times 11 \times 10 \times 9 \times 8 = 95,040 \] **Answer:** **(A) 95,040** --- ### **Question 22** **If \( Kx^{2} - 12x + K = 0 \) has equal roots, find the values of \( K \).** **Options:** - (A) +2 - (B) +3 - (C) +0.6 - (D) +9 **Solution:** - For the quadratic equation \( Kx^2 - 12x + K = 0 \) to have equal roots, the discriminant must be zero: \[ D = b^2 - 4ac = (-12)^2 - 4 \times K \times K = 144 - 4K^2 = 0 \] \[ 4K^2 = 144 \Rightarrow K^2 = 36 \Rightarrow K = \pm6 \] - The possible values of \( K \) are **6** and **-6**. - **Note:** None of the provided options match the correct values. **Answer:** **The values of \( K \) are \( 6 \) and \( -6 \). None of the provided options are correct.** --- If you have any further questions or need more detailed explanations for any of the problems, feel free to ask!