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

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Sure, let's go through each of the questions you've provided and identify the correct answers along with brief explanations.

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### **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!
**Simplified Answers:** 13. **Force on the body:** 98 N 14. **Minimum value of $$ f(x) $$:** -21 15. **$$ \Delta y $$:** 0.20 16. **Sum to infinity of the G.P.:** 5 1⁄5 17. **Magnitude $$ |2\mathbf{s} - \mathbf{r}| $$:** 5.0 18. **Area of the circle:** 9π 19. **Fourth term in the binomial expansion:** -540x³y³ 20. **Value of $$ x $$ at the turning point:** -1 1⁄2 21. **Number of ways to seat 12 people on a bench:** 95,040 22. **Values of $$ K $$:** 6 and -6 (None of the options are correct)

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