Answer
**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)
Solution
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.**
---
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