EMCET Model Question paper & Answers 3

 

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26. Algebra

Q26: Solve the equation $x^2 + 6x + 9 = 0$.

Answer: The equation $x^2 + 6x + 9 = 0$ is a perfect square:

$$(x + 3)^2 = 0$$

Thus, $x + 3 = 0$, so $x = -3$.

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27. Trigonometry

Q27: Find the value of $\sin 60^\circ$.

Answer:

$$\sin 60^\circ = \frac{\sqrt{3}}{2}$$

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28. Coordinate Geometry

Q28: Find the equation of the line passing through the point $P(2, 3)$ and having slope $m = 4$.

Answer: The equation of a line in point-slope form is:

$$y - y_1 = m(x - x_1)$$

Substitute $m = 4$, $x_1 = 2$, and $y_1 = 3$:

$$y - 3 = 4(x - 2)$$

Simplify:

$$y - 3 = 4x - 8$$ $$y = 4x - 5$$

Thus, the equation of the line is $y = 4x - 5$.

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29. Calculus

Q29: Find the derivative of $f(x) = 5x^3 - 2x + 7$.

Answer: Using the power rule for differentiation:

$$f'(x) = 15x^2 - 2$$

Thus, the derivative is $15x^2 - 2$.

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30. Probability

Q30: A coin is flipped twice. What is the probability of getting at least one tail?

Answer: The possible outcomes when a coin is flipped twice are:

$$\{HH, HT, TH, TT\}$$

The favorable outcomes for getting at least one tail are:

$$\{HT, TH, TT\}$$

Thus, the probability is:

$$P(\text{at least one tail}) = \frac{3}{4}$$

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31. Quadratic Equations

Q31: Find the roots of the quadratic equation $x^2 - 6x + 8 = 0$.

Answer: The quadratic equation is $x^2 - 6x + 8 = 0$.

We factor the equation:

$$x^2 - 6x + 8 = (x - 2)(x - 4) = 0$$

Thus, the roots are:

$$x = 2 \quad \text{or} \quad x = 4$$

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32. Set Theory

Q32: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, find $A \cap B$ (the intersection of $A$ and $B$).

Answer: The intersection of sets $A$ and $B$ is the set of elements that are in both sets.

$$A \cap B = \{2, 3\}$$

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33. Matrices

Q33: Find the determinant of the matrix $\begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$.

Answer: The determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is:

$$\text{det} = ad - bc$$

For the matrix $\begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$:

$$\text{det} = (3)(2) - (1)(4) = 6 - 4 = 2$$

Thus, the determinant is 2.

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34. Progressions

Q34: Find the 7th term of an arithmetic progression where the first term $a = 5$ and the common difference $d = 2$.

Answer: The $n$-th term of an arithmetic progression is given by:

$$T_n = a + (n - 1)d$$

Substitute $a = 5$, $d = 2$, and $n = 7$:

$$T_7 = 5 + (7 - 1) \times 2 = 5 + 12 = 17$$

Thus, the 7th term is 17.

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35. Logarithms

Q35: Solve $\log_5 x = 3$.

Answer: Rewriting the logarithmic equation as an exponential equation:

$$x = 5^3 = 125$$

Thus, $x = 125$.

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36. Limits

Q36: Find the limit $\lim_{x \to 0} \frac{\sin 2x}{x}$.

Answer: Using the standard limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$, we rewrite the expression:

$$\lim_{x \to 0} \frac{\sin 2x}{x} = \lim_{x \to 0} \left( 2 \cdot \frac{\sin 2x}{2x} \right)$$

Thus, the limit is:

$$\lim_{x \to 0} \frac{\sin 2x}{x} = 2 \cdot 1 = 2$$

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37. Matrices

Q37: Find the inverse of the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Answer: The inverse of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by:

$$\text{Inverse} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

For $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, the determinant is:

$$\text{det} = (1)(4) - (2)(3) = 4 - 6 = -2$$

Thus, the inverse is:

$$\text{Inverse} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$$

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38. Complex Numbers

Q38: Find the conjugate of the complex number $z = 2 + 3i$.

Answer: The conjugate of $z = 2 + 3i$ is:

$$\overline{z} = 2 - 3i$$

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39. Trigonometry

Q39: Find $\cot 45^\circ$.

Answer:

$$\cot 45^\circ = 1$$

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40. Calculus

Q40: Find the integral of $f(x) = 6x^2 - 4x + 3$.

Answer: Using the power rule for integration:

$$\int (6x^2 - 4x + 3) \, dx = 2x^3 - 2x^2 + 3x + C$$

Thus, the integral is $2x^3 - 2x^2 + 3x + C$, where $C$ is the constant of integration.