EMCET Model Question paper & Answers

1. Algebra

Q1: If $x + 2y = 6$ and $2x - y = 7$, find the values of $x$ and $y$.

Answer: Solve the system of equations:

$$x + 2y = 6 \quad \text{(1)}$$ $$2x - y = 7 \quad \text{(2)}$$

Multiply equation (1) by 2:

$$2x + 4y = 12 \quad \text{(3)}$$

Now subtract equation (2) from equation (3):

$$(2x + 4y) - (2x - y) = 12 - 7$$ $$5y = 5$$ $$y = 1$$

Substitute $y = 1$ into equation (1):

$$x + 2(1) = 6$$ $$x = 4$$

Thus, $x = 4$ and $y = 1$.

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2. Trigonometry

Q2: Find the value of $\sin 30^\circ$.

Answer:

$$\sin 30^\circ = \frac{1}{2}$$

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3. Coordinate Geometry

Q3: Find the distance between the points $A(3, 4)$ and $B(6, 8)$.

Answer: Using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Substitute the values of the coordinates:

$$d = \sqrt{(6 - 3)^2 + (8 - 4)^2}$$ $$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Thus, the distance is 5 units.

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4. Calculus

Q4: Find the derivative of $f(x) = 3x^2 - 5x + 2$.

Answer: Using the power rule for differentiation:

$$f'(x) = 6x - 5$$

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5. Probability

Q5: If a die is rolled, what is the probability of getting a number greater than 4?

Answer: The numbers greater than 4 on a die are 5 and 6, so there are 2 favorable outcomes.

The total number of possible outcomes on a die is 6.

Thus, the probability is:

$$P(\text{greater than 4}) = \frac{2}{6} = \frac{1}{3}$$

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6. Quadratic Equations

Q6: Solve the quadratic equation $x^2 - 4x - 5 = 0$.

Answer: The equation is in the form $ax^2 + bx + c = 0$. Using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Here, $a = 1$, $b = -4$, and $c = -5$:

$$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-5)}}{2(1)}$$ $$x = \frac{4 \pm \sqrt{16 + 20}}{2}$$ $$x = \frac{4 \pm \sqrt{36}}{2}$$ $$x = \frac{4 \pm 6}{2}$$

Thus, the solutions are:

$$x = \frac{4 + 6}{2} = 5 \quad \text{or} \quad x = \frac{4 - 6}{2} = -1$$

Thus, $x = 5$ or $x = -1$.

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7. Set Theory

Q7: If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, find $A \cup B$ (the union of $A$ and $B$).

Answer: The union of sets $A$ and $B$ is the set of all elements that are in either $A$ or $B$, or in both.

$$A \cup B = \{1, 2, 3, 4, 5, 6\}$$

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8. Matrices

Q8: Find the determinant of the matrix $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$.

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

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

For the matrix $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$:

$$\text{det} = (2)(5) - (3)(4) = 10 - 12 = -2$$

Thus, the determinant is $-2$.

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9. Progressions

Q9: Find the sum of the first 10 terms of an arithmetic progression where the first term $a = 2$ and the common difference $d = 3$.

Answer: The formula for the sum of the first $n$ terms of an arithmetic progression is:

$$S_n = \frac{n}{2} [2a + (n-1)d]$$

Substituting the values:

$$S_{10} = \frac{10}{2} [2(2) + (10-1)(3)]$$ $$S_{10} = 5 [4 + 27] = 5 \times 31 = 155$$

Thus, the sum of the first 10 terms is 155.

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10. Logarithms

Q10: Solve $\log_2 32 = x$.

Answer: We know that $32 = 2^5$, so:

$$\log_2 32 = 5$$

Thus, $x = 5$.

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11. Limits

Q11: Find the limit $\lim_{x \to 0} \frac{\sin x}{x}$.

Answer: The limit of $\frac{\sin x}{x}$ as $x \to 0$ is a standard limit, and its value is:

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

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