EMCET Model Question paper & Answers 2

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12. Algebra

Q12: Solve the equation $3x^2 - 12x + 9 = 0$.

Answer: First, divide the equation by 3 to simplify:

$$x^2 - 4x + 3 = 0$$

Now factor the quadratic equation:

$$x^2 - 4x + 3 = (x - 1)(x - 3) = 0$$

Thus, the solutions are:

$$x = 1 \quad \text{or} \quad x = 3$$

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13. Trigonometry

Q13: Find the value of $\cos 45^\circ$.

Answer:

$$\cos 45^\circ = \frac{\sqrt{2}}{2}$$

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14. Coordinate Geometry

Q14: Find the midpoint of the line segment joining the points $P(1, 2)$ and $Q(3, 4)$.

Answer: The formula for the midpoint $M$ of a line segment joining two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Substitute the values of $P(1, 2)$ and $Q(3, 4)$:

$$M = \left(\frac{1 + 3}{2}, \frac{2 + 4}{2}\right) = \left(2, 3\right)$$

Thus, the midpoint is $M(2, 3)$.

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15. Calculus

Q15: Find the integral of $f(x) = 4x^3$.

Answer: The integral of $f(x) = 4x^3$ with respect to $x$ is:

$$\int 4x^3 \, dx = \frac{4x^4}{4} + C = x^4 + C$$

Thus, the integral is $x^4 + C$, where $C$ is the constant of integration.

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16. Probability

Q16: A card is drawn from a standard deck of 52 cards. What is the probability that the card is a queen?

Answer: In a deck of 52 cards, there are 4 queens (one for each suit).

Thus, the probability of drawing a queen is:

$$P(\text{queen}) = \frac{4}{52} = \frac{1}{13}$$

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17. Quadratic Equations

Q17: Find the discriminant of the quadratic equation $x^2 + 4x + 3 = 0$.

Answer: The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$$\Delta = b^2 - 4ac$$

For the equation $x^2 + 4x + 3 = 0$, $a = 1$, $b = 4$, and $c = 3$:

$$\Delta = 4^2 - 4(1)(3) = 16 - 12 = 4$$

Thus, the discriminant is 4.

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18. Progressions

Q18: Find the 5th term of an arithmetic progression where the first term $a = 7$ and the common difference $d = 3$.

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

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

Substitute the given values for $a$, $d$, and $n = 5$:

$$T_5 = 7 + (5-1) \times 3 = 7 + 12 = 19$$

Thus, the 5th term is 19.

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19. Logarithms

Q19: Solve $\log_3 x = 4$.

Answer: To solve for $x$, rewrite the logarithmic equation as an exponential equation:

$$x = 3^4 = 81$$

Thus, $x = 81$.

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20. Limits

Q20: Find the limit $\lim_{x \to 0} \frac{e^x - 1}{x}$.

Answer: This is a standard limit. The value of the limit is:

$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$

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21. Matrices

Q21: 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 $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $a = 1$, $b = 2$, $c = 3$, and $d = 4$. First, calculate the determinant:

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

Now, calculate the inverse:

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

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22. Complex Numbers

Q22: Find the modulus of the complex number $z = 3 + 4i$.

Answer: The modulus of a complex number $z = a + bi$ is given by:

$$|z| = \sqrt{a^2 + b^2}$$

For $z = 3 + 4i$, $a = 3$ and $b = 4$:

$$|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Thus, the modulus is 5.

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23. Trigonometry

Q23: Find the value of $\tan 30^\circ$.

Answer:

$$\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$$

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24. Calculus

Q24: Find the second derivative of $f(x) = 5x^4 - 2x^2 + 3$.

Answer: The first derivative is:

$$f'(x) = 20x^3 - 4x$$

The second derivative is:

$$f''(x) = 60x^2 - 4$$

Thus, the second derivative is $60x^2 - 4$.

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25. Integration

Q25: Find the integral $\int (3x^2 + 5x) \, dx$.

Answer: Using the power rule for integration:

$$\int 3x^2 \, dx = x^3$$ $$\int 5x \, dx = \frac{5x^2}{2}$$

Thus, the integral is:

$$x^3 + \frac{5x^2}{2} + C$$

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These 25 questions and answers give you an idea of what to expect from different topics in EAMCET Mathematics. Let me know if you'd like more questions or explanations on specific topics!