Thoroughly analyzing AP Inter 2nd Year Maths 2A Model Papers and AP Inter 2nd Year Maths 2A Question Paper March 2023 helps students identify their strengths and weaknesses.

## AP Inter 2nd Year Maths 2A Question Paper March 2023

Time: 3 Hours

Maximum Marks: 75

Note: This question paper consists of THREE Sections A, B and C.

Section – A

(10 × 2 = 20 Marks)

**I. Very Short Answer Type Questions.**

- Answer ALL questions.
- Each question carries TWO marks.

Question 1.

Find the square root of the complex number 7 + 24i.

Question 2.

If z_{1} = -1 and z_{2} = i, then find Arg(\(\frac{z_1}{z_2}\)).

Question 3.

If 1, ω, ω^{2} are the cube roots of unity, then find the value of (1 – ω + ω^{2})^{5} + (1 + ω – ω^{2})^{5}.

Question 4.

Form a quadratic equation whose roots are -3 ± 5i.

Question 5.

If the product of the roots of 4x^{3} + 16x^{2} – 9x – a = 0 is 9, then find a.

Question 6.

Find the number of ways of preparing a chain with 6 different coloured beads.

Question 7.

If ^{n}C_{5} = ^{n}C_{6}, then find ^{13}C_{n}.

Question 8.

Find the middle term in the expansion of \(\left(\frac{3 x}{7}-2 y\right)^{10}\).

Question 9.

Find the mean deviation about the median for the following data:

4, 6, 9, 3, 10, 13, 2.

Question 10.

A Poisson variable satisfies P(X = 1) = P(X = 2). Find P(X = 5).

Section – B

(5 × 4 = 20 Marks)

**II. Short Answer Type Questions.**

- Answer ANY FIVE questions.
- Each question carries FOUR marks.

Question 11.

If x + iy = \(\frac{1}{1+\cos \theta+i \sin \theta}\), then show that 4x^{2} – 1 = 0.

Question 12.

Prove that \(\frac{1}{3 x+1}+\frac{1}{x+1}-\frac{1}{(3 x+1)(x+1)}\) does not lie between 1 and 4, if x is real.

Question 13.

If the 6 letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the rank of the word PRISON.

Question 14.

Prove that \(\frac{{ }^{4 n} C_{2 n}}{{ }^{2 n} C_n}=\frac{1 \cdot 3.5 \ldots(4 n-1)}{\left\{1 \cdot 3 \cdot 5 \ldots(2 n-1)^2\right.}\).

Question 15.

Resolve the following fraction into partial fractions \(\frac{x^2-3}{(x+2)\left(x^2+1\right)}\).

Question 16.

Find the probability that a non-leap year contains

(i) 53 Sundays

(ii) 52 Sundays only

Question 17.

A problem in calculus is given to two students A and B whose chances of solving it are \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively. Find the probability of the problem being solved if both of them try independently.

Section – C

(5 × 7 = 35 Marks)

**III. Long Answer Type Questions.**

- Answer ANY FIVE questions.
- Each question carries SEVEN marks.

Question 18.

If α, β are the roots of the equation x^{2} – 2x + 4 = 0, then for any n ∈ N show that \(\alpha^n+\beta^n=2^{n+1} \cos \left(\frac{n \pi}{3}\right)\)

Question 19.

Solve the equation x^{5} – 5x^{4} + 9x^{3} – 9x^{2} + 5x – 1 = 0.

Question 20.

If n is a positive integer and x is any non zero real number, then prove that \(\mathrm{C}_0+\mathrm{C}_1 \cdot \frac{x}{2}+\mathrm{C}_2 \cdot \frac{x^2}{3}+\mathrm{C}_3 \cdot \frac{x^3}{4}+\ldots+\mathrm{C}_{\mathrm{n}} \cdot \frac{x^{\mathrm{n}}}{\mathrm{n}+1}=\frac{(1+x)^{\mathrm{n}+1}-1}{(\mathrm{n}+1) x}\).

Question 21.

If t = \(\frac{4}{5}+\frac{4.6}{5.10}+\frac{4.6 .8}{5.10 .15}\) + ………. ∞, then prove that 9t = 16.

Question 22.

Find the variance and standard deviation of the following frequency distribution.

X_{i} |
4 | 8 | 11 | 17 | 20 | 24 | 32 |

f_{i} |
3 | 5 | 9 | 5 | 4 | 3 | 1 |

Question 23.

State and Prove Baye’s theorem on Probability.

Question 24.

A random variable X has the following probability distribution:

X = x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

P(X = x) | 0 | k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2} + k |

Find: (i) k, (ii) The mean, and (iii) P (0 < X < 5).