Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers and AP Inter 1st Year Maths 1A Question Paper April 2022 helps students identify their strengths and weaknesses.

## AP Inter 1st Year Maths 1A Question Paper April 2022

Time: 3 Hours

Maximum Marks: 60

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 ANY TEN of the following questions.
- Each Question carries TWO marks.

Question 1.

If f: R\{0} → R is defined byf(x) = x^{3} – \(\frac{1}{x^3}\), then show that f(x) + f(\(\frac{1}{x}\)) = 0

Question 2.

Determine whether the following function is even or odd:

f(x) = a^{x} – a^{-x} + sin x

Question 3.

If A = \(\left[\begin{array}{ll}

i & 0 \\

0 & i

\end{array}\right]\), find A^{2}.

Question 4.

Find the rank of the following matrix:

\(\left[\begin{array}{ccc}

1 & 4 & -1 \\

2 & 3 & 0 \\

0 & 1 & 2

\end{array}\right]\)

Question 5.

\(\bar{a}=2 \bar{i}+5 \bar{j}+\bar{k}\) and \(\overline{\mathrm{b}}=4 i+\mathrm{m} \bar{j}+\mathrm{n}\) are collinear vectors, then find m and n.

Question 6.

Find the vector equation of the line passing through the point \(2 \bar{i}+3 \bar{j}+k\) and parallel to the vector \(4 \bar{i}-2 \bar{j}+3 \bar{k}\).

Question 7.

For what values of λ, the vectors \(\bar{i}-\lambda \bar{j}+2 \bar{k}\) and \(8 i-6 \bar{j}-\vec{k}\) are at right angles?

Question 8.

Find a sine function whose period is \(\frac{2}{3}\).

Question 9.

Express \(\frac{\left(\sqrt{3} \cos 25^{\circ}+\sin 25^{\circ}\right)}{2}\) as a sine of an angle.

Question 10.

If sinh x = 3, then show that x = log_{e}(3 + √10).

Section – B

(5 × 4 = 20 Marks)

**II. Short Answer Type Questions.**

- Answer any FIVE questions.
- Each Question carries FOUR marks.

Question 11.

If A = \(\left[\begin{array}{cc}

-1 & 2 \\

0 & 1

\end{array}\right]\) then find AA’. Do A and A’ commute concerning multiplication of matrices.

Question 12.

\(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors. Prove that the following four points are coplanar:

-a + 4b – 3c, 3a + 2b – 5c, -3a + 8b – 5c, -3a + 2b + c

Question 13.

Find a vector of magnitude 3 and perpendicular to both the vectors \(\mathrm{b}=2 \bar{i}-2 \bar{j}+\bar{k}\) and \(\bar{c}=2 \bar{i}+2 \bar{j}+3 \bar{k}\).

Question 14.

If 3 sin θ + 4 cos θ = 5, then find the value of 4 sin θ – 3 cos θ.

Question 15.

Show that \(\cos ^2\left(\frac{\pi}{10}\right)+\cos ^2\left(\frac{2 \pi}{5}\right)+\cos ^2\left(\frac{3 \pi}{5}\right)+\cos ^2\left(\frac{9 \pi}{10}\right)=2\).

Question 16.

Show that, for any θ ∈ R, 4 sin \(\frac{5 \theta}{2}\) cos \(\frac{3 \theta}{2}\) cos 3θ = sin θ – sin 2θ + sin 4θ + sin 7θ.

Question 17.

Prove that \(\tan \frac{A}{2}+\tan \frac{B}{2}+\tan \frac{C}{2}=\frac{b c+c a+a b-s^2}{\Delta}\)

Section – C

(5 × 7 = 35 Marks)

**III. Long Answer Type Questions.**

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

Question 18.

If f(x) = x^{2} and g(x) = |x|, find the following functions.

(i) f + g

(ii) f – g

(iii) fg

(iv) 2f

(v) f^{2}

(vi) f + 3

Question 19.

Solve 3x + 4y + 5z = 18, 2x – y + 8z = 13 and 5x – 2y + 7z = 20 by using ‘Matrix Inversion Method’.

Question 20.

Solve x – y + 3z = 5, 4x + 2y – z = 0, -x + 3y + z = 5 by using Cramer’s Rule.

Question 21.

Show that the line joining the pair of points \(6 \bar{a}-4 \bar{b}+4 \bar{c},-4 \bar{c}\) and the line joining the pair of points \(-a-2 b-3 c,-a+2 b-5 c\) intersect at the point \(-4 \bar{c}\) when \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are non-coplanar vectors.

Question 22.

If \(\bar{a}=7 \bar{i}-2 \bar{j}+3 \bar{k}, \quad b=2 i+8 k\) and \(c=i+j+k\), then compute ab . ac and a(b + c). Verify whether the cross product is distributive over vector addition.

Question 23.

If A, B, C are angles of a triangle, then prove that:

\(\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}-\sin ^2 \frac{C}{2}=1-2 \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{C}{2}\)

Question 24.

Show that: r + r_{3} + r_{1} – r_{2} = 4R cos B