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

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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) = x³ – \(\frac{1}{x^3}\), 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 A².

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) = x² and g(x) = |x|, find the following functions.
(i) f + g
(ii) f – g
(iii) fg
(iv) 2f
(v) f²
(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 + r₃ + r₁ – r₂ = 4R cos B