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

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) = 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) = ax – 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 = loge(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