Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers Set 3 helps students identify their strengths and weaknesses.

## AP Inter 1st Year Maths 1A Model Paper Set 3 with Solutions

Time: 3 Hours

Maximum Marks: 75

Note: The Question Paper consists of three sections A, B, and C.

Section – A

(10 × 2 = 20 Marks)

**I. Very Short Answer Questions.**

- Answer All questions.
- Each Question carries Two marks.

Question 1.

Find the domain and range of the real-valued function f(x) = \(\frac{x^2-4}{x-2}\).

Question 2.

If f : R → R, g : R → R are defined by f(x) = 3x – 2 and g(x) = x^{2} + 1, then find (g o f)(x – 1).

Question 3.

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

1 & 4 & 7 \\

2 & 5 & 8

\end{array}\right]\), B = \(\left[\begin{array}{ccc}

-3 & 4 & 0 \\

4 & -2 & -1

\end{array}\right]\), then show that (A + B)^{T} = A^{T} + B^{T}.

Question 4.

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

2 & 4 \\

-1 & k

\end{array}\right]\) and A^{2} = 0, then find the value of k.

Question 5.

Show that the vectors \(\bar{i}+\bar{j}, \bar{j}+\bar{k},-\bar{k}+\bar{i}\) are linearly dependent.

Question 6.

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

Question 7.

Find the radius of the sphere whose equation is r^{2} = \(2 \bar{r} \cdot(4 \bar{i}-2 \bar{j}+2 \bar{k})\).

Question 8.

Find the period of the function f(x) = \(2 \sin \left(\frac{\pi x}{4}\right)+3 \cos \left(\frac{\pi x}{3}\right)\).

Question 9.

If tan θ = \(\frac{b}{a}\), then prove that a cos 2θ + b sin 2θ = a.

Question 10.

If tanh x = \(\frac{1}{4}\), then prove that x = \(\frac{1}{2} \cdot \log _e\left(\frac{5}{3}\right)\).

Section – B

(5 × 4 = 20 Marks)

**II. Short Answer Questions.**

- Answer any Five questions.
- Each Question carries Four marks.

Question 11.

If θ – φ = \(\frac{\pi}{2}\), show that \(\left[\begin{array}{cc}

\cos ^2 \theta & \cos \theta \sin \theta \\

\cos \theta \sin \theta & \sin ^2 \theta

\end{array}\right]\) \(\left[\begin{array}{cc}

\cos ^2 \phi & \cos \phi \sin \phi \\

\cos \phi \sin \phi & \sin ^2 \phi

\end{array}\right]\) = 0

Question 12.

Prove that the four points \(4 \bar{i}+5 \bar{j}+\bar{k},-(\bar{j}+\bar{k}), 3 \bar{i}+9 \bar{j}+4 \bar{k}\) and \(-4 \bar{i}+4 \bar{j}+4 \bar{k}\) are coplanar.

Question 13.

Prove that \(\sin ^4 \frac{\pi}{8}+\sin ^4 \frac{3 \pi}{8}+\sin ^4 \frac{5 \pi}{8}+\sin ^4 \frac{7 \pi}{8}=\frac{3}{2}\).

Question 14.

Solve cot^{2}x – (√3 + 1) cot x + √3 = 0 in 0 < x < \(\frac{\pi}{2}\).

Question 15.

If \(\cos ^{-1}\left(\frac{p}{a}\right)+\cos ^{-1}\left(\frac{q}{b}\right)=\alpha\), then prove that \(\frac{p^2}{a^2}-2 \frac{p q}{a b} \cos \alpha\) + \(\frac{q^2}{b^2}=\sin ^2 \alpha\).

Question 16.

In ΔABC, If a = (b + c) cos θ, then prove that sin θ = \(\frac{2 \sqrt{b c}}{b+c} \cos \frac{A}{2}\).

Question 17.

Expand sin 5θ, cos 5θ in the powers of sin θ and cos θ.

Section – C

(5 × 7 = 35 Marks)

**III. Long Answer Questions.**

- Answer any Five questions.
- Each Question carries Seven marks.

Question 18.

If f : A → B and g : B → C is bijective, then prove that (gof)^{-1} = (f^{-1}og^{-1}).

Question 19.

Show that 3 . 5^{2n+1} + 2^{3n+1} is divisible by 17 for all n ∈ N, by using mathematical induction.

Question 20.

Show that \(\left|\begin{array}{lll}

b+c & c+a & a+b \\

c+a & a+b & b+c \\

a+b & b+c & c+a

\end{array}\right|=2\left|\begin{array}{lll}

a & b & c \\

b & c & a \\

c & a & b

\end{array}\right|\).

Question 21.

Solve the following equation by using the matrix inversion method.

3x + 4y + 52 = 18

2x – y – 82 = 13

5x – 2y + 72 = 20

Question 22.

\(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) are non-zero and non-collinear vectors and θ ≠ 0, π is the angle between \(\bar{b}\) and \(\bar{c}\). If \((\bar{a} \times \bar{b}) \times \bar{c}=\frac{1}{3}|\bar{b}||\bar{c}| \bar{a}|\), then find sin θ.

Question 23.

Suppose (α – β) is not an odd multiple of \(\frac{\pi}{2}\), m is a non-zero real number such that m ≠ -1 and \(\frac{\sin (\alpha+\beta)}{\cos (\alpha-\beta)}=\frac{1-m}{1+m}\), then prove that \(\tan \left(\frac{\pi}{4}-\alpha\right)=m \tan \left(\frac{\pi}{4}+\beta\right)\).

Question 24.

Show that in a ΔABC, \(\frac{r_1}{b c}+\frac{r_2}{c a}+\frac{r_3}{a b}=\frac{1}{r}-\frac{1}{2 R}\).