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

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)

• 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) = x2 + 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 = AT + BT.

Question 4.
If A = $$\left[\begin{array}{cc} 2 & 4 \\ -1 & k \end{array}\right]$$ and A2 = 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 r2 = $$2 \bar{r} \cdot(4 \bar{i}-2 \bar{j}+2 \bar{k})$$.

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)

• 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 cot2x – (√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)

• Each Question carries Seven marks.

Question 18.
If f : A → B and g : B → C is bijective, then prove that (gof)-1 = (f-1og-1).

Question 19.
Show that 3 . 52n+1 + 23n+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, $$\frac{r_1}{b c}+\frac{r_2}{c a}+\frac{r_3}{a b}=\frac{1}{r}-\frac{1}{2 R}$$.