# AP Inter 1st Year Maths 1A Question Paper March 2023

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## AP Inter 1st Year Maths 1A Question Paper March 2023

Time: 3 Hours
Maximum Marks: 75

Note: This question paper consists of THREE Sections A, B, and C.

Section – A
(10 × 2 = 20 Marks)

I. Very Short Answer Type Questions.

• Each question carries two marks.

Question 1.
If f(x) = 2x – 1, g(x) = $$\frac{x+1}{2}$$ for all x ∈ R, then find (gof)(x).

Question 2.
Find the domain of the real-valued function f(x) = $$\sqrt{x^2-25}$$.

Question 3.
Define a symmetric matrix and give an example.

Question 4.
If A = $$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 2 & 3 & 4 \\ 5 & -6 & x \end{array}\right]$$ and det A = 45 then find x.

Question 5.
If $$\overline{\mathrm{OA}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{AB}}=3 \overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{BC}}=\overline{\mathrm{i}}+2 \overline{\mathrm{j}}-2 \overline{\mathrm{k}}$$ and $$\overline{\mathrm{CD}}-2 \overline{\mathrm{i}}+\overline{\mathrm{j}}+3 \overline{\mathrm{k}}$$ then find the vector $$\bar{OD}$$.

Question 6.
Find the vector equation of the plane passing through the points $$\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+5 \overline{\mathrm{k}},-5 \overline{\mathrm{j}}-\overline{\mathrm{k}}$$ and $$-3 \overline{\mathrm{i}}+5 \overline{\mathrm{j}}$$.

Question 7.
Let $$\bar{a}$$ and $$\bar{b}$$ be non-zero, non-collinear vectors. If $$|\overline{\mathrm{a}}+\overline{\mathrm{b}}|=|\overline{\mathrm{a}}-\overline{\mathrm{b}}|$$, then find the angle between $$\bar{a}$$ and $$\bar{b}$$.

Question 8.
Find the value of sin 330° . cos 120° + cos 210° . sin 300°.

Question 9.
If A – B = $$\frac{3 \pi}{4}$$, then show that (1 – tan A) (1 + tan B) = 2

Question 10.
Show that $$\tanh ^{-1}\left(\frac{1}{2}\right)=\frac{1}{2} \log _e^3$$.

Section – B
(5 × 4 = 20)

• Each question carries four marks.

Question 11.
If A = $$\left[\begin{array}{lll} \mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\ \mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2 \\ \mathrm{a}_3 & \mathrm{~b}_3 & \mathrm{c}_3 \end{array}\right]$$ is a non-singular matrix, then prove that A is invertible and A-1 = $$\frac{{Adj} \mathrm{A}}{{det} \mathrm{A}}$$

Question 12.
If the points whose position vectors are $$3 \mathbf{i}-2 \overline{\mathbf{j}}-\overline{\mathbf{k}}, 2 \overline{\mathbf{i}}+3 \overline{\mathbf{j}}-4 \overline{\mathbf{k}},-\overline{\mathbf{i}}+\overline{\mathbf{j}}+2 \overline{\mathbf{k}}$$ and $$4 \overline{\mathrm{i}}+5 \overline{\mathrm{j}}+\lambda \overline{\mathrm{k}}$$ are coplanar, then show that λ = $$\frac{-146}{17}$$

Question 13.
If $$\overline{\mathrm{a}}=2 \overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}}, \overline{\mathrm{b}}=-\overline{\mathrm{i}}+2 \overline{\mathrm{j}}-4 \overline{\mathrm{k}}$$ and $$\overline{\mathbf{c}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$, then find $$(\overline{\mathbf{a}} \times \overline{\mathrm{b}}) \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$.

Question 14.
For A ∈ R, prove that $$\cos A \cos \left(\frac{\pi}{3}+A\right) \cos \left(\frac{\pi}{3}-A\right)=\frac{1}{4} \cos 3 A$$ and hence deduce that $$\cos \cdot \frac{\pi}{9} \cdot \cos \frac{2 \pi}{9} \cdot \cos \frac{3 \pi}{9} \cdot \cos \frac{4 \pi}{9}=\frac{1}{16}$$.

Question 15.
Solve cot2x – (√3 + 1) cot x + √3 = 0; 0 < x < $$\frac{\pi}{2}$$.

Question 16.
Prove that $${Tan}^{-1} \frac{1}{2}+{Tan}^{-1} \frac{1}{5} {Tan}^{-1} \frac{1}{8}=\frac{\pi}{4}$$.

Question 17.
In a ΔABC, show that $$\frac{b^2-c^2}{a^2}=\frac{\sin (B-C)}{\sin (B+C)}$$.

Section – C
(5 × 7 = 35 Marks)

• Each question carries seven marks.

Question 18.
Let f: A → B be a bijection. Then prove that fof-1 = IB and f-1of = IA.

Question 19.
By using Mathematical Induction, to prove the statement:
$$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots \ldots .+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}$$, ∀ n ∈ N.

Question 20.
Show that $$\left|\begin{array}{lll} \mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|^2=\left|\begin{array}{ccc} 2 \mathrm{bc}-\mathrm{a}^2 & \mathrm{c}^2 & \mathrm{~b}^2 \\ \mathrm{c}^2 & 2 \mathrm{ac}-\mathrm{b}^2 & \mathrm{a}^2 \\ \mathrm{~b}^2 & \mathrm{a}^2 & 2 \mathrm{ab}-\mathrm{c}^2 \end{array}\right|$$ = (a3 + b3 + c3 – 3abc)2.

Question 21.
Solve 2x – y + 3z = 8, -x + 2y + z = 4, 3x + y – 4z = 0 by using matrix inversion method.

Question 22.
Find the shortest distance between the Skew lines $$\overline{\mathrm{r}}=(6 \overline{\mathrm{i}}+2 \overline{\mathrm{j}}+2 \overline{\mathrm{k}})+\mathrm{t}(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+2 \overline{\mathrm{k}})$$ and $$\overline{\mathrm{r}}=(-4 \overline{\mathrm{i}}-\overline{\mathrm{k}})+\mathrm{s}(3 \overline{\mathrm{i}}-2 \overline{\mathrm{j}}-2 \overline{\mathrm{k}})$$.

Question 23.
If A + B + C = $$\frac{\pi}{2}$$, then prove that cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C.

Question 24.
In ΔABC, prove that r + r1 + r2 – r3 = 4R cos C.