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.

  • Answer all the 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.

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

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)

II. Short Answer Type Questions.

  • Answer any five questions.
  • 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 = \(\frac{-146}{17}\)

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

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)

III. Long Answer Type Questions.

  • Answer any five questions.
  • 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 \(\overline{\mathrm{r}}=(-4 \overline{\mathrm{i}}-\overline{\mathrm{k}})+\mathrm{s}(3 \overline{\mathrm{i}}-2 \overline{\mathrm{j}}-2 \overline{\mathrm{k}})\).

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

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.

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