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

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TS 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 questions.
• Each question carries TWO marks.

Question 1.
If A = {-2, -1, 0, 1, 2} and f: A → B is a surjection defined by f(x) = x² + x + 1, then find B.

Question 2.
Find the domain of real valued function f(x) = \(\frac{2 x^2-5 x+7}{(x-1)(x-2)(x-3)}\).

Question 3.
If A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 2 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{lll}
3 & 2 & 1 \\
1 & 2 & 3
\end{array}\right]\), find 3B – 2A.

Question 4.
If A = \(\left[\begin{array}{ccc}
0 & 2 & 1 \\
-2 & 0 & -2 \\
-1 & x & 0
\end{array}\right]\) is a skew symmetric matrix, then find x.

Question 5.
Let \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=\hat{\mathrm{j}}+2 \hat{\mathrm{k}}\). Find the unit vector in the opposite direction of \(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\).

Question 6.
Find the vector equation of the line passing through the point \(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and parallel to the vector \(4 \hat{i}-2 \hat{j}+3 \hat{k}\).

Question 7.
For what values of λ, the vectors \(\hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(8 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) are at right angles?

Question 8.
Find the period for the function cos(\(\frac{4 x+9}{5}\)).

Question 9.
Find the minimum and maximum values of 3 cos x + 4 sin x.

Question 10.
If sin hx = 3 then show that x = log_e(3 + \(\sqrt{10}\)).

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Type Questions.

• Attempt ANY FIVE questions.
• Each question carries FOUR marks.

Question 11.
If A = \(\left[\begin{array}{ccc}
1 & 1 & 3 \\
5 & 2 & 6 \\
-2 & -1 & -3
\end{array}\right]\) then find A³.

Question 12.
Let A, B, C and D be four points with position vectors \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}\), \(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}, \overline{\mathrm{a}}\) and \(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}\) respectively. Express the vectors \(\bar{AC}\), \(\bar{DA}\), \(\bar{BA}\) and \(\bar{BC}\) in terms of \(\bar{a}\) and \(\bar{b}\).

Question 13.
Find the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, -1, 0) and (-1, 0, 1).

Question 14.
Prove that \(\cos ^2 \frac{\pi}{8}+\cos ^2 \frac{3 \pi}{8}+\cos ^2 \frac{5 \pi}{8}+\cos ^2 \frac{7 \pi}{8}\) = 2.

Question 15.
Solve 7 sin²θ + 3 cos²θ = 4.

Question 16.
Prove that \(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{8}{17}=\cos ^{-1} \frac{36}{85}\).

Question 17.
In ΔABC, prove that \(\cot \frac{\mathrm{A}}{2}+\cot \frac{\mathrm{B}}{2}+\cot \frac{\mathrm{C}}{2}=\frac{\mathrm{S}^2}{\Delta}\).

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Type Questions.

• Answer ANY FIVE questions.
• Each question carries SEVEN marks.

Question 18.
If f: A → B, g: B → C are bijective functions then prove that (gof)^-1 = f^-1 o g^-1.

Question 19.
Using mathematical induction, prove that 1.2.3 + 2.3.4 + 3.4.5 + …… upto n terms = \(\frac{n(n+1)(n+2)(n+3)}{4}\) for all n ∈ N.

Question 20.
Show that \(\left|\begin{array}{ccc}
a+b+2 c & a & b \\
c & b+c+2 a & b \\
c & a & c+a+2 b
\end{array}\right|\) = 2(a + b + c)³.

Question 21.
Solve the system of equations x – y + 3z = 5, 4x + 2y – z = 0, -x + 3y + z = 5 by using Cramer’s rule.

Question 22.
If \(\overrightarrow{\mathrm{a}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+8 \hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\), then compute \(\overline{\mathrm{a}} \times \overline{\mathrm{b}}, \overline{\mathrm{a}} \times \overline{\mathrm{c}}\) and \(\bar{a} \times(\bar{b}+\bar{c})\). Verify whether the cross product is distributive over vector addition.

Question 23.
If A, B, and C are angles in a triangle then prove that cos A + cos B + cos C = 1 + 4 \(\sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2}\).

Question 24.
In ΔABC, if a = 13, b = 14, c = 15, then show that R = \(\frac{65}{8}\), r = 4, r₁ = \(\frac{21}{2}\), r₂ = 12 and r₃ = 14.