TS Inter 1st Year Maths 1A Question Paper May 2022

Thoroughly analyzing TS Inter 1st Year Maths 1A Model Papers and TS Inter 1st Year Maths 1A Question Paper May 2022 helps students identify their strengths and weaknesses.

TS Inter 1st Year Maths 1A Question Paper May 2022

Time: 3 Hours
Maximum Marks: 60

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 ANY TEN of the following questions.
• Each Question carries TWO marks.

Question 1.
If f: R → R is defined by f(x) = \(\frac{1-x^2}{1+x^2}\), then show that f(tan θ) = cos 2θ.

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

Question 3.
If A = \(\left[\begin{array}{cc}
-1 & 3 \\
4 & 2
\end{array}\right]\), B = \(\left[\begin{array}{cc}
2 & 1 \\
3 & -5
\end{array}\right]\), X = \(\left[\begin{array}{ll}
x_1 & x_2 \\
x_3 & x_4
\end{array}\right]\) and if A + B = X, then find the values of x₁, x₂, x₃, x₄.

Question 4.
If A = \(\left[\begin{array}{cc}
4 & 2 \\
-1 & 1
\end{array}\right]\), then find A².

Question 5.
If A = \(\left[\begin{array}{ccc}
-1 & 2 & 3 \\
2 & 5 & 6 \\
3 & x & 7
\end{array}\right]\) is a symmetric matrix, then find ‘x’.

Question 6.
If A = \(\left[\begin{array}{cc}
2 & 4 \\
-1 & k
\end{array}\right]\) and A² = 0, then find the value of k.

Question 7.
If the vectors \(-3 \mathrm{i}+4 \mathrm{j}+\lambda \mathrm{k}, \mu \mathrm{i}+8 \mathrm{j}+6 \mathrm{k}\) are collinear vectors, then find λ and µ.

Question 8.
Find the vector equation of the line passing through the point 2i + 3j + k and parallel to the vector 4i – 2j + 3k.

Question 9.
If OA = i + j + k, AB = 3i – 2j + k, BC = i + 2j – 2k and CD = 2i + j + 3k, then find the vector OD.

Question 10.
If a = 2i – 3j + 5k, b = -i + 4j + 2k, then find a × b.

Question 11.
Find the area of the parallelogram having a = 2j – k, b = -i + k as adjacent sides.

Question 12.
Find the period of the function “tan 5x”.

Question 13.
Find the minimum and maximum values of the function “3 cos x + 4 sin x”.

Question 14.
If cosh x = sec θ, then prove that \(\tanh ^2 \frac{x}{2}=\tan ^2 \frac{\theta}{2}\).

Question 15.
If sinh x = \(\frac{3}{4}\), then find sinh (2x).

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Type Questions.

• Answer any FIVE questions.
• Each Question carries FOUR marks.

Question 16.
Find the adjoint and the inverse of the matrix A = \(\left[\begin{array}{cc}
1 & 2 \\
3 & -5
\end{array}\right]\).

Question 17.
\(\left[\begin{array}{ccc}
x-1 & 2 & 5-y \\
0 & z-1 & 7 \\
1 & 0 & a-5
\end{array}\right]=\left[\begin{array}{lll}
1 & 2 & 3 \\
0 & 4 & 7 \\
1 & 0 & 0
\end{array}\right]\), then find the values of x, y and ‘a’.

Question 18.
If A = \(\left[\begin{array}{ccc}
0 & 1 & 2 \\
2 & 3 & 4 \\
4 & 5 & -6
\end{array}\right]\) and B = \(\left[\begin{array}{ccc}
-1 & 2 & 3 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right]\), then find BA and 4A – 5B.

Question 19.
Let a = 2i + 4j – 5k, b = i + j + k and c = j + 2k. Find the unit vector in the opposite direction of a + b + c.

Question 20.
If the position vectors of the points A, B, and C are -2i + j – k, -4i + 2j + 2k, and 6i – 3j – 13k respectively, and AB = λ AC, then find the value of λ.

Question 21.
Find the angle between the vectors i + 2j + 3k and 3i – j + 2k.

Question 22.
If a = i + 2j – 3k and b = 3i – j + 2k, then show that a + b and a – b are perpendicular to each other.

Question 23.
Prove that \(\cot \frac{\pi}{20} \cdot \cot \frac{3 \pi}{20} \cdot \cot \frac{5 \pi}{20} \cdot \cot \frac{7 \pi}{20} \cdot \cot \frac{9 \pi}{20}\) = 1.

Question 24.
If sin α + cosec α = 2, find the value of sinⁿα + cosecⁿα, n ∈ Z.

Question 25.
Prove that for any x ∈ R. sinh (3x) = 3 sinh x + 4 sinh³x.

Question 26.
In ΔABC, if sin θ = \(\frac{a}{b+c}\), then show that cos θ = \(\frac{2 \sqrt{b c}}{b+c} \cos \frac{A}{2}\).

Question 27.
Show that \(\text { b. } \cos ^2 \frac{C}{2}+c \cdot \cos ^2 \frac{B}{2}\) = S. (in ΔABC).

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Type Questions.

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

Question 28.
If f = {(4, 5), (5, 6), (6, -4)} and g = {(4, -4), (6, 5), (8, 5)}, then find
(i) f + g
(ii) f – g
(iii) 2f + 4g
(iv) f + 4
(v) fg
(vi) \(\frac{f}{g}\)
(vii) |f|

Question 29.
By using Cramer’s rule, solve the system of equations:
2x – y + 3z = 8, -x + 2y + z = 4, 3x + y – 4z = 0.

Question 30.
If A = \(\left[\begin{array}{lll}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{array}\right]\), then show that A² – 4A – 5I = 0.

Question 31.
Solve x + y + z = 1, 2x + 2y + 3z = 6 and x + 4y + 9z = 3 by using Matrix inversion method.

Question 32.
Show that the line joining the pair of points 6a – 4b + 4c, -4c and the line joining the pair of points -a – 2b – 3c, a + 2b – 5c intersect at the point -4c when a, b, c are non-coplanar vectors.

Question 33.
a = 3i – j + 2k, b = -i + 3j + 2k, c = 4i + 5j – 2k and d = i + 3j + 5k, then compute the following:
(i) (a b) (c d)

Question 34.
Find the unit vector perpendicular to the plane passing through the points (1, 2, 3), (2, -1, 1) and (1, 2, -4).

Question 35.
If A, B, and C are angles in a triangle, then prove that sin 2A – sin 2B + sin 2C = 4 cos A sin B cos C.

Question 36.
In triangle ABC, if r₁ = 2, r₂ = 3, r₃ = 6 and r = 1, then prove that a = 3, b = 4 and c = 5.

Question 37.
In ΔABC, prove that \(\frac{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}{\cot A+\cot B+\cot C}=\frac{(a+b+c)^2}{a^2+b^2+c^2}\).