AP Inter 1st Year Maths 1A Model Paper Set 2 with Solutions

Thoroughly analyzing AP Inter 1st Year Maths 1A Model Papers Set 2 helps students identify their strengths and weaknesses.

AP Inter 1st Year Maths 1A Model Paper Set 2 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)

I. Very Short Answer Questions.

• Answer All questions.
• Each Question carries Two marks.

Question 1.
Find the domain of f(x) = \(\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right)\).

Question 2.
Find the inverse functions. If f: (0, ∞) → R defined by f(x) = log₂(x).

Question 3.
If \(\left[\begin{array}{ccc}
0 & 1 & 4 \\
-1 & 0 & 7 \\
-x & -7 & 0
\end{array}\right]\) is a skew symmetric matrix, than find x.

Question 4.
Find the rank of the matrix \(\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]\).

Question 5.
ABCDEF is a regular hexagon with centre ‘O’, show that \(\overline{\mathrm{AB}}+\overline{\mathrm{AC}}+\overline{\mathrm{AD}}+\overline{\mathrm{AE}}+\overline{\mathrm{AF}}=3(\overline{\mathrm{AD}})=6(\overline{\mathrm{AO}})\).

Question 6.
Find the vector equation of the plane passing through the points (1, -2, 5), (6, -5, -1) and (-3, 5, 0).

Question 7.
If \(\stackrel{\rightharpoonup}{a}+\stackrel{\rightharpoonup}{b}+\bar{c}=0,|\bar{a}|=3,|\bar{b}|=5\), and |\(\bar{c}\)| = 7, then find the angle between \(\bar{a}\) and \(\bar{b}\).

Question 8.
Find the extreme values of 5 cos x + 3 cos (x + \(\frac{\pi}{3}\)) + 8.

Question 9.
Sketch the region enclosed by y = sin x, y = cos x, and x-axis in the interval [0, π].

Question 10.
Prove that cosh (3x) = 4 cosh³x – 3 cosh x.

Section – B
(5 × 4 = 20 Marks)

II. Short Answer Questions.

• Answer any Five questions.
• Each question carries Four marks.

Question 11.
If A = \(\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\), then show that from all the positive Integers n, Aⁿ = \(\left[\begin{array}{cc}
\cos n \theta & -\sin n \theta \\
-\sin n \theta & \cos n \theta
\end{array}\right]\).

Question 12.
Find the equation of the line parallel to the vector \(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}}\) and which passes through the point A whose position vector is \(3 \bar{i}+\bar{j}-\bar{k}\). If P is a point on this line such that AP = 15, find the position vector of P.

Question 13.
Let \(\bar{a}\) and \(\bar{b}\) be vector, satisfying \(|\bar{a}|=|\bar{b}|\) = 5 and \((\bar{a}, \bar{b})\) = 45°. Find the area of the triangle having \(\bar{a}-2 \bar{b}\) and \(3 \bar{a}+2 \bar{b}\) as two its sides.

Question 14.
Prove that \(\cos \frac{\pi}{11} \cdot \cos \frac{2 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{5 \pi}{11}=\frac{1}{32}\).

Question 15.
If α, β are the solutions of the equation a cos θ + b sin θ = c, where a, b, c ∈ R and if a² + b² > 0, cos α ≠ cos β and sin α ≠ sin β, then show that cos α + cos β = \(\frac{2 a c}{a^2+b^2}\), cos α . cos β = \(\frac{c^2-b^2}{a^2+b^2}\).

Question 16.
Solve the equation \({Tan}^{-1}\left(\frac{x-1}{x-2}\right)+{Tan}^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\).

Question 17.
In ΔABC, If a = 5, b = 4 and cos (A – B) = \(\frac{31}{32}\), then show that c = 6.

Section – C
(5 × 7 = 35 Marks)

III. Long Answer Questions.

• Answer any Five questions.
• Each Question carries Seven marks.

Question 18.
Determine whether the function f : R → R defined by \(\left\{\begin{array}{cc}
x, & \text { if } x>2 \\
5 x-2, & \text { if } x \leq 2
\end{array}\right.\) is an infection a surjection or a bijection.

Question 19.
Using mathematical induction, prove that 1² + (1² + 2²) + (1² + 2² + 3²) + ….. upto n terms = \(\frac{n(n+1)^2(n+2)}{12}\), ∀ n ∈ N.

Question 20.
Show that \(\left[\begin{array}{ccc}
1 & a^2 & a^3 \\
1 & b^2 & b^3 \\
1 & c^2 & c^3
\end{array}\right]\) = (a – b) (b – c) (c – a) (ab + bc + ca).

Question 21.
Examine whether the following system of equations is consistent or inconsistent and if consistent find the complete solutions.
x + y + 2 = 6, x – y + 2 = 2, 2x – y + 32 = 9

Question 22.
If \(\overline{\mathrm{a}}=\overline{\mathrm{i}}-2 \overline{\mathrm{j}}-3 \overline{\mathrm{k}}, \quad \overline{\mathrm{b}}=2 \overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) and \(\bar{c}=\bar{i}+\bar{j}+2 \bar{k}\), then find \(|(\bar{a} \times \bar{b}) \times \bar{c}|\) and \(|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|\).

Question 23.
If A + B + C + D = 360°, prove that cos 2A + cos 2B + cos 2C + cos 2D = 4 cos (A + B) . cos (A + C) . cos (A + D).

Question 24.
In ΔABC, Prove that \(r_1^2+r_2^2+r_3^2+r^2\) = 16R² – (a² + b² + c²).