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

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

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

• Each Question Carries Two marks.

Question 1.
Find the range of the real-valued function $$\sqrt{[x]-x}$$.

Question 2.
Let f = {(1, a), (2, c), (4, d), (3, b)} and g-1{(2, a), (4, b), (1, c), (3, d)} then show that (gof)-1 = f-1og-1.

Question 3.
Define the trace of a matrix.

Question 4.
If A = $$\left[\begin{array}{cc} -2 & 1 \\ 5 & 0 \\ -1 & 4 \end{array}\right]$$ and B = $$\left[\begin{array}{ccc} -2 & 3 & 1 \\ 4 & 0 & 2 \end{array}\right]$$ then find 2A + BT and 3BT – A.

Question 5.
ABCDE is a pentagon. If the sum of the vectors $$\overline{\mathrm{AB}}, \overline{\mathrm{AE}}, \overline{\mathrm{BC}}, \overline{\mathrm{DC}}, \overline{\mathrm{ED}}$$ and $$\bar{AC}$$ is λ($$\bar{AC}$$), then find the value of λ.

Question 6.
$$\bar{a}, \bar{b}, \bar{c}$$ are pair wise non-zero and non collinear vectors. If $$\bar{a}+\bar{b}$$ is collinear with $$\bar{c}$$ and $$\bar{b}+\bar{c}$$ is collinear with $$\bar{a}$$, then find the vector $$\bar{a}+\bar{b}+\bar{c}$$.

Question 7.
Find the angle between the planes $$\overline{\mathrm{r}} \cdot(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}})=3$$ and $$\bar{r} \cdot(3 \bar{i}+6 \bar{j}+\bar{k})=4$$.

Question 8.
Prove that tan 50° – tan 40° = 2 tan 10°.

Question 9.
Find the value of cot $$67 \frac{1}{2}^{\circ}$$.

Question 10.
For any n ∈ R, prove that (cosh x – sinh x)n = cosh (nx) – sinh (nx).

Section – B
(5 × 4 = 20 Marks)

• Each Question carries Four marks.

Question 11.
If I = $$\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$$ and E = $$\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right]$$ then show that (aI + bE)3 = a3I + 3a2bE.

Question 12.
The median AD of ∆ABC is bisected at E and BE is produced to meet the side AC in F. Show that $$\overline{\mathrm{AF}}=\frac{1}{3}(\overline{\mathrm{AC}})$$ and $$\overline{\mathrm{EF}}=\frac{1}{4} \overline{\mathrm{BF}}$$.

Question 13.
Let $$\overline{\mathrm{a}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}$$ and $$\overline{\mathrm{b}}=2 \overline{\mathrm{i}}+3 \overline{\mathrm{j}}+\overline{\mathrm{k}}$$. Find
(i) The projection vector of $$\bar{b}$$ on $$\bar{a}$$ and its magnitude.
(ii) The component vector of $$\bar{b}$$ in the direction of a and perpendicular to $$\bar{a}$$.

Question 14.
If 8α is not an integral multiple of π, then prove that tan α + 2 tan 2α + 4 tan 4α + 8 cot 8α = cot α.

Question 15.
If $$\tan \left(\frac{\pi}{2} \sin \theta\right)=\cot \left(\frac{\pi}{2} \cos \theta\right)$$, then prove that $$\sin \left(\theta+\frac{\pi}{4}\right)= \pm \frac{1}{\sqrt{2}}$$.

Question 16.
Prove that $$\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)+\tan \left(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \frac{a}{b}\right)=\frac{2 b}{a}$$.

Question 17.
In ∆ABC, prove that $$\left[\frac{b-c}{b+c}\right] \cot \left(\frac{A}{2}\right)+\frac{b+c}{b-c} \tan \left(\frac{A}{2}\right)$$ = 2 cosec (B – C).

Section – C
(5 × 7 = 35 Marks)

• Each question carries Seven marks.

Question 18.
If f = {(4, 5), (5, 6), (6, -4)} and g = {(4, -4), (6, 5), (8, 5)} then find
(i) f – g
(ii) fg
(iii) $$\frac{f}{g}$$
(iv) |f|
(v) 2f + 4g

Question 19.
Using mathematical induction, for all n ∈ N, prove that 2.3 + 3.4 + 4.5 + …… up to n terms = $$\frac{n\left(n^2+6 n+11\right)}{3}$$.

Question 20.
Show that $$\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|^2=\left|\begin{array}{ccc} 2 b c-a^2 & c^2 & b^2 \\ c^2 & 2 a c-b^2 & a^2 \\ b^2 & a^2 & 2 a b-c^2 \end{array}\right|$$

Question 21.
Solve the following system of equations by using the Gauss-Jordan method.
2x + 4y – z = 0, x + 2y + 2z = 5, 3x + 6y – 7z = 2.

Question 22.
Let $$\overline{\mathrm{OA}}=\overline{\mathrm{a}}, \overline{\mathrm{OB}}=10, \overline{\mathrm{a}}+2 \overline{\mathrm{b}}$$ and $$\overline{\mathrm{OC}}=\overline{\mathrm{b}}$$ where O, A, B and C are non-collinear points. Let λ denote the area of the quadrilateral OABC and Let µ denote the area of the parallelogram with $$\bar{OA}$$ and $$\bar{OC}$$ as adjacent sides. Prove that λ = 6µ.

Question 23.
In ΔABC, prove that $$\cos \frac{A}{2}+\cos \frac{B}{2}+\cos \frac{C}{2}=4 \cos \left(\frac{\pi-A}{4}\right) \cos \left(\frac{\pi-B}{4}\right)$$ $$\cos \left(\frac{\pi-C}{4}\right)$$.

Question 24.
If r1 = 36, r2 = 18 and r3 = 12, then prove that a = 30, b = 24, c = 18 and R = 15.