AP Inter 1st Year Mathematics 1A Guess Paper 2025

1

Find the range of the real-valued function √ [x] -  x

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.

3

Define the trace of a matrix.

4

ABCDE is a pentagon. If the sum of the vectors AB¯¯¯¯¯¯¯, AE¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯, DC¯¯¯¯¯¯¯, ED¯¯¯¯¯¯¯, and AC¯ is λ(AC¯), then find the value of λ.

5

a¯,b¯,c¯ are pairwise non-zero and noncollinear vectors. If a¯+b¯ is collinear with c¯ and b¯+c¯ is collinear with a¯, then find the vector a¯+b¯+c¯.

6

Find the angle between the planes r¯⋅(2i¯−j¯+2k¯¯¯)=3 and r¯⋅(3i¯+6j¯+k¯)=4.

7

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

8

The median AD of ∆ABC is bisected at E and BE is produced to meet the side AC in F. Show that AF¯¯¯¯¯¯¯=13(AC¯¯¯¯¯¯¯) and EF¯¯¯¯¯¯¯=14BF¯¯¯¯¯¯¯.

9

Let a¯¯¯=i¯+j¯+k¯¯¯ and b¯¯¯=2i¯+3j¯+k¯¯¯. Find

(i) The projection vector of b¯ on a¯ and its magnitude.

(ii) The component vector of b¯ in the direction of a and perpendicular to a¯.

10

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.

11

If f = {(4, 5), (5, 6), (6, -4)} and g = {(4, -4), (6, 5), (8, 5)} then find

(i) f – g

(ii) fg

(iii) fg

(iv) |f|

(v) 2f + 4g

12

Using mathematical induction, for all n ∈ N, prove that 2.3 + 3.4 + 4.5 + …… up to n terms = n(n2+6n+11)3.

13

Solve the following system of equations by using the Gauss-Jordan method.

2x + 4y – z = 0, x + 2y + 2z = 5, 3x + 6y – 7z = 2.

14

Let OA¯¯¯¯¯¯¯¯=a¯¯¯, OB¯¯¯¯¯¯¯=10,a¯¯¯+2b¯¯¯ and OC¯¯¯¯¯¯¯¯=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 OA¯ and OC¯ as adjacent sides. Prove that λ = 6µ.

15

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