Kerala Class 12 First Term Question Paper 2025 Maths

SECOND YEAR HIGHER SECONDARY EXAMINATION AUGUST 2025
PARTIII
MATHEMATICS(COMMERCE)
HSE II Max Mark: 60
Time :2hrs
Cool-off Time :15mts

Part A
Answer any 6 questions from 1 to 8. Each carries 3 score. (6 × 3 = 18)

1. (a) Consider the function f : R → R is defined by f (x) = 2x − 1. Choose the correct answer (1)
A. f is one-one B. f is many- one onto
C. f is one-one but not onto D. f is neither one-one nor onto
(b) Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but (2)
neither reflexive nor transitive.
   
2 −2 2 3
2. Find the matrice X and Y such that X + Y = , and X − Y = (3)
−1 5 4 0
   
3 −2 1 0
3. if A= and I= , then find k so that A2 = kA − 2I. (3)
4 −2 0 1

0 −1 x − 2
 

4. (a) if A=1 0 4  is a skew-symmetric matrix , then find the value of x. (1)
3 −4 0
 
4
(b) Let A = −1 2 3 and B= −2 , Verify that (AB)0 = B 0 A0
 
 (2)
0

5. Using the determinants, find the equation line passing through the poin A(1, 2) and B(3, 6) (3)

6. Show that the relation R on Z defined by R = {(a, b) : |a − b| is even } is an equivalence relation . (3)
(
x + 2 if x ≤ 1
7. Discuss the continuity of the function f defined by f (x) = (3)
x − 2 if x >1
 
−1 −1
8. (a) Find the principal value of sin . (1)
2
   
−1 1 −1 1
(b) Evaluate cos +2 sin (2)
2 2

Part B
Answer any 6 questions from 9 to 16. Each carries 4 score. (6 × 4 = 24)

d2 y
9. Find (i) y = x3 + 3x2 + 5x (ii) y = sin(3x + 7) (4)
dx2
dy
10. (a) Find if y = 3x (1)
dx
(b) Prove that sin(x2 ) is a continuous function. (3)

dy
11. (a) If y = sin−1 x then =−−−−− (1)
dx
(
2x , if x ≤ 5
(b) Find the value of k so that the function f (x) = (3)
k , if x > 5

x−2
12. Let A=R-{3} and B= R- {1} consider the function f : A → B defined by f (x) = , Verify (4)
x−3
that f is bijiective.

13. (a) Which among the following is a possible number of elements of a square matrix ? (1)
(A) 18 (B) 36 (C) 45 (D) 24
     
a b a 0 4 3
(b) If + = find the value a,b,c and d (3)
c d 0 d 2 6

  (i + j)2
14. Construct a 2x2 matrix A= aij where aij = (2)
2
(a) write the tanspose of A (1)
(b) Show that A is symmetric (1)
d2 y
15. If y = 5 cos x + 3 sin x , Prove that +y =0 (4)
dx2
16. (a) The principal value branch of sin−1 x = − − − − − (1)
√ −1 1
(b) Show that sin−1 (2x 1 − x2 ) = 2 sin−1 x, √ ≤ x ≤ √ (3)
2 2

Part C
Answer any 3 questions from 17 to 20. Each carries 6 score. (6 × 3 = 18)

17. Solve the system of linear equations by matrix method. (6)
3x − 2y + 3z = 8
2x + y − z = 1
4x − 3y + 2z = 4
 
1 3 4
18. (a) Express the matrix A = 3 2 4 as the sum of symmetric and skew symmetric matrices. (3)
5 0 6
 
2 2 0
(b) Find A − 5A + 6 if A = (3)
1 −1

dy
19. Find for the following
dx
(a) 2x + 3y = sinx (2)
(b) x = at2 y = 2at (2)
(c) y = log(logx) , x > 1 (2)
dy
20. (a) Find dx if xy = y x (3)
dy
(b) Find dx if x = a (θ − sin θ) , y = a (1 + cos θ) (3)

HIGHER SECONDARY FIRST TERMINAL EXAMINATION - 2025
Max. Score : 60
PART – III Time : 2 Hrs
Second Year MATHEMATICS (SC 60) Cool-off Time : 15 Mts
==================================================================
Answer any 6 questions from 1 to 8. Each carries 3 scores.
1. Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝑅 = {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑐, 𝑑)} be a relation on A
(i) The relation 𝑅 is
(A) Reflexive and Symmetric (B) Symmetric and Transitive
(C) Reflexive and Transitive (D) Transitive only [1]
(ii) Make the relation R equivalence by adding exactly two elements [1]
(iii) Find the number of elements in the largest equivalence relation on A [1]
2. Find the value of cot −1 (−1) + cosec −1 (−√2) + sec −1 (2) [3]
3. (i) The number of all possible 2 x 2 matrices with entries 0 or 1 is
(A) 8 (B) 9 (C) 16 (D) 25 [1]
1 3 𝑦 0 5 6
(ii) Find the value of 𝑥 and 𝑦 if 2 [ ]+[ ]=[ ] [2]
0 𝑥 1 2 1 8
𝑥 3
4. (i) If | | = 5 , then x = …………….. [1]
5 2
(ii) If the area of a triangle with vertices (k, 0), (4, 0), (0, 2) is 4 square units , find k
5. (i) Identify the following function [2]

(A) |𝑥| (B) |𝑥| + 1 (C) |𝑥 − 1| (D) |𝑥 + 1| [1]
(ii) Discuss the continuity of the function [1]
(iii) Examine the differentiability of the function [1]
(1) P.T.O

6. (i) Identify the function which is one – one but not onto among the following
(A) 𝑓 ∶ ℝ → ℝ , 𝑓(𝑥) = 3 − 4𝑥 (B) 𝑓 ∶ ℝ → ℝ , 𝑓(𝑥) = 𝑥 2
(C) 𝑓 ∶ ℤ → ℤ , 𝑓(𝑥) = (𝑥 − 2)2 (D) 𝑓 ∶ ℕ → ℕ , 𝑓(𝑥) = 𝑥 3 [1]
(ii) Let 𝑓 ∶ ℝ → ℝ and 𝑔 ∶ ℝ → ℝ defined by 𝑓(𝑥) = cos 𝑥 and 𝑔(𝑥) = 3𝑥 2
Show that fog ≠ gof [2]
√1+𝑥 2 −1
7. Write the simplest form of tan−1 ( ),𝑥≠0 [3]
𝑥
𝑑𝑦
8. Find 𝑑𝑥 (i) 𝑦 = √tan 𝑥 [1]

(ii) 𝑥 2 + 𝑥𝑦 + 𝑦 2 = 100 [2]
Answer any 6 questions from 9 to 16. Each carries 4 scores.
9. (i) Let 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| is even} be a relation on 𝐴 = {1, 2, 3, 4, 5}
Show that 𝑅 is an equivalence relation [3]
(ii) Write the set of all elements related to 5 in 𝐴 under the relation 𝑅 [1]
10. Match the following : [1 x 4 = 4]
Functions Principal Value Branch
𝜋 𝜋
(a) cos−1 𝑥 (i) (− , )
2 2

(b) sec −1 𝑥 (ii) (0, 𝜋)
(c) tan−1 𝑥 (iii) [0, 𝜋]
𝜋
(d) cot −1 𝑥 (iv) [0, 𝜋] − {2 }
𝜋 𝜋
(v) [− 2 , 2 ] − {0}
−2
11. If 𝐴 = [ 4 ] and 𝐵 = [1 3 6]
5
(i) What is the order of AB [1]
(ii) Verify that (𝐴𝐵)′ = 𝐵 ′ . 𝐴′ [3]
2+𝑥 3 4
12.(i) Given that [ 1 −1 2] is a singular matrix. Find the value of 𝑥 [2]
𝑥 1 5
1 −1
(ii) If 𝐴 = [ ] , prove that A(Adj A) = |A|I [2]
2 3
0 2𝑏 −2
13.(i) If 𝐴 = [ 3 1 3 ] is a symmetric matrix, find the values of 𝑎 and 𝑏 [1]
3𝑎 3 −1
3 1
(ii) If 𝐴 = [ ] , show that A2 − 5A + 7I = 0 [3]
−1 2
(2) P.T.O

𝑑2 𝑦 𝑑𝑦
14.(i) If 𝑦 = sin−1 𝑥 , show that (1 − 𝑥 2 ) =𝑥 [2]
𝑑𝑥 2 𝑑𝑥
𝑑𝑦 1−𝑥 2
(ii) Find 𝑑𝑥 , where 𝑦 = cos−1 (1+𝑥 2) , 0 < 𝑥 < 1 [2]
2𝜋
15.(i) sin−1 (sin 3 ) = ……………………….. [1]
8 3 77
(ii) Prove that sin−1 17 + sin−1 5 = tan−1 36 [3]

16. (i) Show that the function f(x) = cos(𝑥 2 ) is a continuous
function [2]
𝑘𝑥 + 1 , 𝑖𝑓 𝑥 ≤ 5
(ii) Find the value of 𝑘 so that the function 𝑓(𝑥) = {
3𝑥 − 5 , 𝑖𝑓 𝑥 > 5
is continuous at x = 5 [2]
Answer any 3 questions from 17 to 20. Each carries 6 scores.
17.(i) Number of onto functions that can be defined from {1, 2, 3} to {4, 5, 6, 7} is ….
(A) 0 (B) 3 (C) 81 (D) 64 [1]
(ii) Consider the graph of function 𝑓 ∶ ℝ → ℝ by 𝑓(𝑥) = (x − 1)2 + 1

Make 𝑓(𝑥) bijective by redefining its domain and co domain [2]
𝑥−2
(iii) Show that the function 𝑓 ∶ ℝ − {3} → ℝ − {1} defined by 𝑓(𝑥) = 𝑥−3 is both

one – one and onto. [3]
18.(i) Construct a 3 X 3 matrix A whose (𝑖, 𝑗)𝑡ℎ element 𝑎𝑖𝑗 = 2𝑖 − 𝑗 [2]
(ii) Express A as the sum of a symmetric and skew symmetric matrices [4]
19. Solve the following system of linear equations using matrix method
x–y+z=4; 2x + y – 3z = 0 ; x + y + z = 2 [6]
𝑑
20.(i) 𝑑𝑥 (𝑎 𝑥 ) = ………………… [1]
𝑑𝑦
(ii) Find 𝑑𝑥 of the following

(a) 𝑥 = 𝑎 𝑐𝑜𝑠 𝜃 , 𝑦 = 𝑎 (𝜃 + 𝑠𝑖𝑛 𝜃) [2]
(b) 𝑦 = 𝑥 𝑥 + 𝑥 sin 𝑥 [3]
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