GOVERNMENT PU COLLEGE (AN0089) , YELAHANKA BANGALORE NORTH

CLASS : II PU FIRST TEST MODEL PAPER 3

TIME : 1 hr 30 min SUB: MATHEMATICS ( 35 ) MAX MARKS : 40
I ANSWER ALL MULTIPLE CHOICE QUESTIONS 7X1=7
1. Let R be the relation in the set { 1 , 2 , 3 } given by R = { (1 , 2) , (2 , 1) } .
Choose the correct answer .
(A) R is an equivalence relation .
(B) R is reflexive but neither symmetric nor transitive .
(C) R is symmetric but neither reflexive nor transitive .
(D) R is transitive but neither reflexive nor symmetric.
2. Let f : R→ R defined as f(x) = x 4 . Choose the correct answer .
(A) f is neither one – one nor onto (B) f is one one onto
(C) f is one-one but not onto (D) f is many one onto
3. sin -1 ( 1 – x ) – 2 sin -1 x = π
2
1 1 1
(A) 0 , (B) 1 , (C) 0 (D)
2 2 2
x x− y
4. tan-1 ( ) - tan -1 ( ) is equal to
y x+ y
−3 π
(A) π (B) π (C) π (D)
2 3 4 4
5. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is
(A) 27 (B) 18 (C) 81 (D) 512
6. If A is an invertible matrix of order 2 , then det (A-1 ) is equal to
1
(A) det ( A ) (B) (C) 1 (D) 0
det ( A)
7. Choose which of the following is incorrect .
(A) The function f(x) = 2x + 3 is continuous at x = 1 .
(B) The left hand limit of the function f(x) = | x | at x = 0 is 0 .
(C) The function f(x) = | x – 5 | is discontinuous at x = 5 .
(D) Every rational function is continuous .

II . FILL IN THE BLANKS BY CHOOSING FROM GIVEN BOX 3X1=3

[ −π , 0 , 1 , 2 ]
3
8. The number of equivalence relation in the set { 1 , 2 , 3 } containing (1 , 2) and ( 2 , 1 ) is ___

9. The value of tan -1 √ 3 – sec -1 ( – 2 ) is -------

10. A square matrix A is said to be singular if | A | = ___
III ANSWER ANY THREE QUESTIONS : 3 X2=6
−1 1
11. Show that sin -1 ( 2 x √ 1− x 2 ) =2 sin -1 x , ≤x≤ .
√2 √2
1 1
12. Find the value of cos -1 + 2 sin -1 .
2 2

13. If A = [12 2 3
3 1 ] and B = [−13 −1 3
0 2 ] , then find 2A – B .
14. Find the equation of the line joining the A (1, 3) and B(0 , 0) using determinants.
15. Differentiate sin ( cos (x 2 ) ) with respect to x .
IV ANSWER ANY THREE QUESTIONS : 3X3=9
16. Check whether the relation R defined in the set { 1 , 2 , 3 , 4 , 5 , 6 } as
R = { ( a , b ) : b = a + 1 } is reflexive , symmetric or transitive .
17. Show that the relation R in the set A of all the books in the library of a college ,
given by R= { (x , y ) : x and y have same number of pages } is an equivalence relation.
18. Solve : 2 tan -1 ( cos x ) = tan -1 ( 2 cosec x )

19. Express the matrix [16 57 ] as the sum of symmetric and skew symmetric matrix.
20. If A = [13 24] then verify that A ( adj A ) = ( adj A ) A = |A| I .

V ANSWER ANY THREE QUESTIONS : 3 X 5 = 15

21. Let f : ℕ → Y be a function defined as f(x) = 4x + 3 , where Y = { y ∈ℕ : y = 4x +3 for

some x ∈ℕ } . Show that f is invertible . Find the inverse .

[ ]
1 2 3
22. If A = 3 −2 1 , then show that A3 – 23 A – 40 I = O .
4 2 1

[]
0
23. If A = 1 B=[1 5 7] Verify that ( AB )1 = B1 A1
2
24. Solve the system of linear equations , using matrix method
3x – 2y +3z = 8 , 2x + y – z = 1 , 4x – 3y + 2z = 4
5 , if x ≤ 2
25 . Find the values of a & b such that the function defined by f(x) = { ax +b , if 2< x <10
21 , if x ≥ 10
is continuous function .

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 BANGALORE RURAL DISTRICT MATHEMATICS FORUM
CLASS : II PU FIRST TEST MODEL PAPER 4
TIME : 1 hr 30 min SUB: MATHEMATICS ( 35 ) MAX MARKS : 40
I ANSWER ALL MULTIPLE CHOICE QUESTIONS 7X1= 7
1. Which of the relation in the set { 1 , 2 , 3 } is symmetric and transitive but not reflexive ?
(A) { ( 1, 2 ) , ( 2 , 1 ) ( 1 , 1 ) } ( B) { ( 1, 2 ) , ( 2 , 1 )}
(C) { ( 2 , 2 ) } (D) { ( 1 , 1 ) , ( 2 , 2) ( 3 , 3 ) }
2. If f : A → B is injective then
(A) n(A ) ≤ n(B) (B) n(A ) ≥ n(B) (C) n(A ) = n(B) (D) A = B
3. The domain of sec -1 x is
(A) ( - ∞ , ∞ ) (B) [ -1 , 1 ] (C) R - ( -1 , 1 ) (D) ( - ∞ , -1 ) U ( 1 , ∞)
4. sin( tan -1 x ) , | x | < 1 is equal to
x 1 1 x
(A) (B) (C) (D)
√ 1− x 2 √ 1− x 2 √ 1+ x 2
√ 1+ x 2
5. If A = α β [
γ −α ]
is such that A2 = I , then
(A) 1 + α 2 + βγ (B) 1 – α 2 + βγ (C) 1 – α 2 – βγ (D) 1+ α 2 – βγ
6. If A be a non-singular matrix of order 3 , then | adj A | is equal to

(A) | A | (B) |A| 2 (C) |A|3 (D) 3 |A|

dy
7. If y = cos ( √ x ) , then =
dx

−sin( √ x ) sin( √ x )
(A) cos ( √x ) (B) - sin ( √x ) (C) (D)
2√x 2√x
II . FILL IN THE BLANKS BY CHOOSING FROM GIVEN BOX 3X1=3
[ 405 , 1 , 0 , 675 ]
8. Let A = { 1,2,3} . Then number of relations containing ( 1 , 2 ) and (1 , 3 ) which are reflexive and
symmetric but not transitive is ___
9. sin -1 ( 1 – x ) – 2 sin -1 x = π then x is equal to _____
2
10. If A and B are square matrices of order 3 and |A| = 5 , | B | = 3 then | 3 A B | = ____
III ANSWER ANY THREE QUESTIONS : 3X2=6
1
11. Write tan-1 ( ) , |x|> 1 in the simplest form .
√ x 2−1
3 24
12. Prove that 2 sin -1 = tan -1
5 7

13. If A = [−sin
cos α
α
sin α
cos α ] , then verify that A 1 A = I .
 14. If the area of the triangle whose vertices are ( -2 , 0 ) ,( 0 , 4 ) and ( 0 , k ) is 4 sq. units.

Find the values of k.

15. Differentiate sec ( tan ( √ x ) ) with respect to x .

IV ANSWER ANY THREE QUESTIONS : 3X3=9
16. Show that the relation R in the set A = { x ∈ℤ : 0 ≤ x ≤ 12 } , given by
R = { (a, b) : | a – b | is a multiple of 4 } is an equivalence relation .
17. Find gof and fog , if f : R→R and g : R→R are given by f(x) = cos x and g(x) = 3 x2 .

Show that gof ≠ fog .

18. Write the simplest form of tan - 1 ( √ 1+ x 2−1 ) ,x≠0

x
19. For any square matrix A with real numbers entries , Prove that A+ A1 is a symmetric matrix
and A - A1 is a skew symmetric matrix .

20. If the matrix A = [21 32] satisfies the equation A2 – 4 A + I = O then find A -1 .

V ANSWER ANY THREE QUESTIONS : 3 X 5 = 15

x−2
21. Let A = R – {3} and B = R – {1} . Consider f : A→ B defined by f(x) = .
x−3

Is f one-one and onto ? Justify your answer.

[ ]
1 0 2
22. If A = 0 2 1 , prove that A3 - 6 A2 + 7 A + 2I = 0
2 0 3

[]
1
23. If A = −4 B = [ -1 2 1] Verify that ( AB )1 = B1 A1
3
24. Solve system of linear equations using matrix method ,
x-y+z=4
2x + y - 3 z = 0
x+y+z=2 .

|x|+3 , if x ≤−3
25 . Find all points of the discontinuity of the function f(x) = { −2 x , if −3 < x<3
6 x+2 , if x ≥ 3

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