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SAMPLE CASPAM REGIONAL STUDENTS OLYMPIAD OF MATHEMATICS

CASPAM BAHAUDDIN ZAKARTYA UNIVERSITY MULTAN

MATCH (QF1)

1. How many real roots can the polynomial equation 2𝑥5 + 8𝑥 − 7 = 0 have?

a. None

b. One

c. Two

d. Four

2. Suppose that 𝑓 is differentiable on [0,1] and that its derivative is never zero. Then, which of the

following MUST be true?

A. 𝑓(0) ≠ 𝑓(1) B. 𝑓(0) = 𝑓(1) C. 𝑓(0) = 𝑓(𝑥) for some 𝑥 ≠ 0 D. None

3. In the xy-plane, the curve with parametric equations 𝑥 = 𝑐𝑜𝑠𝑡 and 𝑦 = 𝑠𝑖𝑛𝑡 0 ≤ 𝑡 ≤ 𝜋, has length

a. 3

b. 𝜋

c. 3𝜋

d. 𝜋2⁄

4. ∫ 𝑓(𝑥 − 1)6

0𝑑𝑥 =

a) ∫ 𝑓(𝑥)7

−1𝑑𝑥 =

b) ∫ 𝑓(𝑥)5

−1𝑑𝑥 =

c) ∫ 𝑓(𝑥 + 1)5

−1𝑑𝑥 =

d) ∫ 𝑓(𝑥)5

1𝑑𝑥 =

5. Which property of a group does not hold in set rational numbers with respect to

multiplication

a) Associative

b) Identity

c) Inverse

d) Closure

6. A group of order 13 must be: (A) abelian (B) cyclic (C) both A and B (D) none of these.

7. In a metric space X, which statement is not true in general

a) Infinite union of closed sets is closed

b) Infinite intersection of closed sets is closed

c) Infinite union of open sets is open

d) finite intersection of open sets is open.

8. In which of the following, the variables cannot be separated?



(a) sin( ) 0xye dx x y dy (b) sin cos 0x ye dx x y dy

(c) ( 1) 0xy y x dx ydy (d) None.

9. What is the correct transformation of the equation ln 𝑥 +1

𝑥2 = 5 to the form 𝑥 = 𝑔(𝑥)?

a) 𝑥 =1

5−

1

ln 𝑥 b) 𝑥 = √

1

5−

1

ln 𝑥

c) 𝑥 = 𝑒5 − 𝑒1

𝑥2 d) 𝑥 = 𝑒5+

1

𝑥2

10 A pair of fair dice is thrown twice. What is the probability of getting totals of 5 and 11. ?

(a) 1/81 (b) 2/81 (c) 3/81 (d) 4/81 (e) None of these



MATCH QF2

1. Which of the following is true for the behavior of 𝑓(𝑥) =𝑥3+8

𝑥2−4 as 𝑥 → 2 ?

a. The limit is 1

b. The limit is 0

c. The graph of the function has vertical asymptote at 2

d. The function has unequal, finite left hand and right hand limits.

2. If 𝑓(𝑥) = |𝑥| + 3𝑥2 for all real 𝑥, then 𝑓′(−1) =______

a. -7

b. -5

c. 5

d. 7

3. Let A be the region bounded by 𝑦 = 𝑙𝑛𝑥, the x-axis and the line e. Which of the following

represents the volume of solid generated when A is revolved around the y-axis?

a) 𝜋 ∫ (𝑒2 − 𝑒2𝑦)1

0𝑑𝑦

b) 𝜋 ∫ (𝑒2 − 𝑒𝑦)1

0𝑑𝑦

c) 𝜋 ∫ (𝑒2𝑦)1

0𝑑𝑦

d) 2𝜋 ∫ (𝑒2 − 𝑒𝑦)𝑒

0𝑑𝑦

4. ∫ 𝑥2 cos(𝑥3) 𝑑𝑥 =

a) −1

3sin(𝑥3) + 𝐴

b) −𝑥3

3sin(𝑥3) + 𝐴

c) 1

3sin(𝑥3) + 𝐴

d) 𝑥3

3sin(𝑥3) + 𝐴

5. Let a subset “S” of a vector space “V(F)” is a basis of “V”, then “S” is

a) Linearly dependent

b) Linearly independent

c) Subspace of “V”

d) Both (b) and (c).

6. If “U” and “W” are finite dimensional subspaces of a vector space “V” over the field “F”,

then dim(U+W)=

a) dim (U)+dim(W)

b) dim(U)-dim(W)

c) dim(U)+dim(W)-dim(UW)



d) dim(U)+dim(W)-dim(UW)

7. Each singleton in usual metric space R is

a) Open

b) Closed

c) Both open and closed

d) Neither open nor closed

8. Trapezoidal’s rule is applicable to

(a) Odd number of intervals

(b) Even number of intervals

(c) Any number of intervals

(d) None of the above

9. If two fair dices are thrown, what is the probability of getting a sum of 8 or more dots?


10. For limiting case ( 0y ) the differential equation ( ) 1 0y Sin y becomes

(a) Linear and homogeneous (b) Linear and non-homogeneous

(c) Non-linear and homogeneous (d) Non-linear and non-homogeneous



MATCH QF3

1. As the graph moves from left to right through the point 𝑐 = 2, the graph of 𝑓(𝑥) = 𝑥3 −

3𝑥 + 2 is

A. Rising B. Falling C. Constant D. not possible.

2. A function 𝑓(𝑥) has relative maximum at 𝑥 = 𝑎, if 𝑓′(𝑎) = 0 and

______………………………………

A. 𝑓′′(𝑎) = 0 B. 𝑓′′(𝑎) > 0 C. 𝑓′′(𝑎) < 0 D. None of these

3. The domain of 2 2

1( , )

4f x y

x y

is

( ) 1 1( ) 2 2 ( ) ( ,0) ( )a x b x y x c d None

4. What are the values of k such that ∫ 𝑥3𝑘

−2𝑑𝑡 = 0?

a) ‒2 and 2

b) 0

c) 2

d) 0 and 2

5. An integer is chosen at random from the first 100 positive integers. What is the probability

that the integer chosen is divisible by 4 or by 5 ?


6. If 1 2( (x), (x) ) 0W y y , then the functions 1(x)y and 2 (x)y is said to be

(a) Linearly independent (b) Linearly dependent (c) Separable (d) None

7. If 𝑈 and 𝑉 are 3-dimensional subspaces of ℝ5, what are the possible dimensions of the

subspace 𝑈 ∩ 𝑉

a. 0,1,2

b. 0, 1

c. 0,1,2, 3

d. 0

8. Let 𝑇: ℝ2 → ℝ2 be a linear transformation and 𝑇(2,3) = (0,0) then 𝑇 is

a. One-one

b. Onto

c. Non singular



d. None of these

9. The Fourier series of an even function on the interval (-p,p) is the

(a) Fourier cosine series (b) Fourier sine series (c) inverse Fourier series (d) none

10. In usual metric space R open sphere is a

a) Open disc

b) Open ball

c) Open interval

d) Closed interval



MATCH QF4

1. The 𝑓(𝑥) = −3𝑥2 + 4𝑥 + 5 has relative minimum value at 𝑥 = ______

A. 2

3 B)

2

6 C.

3

2 D. None of A, B & C

2. If at every point of the curve C the tangent line has a slope 2x and C passes through (0,1). Then the value

of C at k+1 is

a. 𝑘2 + 𝑘 + 2

b. 𝑘2 + 2𝑘 + 2

c. 𝑘2 − 𝑘 + 2

d. Does not exists

3. An equation of y in terms of x satisfy 𝑑𝑦 = 𝑥2𝑑𝑥 and the curve passes through (1, 1). If x=2 then

y=______

a. 3

b. 73⁄

c. 103⁄

d. 11

4. The table gives values of 𝑓, 𝑓′, 𝑔 𝑎𝑛𝑑 𝑔′ for selected values of x.

If ∫ 𝑓′(𝑥)𝑔(𝑥)1

0𝑑𝑥 = 5, then ∫ 𝑓(𝑥)𝑔′(𝑥)

1

0𝑑𝑥 =

a) ‒14

b) ‒13

c) 7

d) ‒2

5. If A and B are invertible matrices. Then which must be invertible (a) AB (b) A+B (c) BA (d) Both (a) and (c)

6. If A is Hermitian Matrix then which must be true

(a) A is symmetric. (b) 𝐴 + �̅�𝑡 is Hermitian. (c) A is positive definite. (d) All are true.

7. Three players A, B, C are in a race. A is twice as likely to win as B and B is twice as likely to win as C.

What is the probability that B wins ?

(b) (a) 1/7 (b) 2/7 (c) 3/7 (d) 4/7 (e) None of these



8. Let d be a discrete metric on 2R and 2, Rba at )1,1(a & )6,4(b then ?)( abd

a) 6

b) 0

c) 1

d)

e) None

9. If 33 8 4sinxy y e x , then by undetermined coefficient method py

(a) 3 cos sinxAe B x C x (b) 2 3 cos sinxA x e B x C x

(c) 3 cos sin 2xAe B x C x (d) None.

10. The Fourier series of an even function on the interval (-p,p) is the

(a) Fourier cosine series (b) Fourier sine series (c) inverse Fourier series (d) none



MATCH SF1

1. If 𝑓′(𝑥) = 𝑔′(𝑥) for x in an interval (a, b), then which of the following is true?

a. 𝑓 − 𝑔 is constant on (a, b)

b. 𝑓 − 𝑔 is continuous on (a, b)

c. 𝑓 − 𝑔 is differentiable on (a, b)

d. Both b and c

2. If 𝑓′′(𝑥) = 𝑓′(𝑥), for all real x, and 𝑓(0) = 0, 𝑓′(0) = −1, then 𝑓(𝑥) =______

a. 1 − 𝑒𝑥

b. 𝑒𝑥 − 1

c. 𝑒−𝑥 − 1

d. 𝑒𝑥

3. An equation of y in terms of x satisfy 𝑑𝑦 = 𝑥2𝑑𝑥 and the curve passes through (1, 1). If x=2 then

y=______

i. 3

ii. 73⁄

iii. 103⁄

iv. 11

4. Let f and g be continuous functions for a ≤ x ≤ b. If a < c < b ∫ 𝑓(𝑥)𝑏

𝑎𝑑𝑥 = 𝑃, ∫ 𝑓(𝑥)

𝑑

𝑐𝑑𝑥 =

𝑄, ∫ 𝑔(𝑥)𝑏

𝑎𝑑𝑥 = 𝑅 and ∫ 𝑔(𝑥)

𝑑

𝑐𝑑𝑥 = 𝑆, then ∫ (𝑓(𝑥) − 𝑔(𝑥))

𝑐

𝑎𝑑𝑥 =

a) P‒Q+R‒S

b) P‒Q‒R+S

c) P‒Q‒R‒S

d) P+Q‒R+S

5. If a number is correct to n-decimal places, then absolute error is:

a) ≤1

210−𝑛 b) ≤

1

2101−𝑛

c) ≤1

210𝑛+1 d) None of them

6. Two mutually exclusive events

(a) Always occur together (b) can never occur together (c) sometime occur together (d) None of these

7. In a metric space each open sphere is

a) Open set

b) Closed set



c) Both open and closed

d) Neither open nor closed

8. In Fourier series ( )f x is real and ( ) 0f x as

( ) 0 ( ) ( ) 0 ( )a x b x c x d none

9. If 1 2( ) cos3 and y ( ) sin 3y x x x x ,then their Wronskian 1 2( (x), (x) )W y y

(a) cos x (b) sin x (c) 3 (d) None

10. The set of integers is a cyclic group under addition and its generator is:

(A) 0 (B) 1 (C) 2 (D) None of these.



MATCH SF2

1. Which of the following functions is increasing?

(A) 𝑓(𝑥) = −ln𝑥 (B) 𝑓(𝑥) = 𝑒𝑥 (C) 𝑓(𝑥) = 𝑥2

(D) All of of these.

2. The line that is normal to the curve 𝑥2 + 2𝑥𝑦 − 3𝑦2 =0 at (1,1) is______.

A. 𝑥 + 𝑦 + 2 = 0 B. 𝑥 + 𝑦 − 2 = 0 C. 𝑥 − 𝑦 + 2 = 0 D. 𝑥 + 𝑦 + 1 = 0

3. Which one is the domain of the function 𝑓(𝑥, 𝑦) = √𝑙𝑛(𝑥2 + 𝑦2)

a- 𝑅2 b- 𝑅2\{(0,0)} c- 𝑅2\{(𝑥, 𝑦): 𝑥2 + 𝑦2 ≤ 1} d- 𝑅2\{(𝑥, 𝑦): 𝑥2 + 𝑦2 ≥ 1}

4. What is the average value of 𝑦 = 𝑠𝑖𝑛2𝑥 over [𝜋

4,

𝜋

3].

a) 6

𝜋

b) −1

6𝜋

c) 3

𝜋

d) 6

𝜋

5. What is the rate of convergence of Secant’s method?

a) 0.6 b) 2.0

c) 1.0 d) 1.6

6. In Fourier series f and f are

(a) Uniform continuous (b) piecewise continuous (c) discontinuous (d) none

7. In a metric space ),( dX and XBA , then which one is not true in general

a) )()()( BClAClBACl

b) )()()( BClAClBACl

c) )()()( BClAClBACl

d) )()()( BClAClBACl

8. If the function f has the property ( , ) ( , )nf tx ty t f x y for some real number n , then f is said to

be

(a) Separable function (b) Homogeneous function

(c) Non homogeneous function (d) None



9. Which of the following sets forms a basis for the Euclidean space E2?

(A) {(2; 1); (4; 2)} (B) {(0; 1); (0; 2)} (C) {(2; 1); (1; 2)} (D) {(0; 0); (x; y)}.

10. The number of linear independent vectors in the vector space of matrices M2_2

can not exceed:

(A) 1 (B) 2 (C) 3 (D) 4.



FINAL MACTH

1.. .Let 𝑓 be a continuous real-valued function defined on the closed interval [-2, 3]. Which of the

following is not necessarily true?

a. 𝑓 is bounded.

b. ∫ 𝑓(𝑡)𝑑𝑡3

−2 exists

c. For each k between f(-2) and f(3), there is an 𝑥 ∈ [−2,3] such that f(x)=k

d. limℎ→0

𝑓(ℎ)−𝑓(0)

ℎ exists

2. Let f be the function defined on the real line by

𝑓(𝑥) = {𝑥2⁄ if x is rational and 𝑥 3⁄ if x is irrational}.

If D is the set of points of discontinuity of 𝑓, then D is the

a. Set of real numbers b. Set of rational numbers c. Set of irrational numbers d. Set of nonzero real numbers

3. Describe and sketch a solid with the following properties.

When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are

parallel to the y-axis, its shadow is a rectangle. If the rays are parallel to the x-axis, its shadow

is an isosceles triangle.

Answer: This is a tube like a cream tube whose top face lies in the xy-plane and is closed and circular;

diameter of the top body is equal to that of the top face; bottom is a sharp edge parallel to x-axis

having length equal to the diameter of the top face.

4. Let 𝑔(𝑥) = ∫ 𝑓(𝑡)𝑥

𝑎𝑑𝑡, where a ≤ x ≤ b. The figure shows the graph of g on [a,b].

Which of the following could be the graph of f on [a,b].



Ans: a

5. Let R be the region bounded by 𝑦 = 3 − 𝑥2, 𝑦 = 𝑥3 + 1 and x=0. If R is rotated about the x-

axis, the volume of the solid formed could be determined by:

a) 𝜋 ∫ ((𝑥3 + 1)2 − (3 − 𝑥2)2)1

0𝑑𝑥

b) 𝜋 ∫ ((𝑥3 + 1)2 − (3 − 𝑥2)2)0

1𝑑𝑥

c) 2𝜋 ∫ 𝑥(−𝑥3 − 𝑥2 + 2)0

1𝑑𝑥

d) 𝜋 ∫ ((𝑥3 + 1)2 − (3 − 𝑥2)2)0

−1𝑑𝑥

6. The area of the shaded region in the preceding diagram is:



a) ∫ (𝑓(𝑥) − 𝑔(𝑥))𝑏

𝑎𝑑𝑥

b) ∫ (𝑔(𝑥) − 𝑓(𝑥))𝑏

𝑎𝑑𝑥

c) ∫ (𝑔(𝑥) + 𝑓(𝑥))𝑎

𝑏𝑑𝑥

d) ∫ (𝑓(𝑥) − 𝑔(𝑥))𝑎

𝑏𝑑𝑥

7. Let (X,d) be a metric space and {𝑥𝑛} and {𝑦𝑛} be any two sequences from “X”

then {d(𝑥𝑛,𝑦𝑛)} is a sequence in

a) 𝑅 × 𝑅

b) 𝑅

c) 𝐶 × 𝐶

d) 𝐶

8. Let d be a discrete metric on 2R and 2, Rba at )1,1(a & )6,4(b then ?)( abd

a. 6

b. 0

c. 1

d. None

9. A coin is tossed 4 times in succession. What is the probability that at least one head occurs. ?


10. Two coins are tossed. What is the conditional probability that two heads result, given that there is at

least one head ?

(a) 1/3 (b) 2/3 (c) 3/8 (d) 1 (e) None of these



11. Let 𝐴 and 𝐵 be two matrices of order 2 × 3 such that rank(𝐴) = 2 and rank(𝐵) = 2 then

rank(𝐴 + 𝐵) can be_______

a. 1

b. 3

c. 2

d. 4

12. Let 𝐴 be a 3 × 5 matrix over ℝ then 𝐴 is linear transformation__________

a. 𝐴: ℝ3 → ℝ5

b. 𝐴: ℝ5 → ℝ3

c. One-one

d. a and c

e. b and c

13. Given the two points [a, f(a)], [b,f(b)] the linear Lagrange polynomial 𝑓1(𝑥) that passes through these

two points is given by:

a) 𝑥−𝑏

𝑥−𝑎𝑓(𝑎) +

𝑥−𝑎

𝑎−𝑏𝑓(𝑏) b)

𝑥

𝑏−𝑎𝑓(𝑎) +

𝑥

𝑎−𝑏𝑓(𝑏)

c) 𝑓(𝑎) +𝑓(𝑏)−𝑓(𝑎)

𝑏−𝑎(𝑏 − 𝑎) d)

𝑥−𝑏

𝑎−𝑏𝑓(𝑎) +

𝑥−𝑎

𝑏−𝑎𝑓(𝑏)

14. Which of the following is most accurate for numerical integration?

(a) Boole’s rule

(b) Simpson’s rule

(c) Trapezoidal rule

(d) None of the above

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