# Pure Mathematics 2019 — CSS Past Paper

## PURE MATHEMATICS

TIME ALLOWED: THREE HOURS
PART-I(MCQS): MAXIMUM 30 MINUTES
PART-I (MCQS) MAXIMUM MARKS = 20
PART-II MAXIMUM MARKS = 80

NOTE:

• (i) Attempt ONLY FIVE questions from PART-II. ALL questions carry EQUAL marks.
• (ii) All the parts (if any) of each Question must be attempted at one place instead of at different places.
• (iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
• (iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
• (v) Extra attempt of any question or any part of the attempted question will not be considered.
• (vi) Use of calculator is allowed.

PART-II

SECTION-A

Q. No. 1.

(a)

Show that the order and the index of a subgroup divides the order of a finite group. (10)

(b)

Show that every finite integral domain is a field. (10) (20)

Q. No. 2.

(a)

Show that the characteristic of an integral domain is R is either zero or a prime. (10)

(b)

Determine whether or not the set {(1,2,−1),(0,3,1),(1,−5,3)} of vectors is a basis for 𝑅3. (10) (20)

Q. No. 3.

(a)

Show that a one-to-one linear transformation preserves basis and dimension. (10)

(b)

Solve the system of linear equations: 12312312325432225821.xxxxxxxxx++=−+=−+= (10) (20)

SECTION-B

Q. No. 4.

(a)

Solve ∫sin26𝑥cos43𝑥𝑑𝑥𝜋20. (10)

(b)

Find the area enclosed by 𝑦=62−cos𝜃. (10) (20)

Q. No. 5.

(a)

Show that in any conic semi-latusrectum is the harmonic mean between the segments of focal chord. (10)

(b)

Prove that the evolute of hyperbola
2𝑥𝑦=𝑎 is (𝑥+𝑦)23−(𝑥−𝑦)23=2𝑎23. (10) (20)

Q. No. 6.

(a)

Define Supremum and Infimum of a sequence. Find the supremum and infimum of the set
􁉄(−1)𝑛􁉀1−1𝑛􁉁,𝑛=1,2,3…􁉅. (10)

(b)Evaluatelim𝑥→0(1+𝑥)1𝑥−𝑒𝑥. (10) (20)

SECTION-C

Q. No. 7.

(a)

Show that 𝐿𝑜𝑔(1+cos𝜃+𝑖sin𝜃)=ln(2cos𝜃2)+𝑖𝜃2. (10)

(b)

Find 𝑣 such that 𝑓(𝑧)=𝑢+𝑖𝑣 is analytic. (10) (20)

Q. No. 8.

(a)

Prove that the series 𝑧(1−𝑧)+𝑧2(1−𝑧)+𝑧3(1−𝑧)+⋯ converges for |𝑧|<1, and find its sum. (10)

(b)

Find the residues of 𝑓(𝑧)=𝑧2−2𝑧(𝑧+1)2(𝑧2+4)at all its poles in the finite plane. (10)

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