CBSE Assistant Secretary 2024: Assistant Secretary, CBSE Recruitment Examination, Tier 2 (RAS-1124), Academics and Training, Mathematics
Text of page 1
M A S T E R C O P Y
No. of Printed Pages : 11
RAS-1124 •ŸÈ∑˝§◊Ê¥∑§
Roll No.
ATS
MATHEMATICS / ªÁáÊÃ
Question Paper / ¬˝‡Ÿ-¬òÊ
Time allowed : Three Hours Maximum Marks : 320
‚◊ÿ — ÃËŸ ÉÊ¥≈U •Áœ∑§Ã◊ •¥∑§ — 320
Question Paper may be detached which can be taken away by the candidate after examination.
¬˝‡Ÿ-¬òÊ •‹ª (ÁflÿÊíÿ) Á∑§ÿÊ ¡Ê ‚∑§ÃÊ „Ò •ÊÒ⁄U ¬⁄UˡÊÊÕ˸ ¬⁄UˡÊÊ ∑§ ©¬⁄Uʥà •¬Ÿ ‚ÊÕ ‹ ¡Ê ‚∑§Ã „Ò¥–
Part ‘A’ / π¥«U “•”
Part ‘A’ has 15 questions. All questions are compulsory. Each question carries 2 marks.
π¥«U “•” ◊¥ 15 ¬˝‡Ÿ „Ò¥, ÿ ‚÷Ë ¬˝‡Ÿ •ÁŸflÊÿ¸ „Ò¥– ¬˝àÿ∑§ ¬˝‡Ÿ 2 •¥∑§ ∑§Ê „Ò–
1. The equation of the type f1(x, p)=f2(y, p) does not have any __________ solution.
f1(x, p)=f2(y, p) ¬˝∑§Ê⁄U ∑§Ë ‚◊Ë∑§⁄UáÊ ∑§Ê ∑§Ê߸ __________ „‹ Ÿ„Ë¥ „ÊÃÊ „Ò–
2. If S is a closed surface, then ∫∫ r⋅n ds is...
s
ÿÁŒ S ∞∑§ ‚¥flÎà ¬Îc∆U „Ò, ÃÊ ∫∫ sr⋅n ds ...„Ò–
3. Can Newton-Raphson iteration method be used to solve the equation x1/3=0 ? Give reasons for
your answer.
ÄÿÊ ‚◊Ë∑§⁄UáÊ x1/3=0 ∑§Ê „‹ ∑§⁄UŸ ∑§ Á‹∞, ãÿÍ≈UŸ-⁄UÊ»§‚Ÿ ¬ÈŸ⁄UÊflÎÁûÊ ∑§Ê ©¬ÿʪ Á∑§ÿÊ ¡Ê ‚∑§ÃÊ „Ò? •¬Ÿ ©ûÊ⁄UÊ¥ ∑§
Á‹∞ ∑§Ê⁄UáÊ ŒËÁ¡∞–
4. A vector field is given by u=x2i+xyj+2t2k m/s. The acceleration at the point (2, 1, 0) meters and
t=2 s is ___________.
∑§Ê߸ ‚ÁŒ‡Ê ˇÊòÊ u=x2i+xyj+2t2k m/s mÊ⁄UÊ ÁŒÿÊ ¡ÊÃÊ „Ò– Á’¥ŒÈ (2, 1, 0) ◊Ë≈U⁄U •ÊÒ⁄U t=2 s ¬⁄U àfl⁄UáÊ ___________„Ò–
5. A vector field is given by u=y2i+2xyj−z2k. This vector field is :
(1) Irrotational and Solenoidal (2) Irrotational but not Solenoidal
(3) Not Irrotational but Solenoidal (4) Neither Irrotational nor Solenoidal
∑§Ê߸ ‚ÁŒ‡Ê ˇÊòÊ u=y2i+2xyj−z2k mÊ⁄UÊ ÁŒÿÊ ¡ÊÃÊ „Ò– ÿ„ ‚ÁŒ‡Ê ˇÊòÊ „Ò —
(1) •ÉÊÍáÊ˸ •ÊÒ⁄U ¬Á⁄UŸÊÁ‹∑§Ëÿ (2) •ÉÊÍáÊ˸ ¬⁄¥UÃÈ ¬Á⁄UŸÊÁ‹∑§Ëÿ Ÿ„Ë¥
(3) •ÉÊÍáÊ˸ Ÿ„Ë¥ ¬⁄¥UÃÈ ¬Á⁄UŸÊÁ‹∑§Ëÿ (4) Ÿ ÃÊ •ÉÊÍáÊ˸ •ÊÒ⁄U Ÿ „Ë ¬Á⁄UŸÊÁ‹∑§Ëÿ
RAS-1124 / ATS 1 Mathematics
Text of page 2
M A S T E R C O P Y
6. If the curvature (dT/ds) of a curve is zero then it will be __________.
ÿÁŒ Á∑§‚Ë fl∑˝§ ∑§Ë fl∑˝§ÃÊ (dT/ds) ‡ÊÍãÿ „Ò, ÃÊ fl„ __________„ʪ˖
7. ?R'×R''?/?R'?^3 is the formula for __________.
?R'×R''?/?R'?^3 __________ ∑§ Á‹∞ ‚ÍòÊ „Ò–
8. Let T : Rn → Rn be a linear transformation. Which of the following statements implies that T is
bijective ?
(1) Nullity (T)=n (2) Rank (T)=Nullity (T)=n
(3) Rank (T)+Nullity (T)=n (4) Rank (T)−Nullity (T)=n
◊ÊŸ ‹ËÁ¡∞ Á∑§ T : Rn → Rn ∞∑§ ⁄ÒUÁπ∑§ M§¬Ê¥Ã⁄UáÊ „Ò– ÁŸêŸÁ‹Áπà ◊¥ ‚ Á∑§‚ ∑§ÕŸ ∑§Ê ÁŸÁ„ÃÊÕ¸ „Ò Á∑§ T ∞∑Ò§∑§Ë
•Êë¿UÊŒË „Ò?
(1) ‡ÊÍãÿÃÊ (T)=n (2) ⁄Ò¥U∑§ (T)=‡ÊÍãÿÃÊ (T)=n
(3) ⁄Ò¥U∑§ (T)+‡ÊÍãÿÃÊ (T)=n (4) ⁄Ò¥U∑§ (T)−‡ÊÍãÿÃÊ (T)=n
9. The system of equations x+y+z=1, 2x+3y−z=5, x+2y−kz=4. Where k∈R, has an infinite
number of solutions for :
(1) k=0 (2) k=1 (3) k=2 (4) k=3
‚◊Ë∑§⁄UáÊÊ¥ x+y+z=1, 2x+3y−z=5, x+2y−kz=4 ∑§ ÁŸ∑§Êÿ, ¡„Ê° k∈R „Ò, ∑§ ÁŸêŸÁ‹Áπà ∑§ Á‹∞ •¬Á⁄UÁ◊Ã
‚¥ÅÿÊ ∑§ „‹ „Êà „Ò¥ —
(1) k=0 (2) k=1 (3) k=2 (4) k=3
10. Which of the following subset of R2 are Compact ?
(1) {(x, y) : ?x?≤1, ?y?/2} (2) {(x, y) : ?x?≤1, ?y?2≤2}
(3) {(x, y) : x2+3y2≤5} (4) {(x, y) : x2≤y2+5}
R2 ∑§Ê ÁŸêŸÁ‹Áπà ◊¥ ‚ ∑§ÊÒŸ-‚Ê ©¬‚◊ÈìÊÿ ‚¥„à „Ò?
(1) {(x, y) : ?x?≤1, ?y?/2} (2) {(x, y) : ?x?≤1, ?y?2≤2}
(3) {(x, y) : x2+3y2≤5} (4) {(x, y) : x2≤y2+5}
11. The function f : C→ C defined by f=ez+e−z has :
(1) Finitely many zeros (2) No zeros
(3) Only real zeros (4) Has infinitely many zeros
f=ez+e−z ‚ ¬Á⁄U÷ÊÁ·Ã »§‹Ÿ f : C→ C —
(1) ∑§ ¬Á⁄UÁ◊à M§¬ ‚ •Ÿ∑§ ‡ÊÍãÿ∑§ „Ò¥– (2) ∑§Ê ∑§Ê߸ ‡ÊÍãÿ∑§ Ÿ„Ë¥ „Ò–
(3) ∑§ ∑§fl‹ flÊSÃÁfl∑§ ‡ÊÍãÿ∑§ „Ò¥– (4) ∑§ •¬Á⁄UÁ◊à M§¬ ‚ •Ÿ∑§ ‡ÊÍãÿ∑§ „Ò¥–
RAS-1124 / ATS 2 Mathematics
Text of page 3
M A S T E R C O P Y
12. Let f(x) be non-constant entire function and z=x+iy. Let u(x, y), v(x, y) denote its real and
imaginary parts respectively. Which of the following statement is FALSE ?
(1) ux=vy and uy=vx (2) uy=vx and ux=−vy
(3) ? f '(x+iy)? =u x(x, y)+v x(x, y) (4) ? f '(x+iy)?2=u2y(x, y)+v2y(x, y)
2 2 2
◊ÊŸ ‹ËÁ¡∞ Á∑§ f(x) ∞∑§ ‚¥¬Íáʸ »§‹Ÿ „Ò, ¡Ê •ø⁄U Ÿ„Ë¥ „Ò ÃÕÊ z=x+iy „Ò– ÿ„ ÷Ë ◊ÊŸ ‹ËÁ¡∞ Á∑§ u(x, y) •ÊÒ⁄U
v(x, y) ∑˝§◊‡Ê— ß‚∑§ flÊSÃÁfl∑§ •ÊÒ⁄U ∑§ÊÀ¬ÁŸ∑§ ÷ʪʥ ∑§Ê √ÿÄà ∑§⁄Uà „Ò¥– ÁŸêŸÁ‹Áπà ◊¥ ‚ ∑§ÊÒŸ-‚Ê ∑§ÕŸ •‚àÿ „Ò?
(1) ux=vy •ÊÒ⁄U uy=vx (2) uy=vx •ÊÒ⁄U ux=−vy
(3) ? f '(x+iy)?2=u2x(x, y)+v2x(x, y) (4) ? f '(x+iy)?2=u2y(x, y)+v2y(x, y)
13. Let f : C → C be a complex valued function of the form f (x, y)=u(x, y)+iv(x, y). Suppose that
u(x, y)=3x2y. Then :
(1) f can not be holomorphic on C for any choice of v
(2) f is holomorphic on C for a suitable choice of v
(3) f is holomorphic on C for all choice of v
(4) u is not differentiable
◊ÊŸ ‹ËÁ¡∞ Á∑§ f : C → C ∞∑§ ‚Áê◊üÊ ◊ÊŸ »§‹Ÿ f (x, y)=u(x, y)+iv(x, y) ∑§ M§¬ ∑§Ê „Ò– ◊ÊŸ ‹ËÁ¡∞ Á∑§
u(x, y)=3x2y „Ò– Ã’,
(1) v ∑§ Á∑§‚Ë ÷Ë Áfl∑§À¬ ∑§ Á‹∞, C ¬⁄U f „Ê‹Ê◊ÊÁ»¸§∑§ Ÿ„Ë¥ „Ê ‚∑§ÃÊ–
(2) v ∑§ ∞∑§ ©¬ÿÈÄà Áfl∑§À¬ ∑§ Á‹∞, C ¬⁄U f „Ê‹Ê◊ÊÁ»¸§∑§ „Ò–
(3) v ∑§ ‚÷Ë Áfl∑§À¬Ê¥ ∑§ Á‹∞ C ¬⁄U f „Ê‹Ê◊ÊÁ»¸§∑§ „Ò–
(4) u •fl∑§‹ŸËÿ Ÿ„Ë¥ „Ò–
14. A heavy body rests on a smooth plane inclined to the horizon at an angle α, being held by a string
to which a force P is applied. If the normal reaction of the plane is equal to the force P in magnitude,
the inclination of P to the inclined plane is :
(1) 2α (2) 90 - 2α (3) α (4) 90 - α
∞∑§ ÷Ê⁄UË Á¬¥«U ∞∑§ Áø∑§Ÿ ‚◊Ë ¬⁄U Á≈U∑§Ê „È•Ê „Ò, Á¡‚ ∞∑§ «UÊ⁄UË ‚ ’Ê¥œÊ „È•Ê „Ò, Á¡‚ ¬⁄U ∞∑§ ’‹ P •ŸÈ¬˝ÿÊÁªÃ Á∑§ÿÊ
ªÿÊ „Ò– ÿ„ ‚◊Ë ˇÊÒÁá ‚ ∑§ÊáÊ α ¬⁄U ŸÁà (¤ÊÈ∑§Ê „È•Ê) „Ò– ÿÁŒ ‚◊Ë ∑§Ë •Á÷‹¥’ ¬˝ÁÃÁ∑˝§ÿÊ ¬Á⁄U◊ÊáÊ ◊¥ ’‹ P ∑§
’⁄UÊ’⁄U „Ò, ÃÊ ¤ÊÈ∑§ „È∞ ‚◊Ë ‚ P ∑§Ê ŸÁà ∑§ÊáÊ „Ò —
(1) 2α (2) 90 - 2α (3) α (4) 90 - α
15. Let G be a group with ?G?=96. Suppose H and K are subgroups of G with ?H?=12 and ?K?=16.
Then :
(1) H ∩ K={e} (2) H ∩ K ≠ {e}
(3) H ∩ K is Abelian (4) H ∩ K is not Abelian
◊ÊŸ ‹ËÁ¡∞ Á∑§ G ∞∑§ ‚◊Í„ „Ò ÃÊÁ∑§ ?G?=96 „Ò– ◊ÊŸ ‹ËÁ¡∞ Á∑§ ?H?=12 •ÊÒ⁄U ?K?=16 ∑§ ‚ÊÕ G ∑§ H •ÊÒ⁄U K
©¬‚◊Í„ „Ò¥– Ã’ —
(1) H ∩ K={e} (2) H ∩ K ≠ {e}
(3) H ∩ K •Ê’‹Ë „Ò (4) H ∩ K •Ê’‹Ë Ÿ„Ë¥ „Ò
RAS-1124 / ATS 3 Mathematics
Text of page 4
M A S T E R C O P Y
Part ‘B’ / π¥«U “’”
Part ‘B’ has 25 questions. Out of which any 20 questions are to be answered. Each question carries
5 marks. The question may be answered in maximum 30 words.
π¥«U “’” ◊¥ 25 ¬˝‡Ÿ „Ò¥, Á¡Ÿ◊¥ ‚ Á∑§ã„Ë¥ 20 ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒŸ „Ò¥– ¬˝àÿ∑§ ¬˝‡Ÿ 5 •¥∑§ ∑§Ê „Ò– ¬˝‡Ÿ ∑§Ê ©ûÊ⁄U •Áœ∑§Ã◊
30 ‡ÊéŒÊ¥ ◊¥ ÁŒÿÊ ¡Ê ‚∑§ÃÊ „Ò–
2 x
16. Change the order of integration of ∫ ∫ dy dx .
−1 x 2−2
2 x
∫−1 ∫ x2−2 dy dx ∑§ ‚◊Ê∑§‹Ÿ ∑§ ∑˝§◊ ∑§Ê ’ŒÁ‹∞–
17. Find whether the series
3 4 5
2 + x + x 2 + x 3+.............
2 3 4
is convergent or divergent for positive values of x.
ôÊÊà ∑§ËÁ¡∞ Á∑§ üÊáÊË
3 4 5
2 + x + x 2 + x 3+.............
2
3 4
x ∑§ œŸÊà◊∑§ ◊ÊŸÊ¥ ∑§ Á‹∞ •Á÷‚Ê⁄UË „Ò ÿÊ •¬‚Ê⁄UË „Ò–
18. State Lami’s theorem.
‹Ê◊Ë ∑§Ë ¬˝◊ÿ ∑§Ê ∑§ÕŸ ŒËÁ¡∞–
19. Let V be a vector space of all n×n matrices over the field F, and let B be a fixed matrix.
If T(A)=AB−BA∀A∈V verify that T is a linear transformation from V into V.
◊ÊŸ ‹ËÁ¡∞ Á∑§ ˇÊòÊ F ¬⁄U V ‚÷Ë n×n •Ê√ÿÍ„Ê¥ ∑§Ë ∞∑§ ‚ÁŒ‡Ê ‚◊Ác≈U „Ò ÃÕÊ B ∞∑§ ÁSÕ⁄U •Ê√ÿÍ„ „Ò– ÿÁŒ
T(A)=AB−BA∀A∈V „Ò, ÃÊ ‚àÿÊÁ¬Ã ∑§ËÁ¡∞ Á∑§ V ‚ V Ã∑§ T ∞∑§ ⁄ÒUÁπ∑§ M§¬Ê¥Ã⁄UáÊ „Ò–
20. If λ1, λ2, .....λn are the eigen values of A, then show that kλ1, kλ2, ........ kλn are the eigen values
of kA.
ÿÁŒ A ∑§ λ1, λ2, .....λn •ÊߪŸ ◊ÊŸ „Ò¥, ÃÊ Œ‡Êʸß∞ Á∑§ kA ∑§ kλ1, kλ2, ........ kλn •ÊߪŸ ◊ÊŸ „Ò¥–
21. Write down the matrix of the quadratic form x2−18xy+5y2 and verify that they can be written as
matrix products XtAX.
ÁmÉÊÊà M§¬ x2−18xy+5y2 ∑§Ê •Ê√ÿÍ„ Á‹Áπ∞ ÃÕÊ ‚àÿÊÁ¬Ã ∑§ËÁ¡∞ Á∑§ ßã„¥ •Ê√ÿÍ„ ªÈáÊŸ»§‹Ê¥ XtAX ∑§ M§¬ ◊¥ Á‹πÊ
¡Ê ‚∑§ÃÊ „Ò–
RAS-1124 / ATS 4 Mathematics
Text of page 5
M A S T E R C O P Y
22. Let (X, d) be a metric space and A ⊆ X, B ⊆ X. Then (A ∩ B)0=A0 ∩ B0.
◊ÊŸ ‹ËÁ¡∞ Á∑§ (X, d) ∞∑§ ŒÍ⁄UË∑§ ‚◊Ác≈U „Ò ÃÕÊ A ⊆ X, B ⊆ X „Ò– Ã’, (A ∩ B)0=A0 ∩ B0 „Ò–
23. Show that Q' (all irrational) with the usual metric is a separable metric space.
Œ‡Êʸß∞ Á∑§ ‚Ê◊Êãÿ ŒÍ⁄UË∑§ ∑§ ‚ÊÕ Q' (‚÷Ë •¬Á⁄U◊ÿ) ∞∑§ ¬ÎÕÄ∑§⁄UáÊËÿ ŒÍ⁄UË∑§ ‚◊Ác≈U „Ò–
24. Find the semi-major axis, semi-minor axis and apsidal distances of the Earth, assuming its period
to be 365.25 days. You may assume that the eccentricity of the Earth’s orbit to be 0.0167 and the
mass of the Sun to be 1.989×1030 kg.
¬ÎâflË ∑§Ë •œ¸-ŒËÉʸ •ˇÊ, •œ¸-‹ÉÊÈ •ˇÊ ÃÕÊ SÃÁéœ∑§Ê ŒÍÁ⁄UÿÊ° ôÊÊà ∑§ËÁ¡∞; ¬Á⁄U∑§À¬ŸÊ ∑§ËÁ¡∞ Á∑§ ß‚∑§Ê •ÊflÃ˸ ∑§Ê‹
365.25 ÁŒŸ „Ò– •Ê¬ ¬Á⁄U∑§À¬ŸÊ ∑§⁄U ‚∑§Ã „Ò¥ Á∑§ ¬ÎâflË ∑§ ¬Á⁄U¬Õ ∑§Ë ©à∑§ãŒ˝ÃÊ 0.0167 „Ò ÃÕÊ ‚Íÿ¸ ∑§Ê Œ˝√ÿ◊ÊŸ
1.989×1030 kg „Ò–
25. Estimate the mass of the Sun, assuming the orbit of the Earth around the Sun to be a circle. The
distance between the Sun and the Earth is 1.49×1011 m, and G=6.67×10−11 m3kg−1.
‚Íÿ¸ ∑§ øÄ∑§⁄U ‹ªÊà ‚◊ÿ, ¬ÎâflË ∑§ ¬Á⁄U¬Õ ∑§Ê ∞∑§ flÎûÊ ◊ÊŸÃ „È∞, ‚Íÿ¸ ∑§ Œ˝√ÿ◊ÊŸ ∑§Ê •Ê∑§‹Ÿ ∑§ËÁ¡∞– ‚Íÿ¸ •ÊÒ⁄U ¬ÎâflË
∑§ ’Ëø ∑§Ë ŒÍ⁄UË 1.49×1011 m „Ò ÃÕÊ G=6.67×10−11 m3kg−1 „Ò–
∂2 z
26. If xxyyzz=C, then show that =−(xlog ex )−1 , when x=y=z.
∂x∂y
∂2 z −1
ÿÁŒ xxyyzz=C „Ò, ÃÊ Œ‡Êʸß∞ Á∑§ ∂x∂y =−(xlog ex ) „Ò, ¡’ x=y=z „Ò–
27. Use the factorization :
s4+4a4=(s2−2as+2a2) (s2+2as+2a2) and apply inverse Laplace transform to show that :
1 1
L−1 4 4
= 3 (cosh at sin at−sinh at cos at)
s + 4a 4a
ÿ„ Œ‡ÊʸŸ ∑§ Á‹∞ Á∑§
1 1
L−1 4 4
= 3 (cosh at sin at−sinh at cos at) „Ò , ªÈ á ÊŸπ¥ « U Ÿ s 4 +4a 4 =(s 2 −2as+2a 2 )
s + 4a 4a
(s2+2as+2a2) ∑§Ê ©¬ÿʪ ∑§ËÁ¡∞ ÃÕÊ ¬˝ÁËÊ◊ ‹Êå‹Ê‚ M§¬Ê¥Ã⁄U ∑§Ê •ŸÈ¬˝ÿʪ ∑§ËÁ¡∞–
RAS-1124 / ATS 5 Mathematics
Text of page 6
M A S T E R C O P Y
28. Consider the data set
x −3 1 2 5
y −23 −11 −23 1
As the polynomials f (x)=x3−3x2−10x+1 and g(x)=−23+(x+3) (x−1)+(x−2) (x+3) (x−1)
both interpolate all of the data. Why does this not contradict the uniqueness part of the theorem
on existence and uniqueness of polynomial interpolation ?
ÁŸêŸÁ‹Áπà •Ê°∑§«∏Ê¥ ∑§ ‚◊ÈìÊÿ ¬⁄U ÁfløÊ⁄U ∑§ËÁ¡∞ —
x −3 1 2 5
y −23 −11 −23 1
¡’ Á∑§ ŒÊŸÊ¥ ’„ȬŒ f (x)=x3−3x2−10x+1 •ÊÒ⁄U g(x)=−23+(x+3) (x−1)+(x−2) (x+3) (x−1) ‚÷Ë
•Ê°∑§«∏Ê¥ ∑§Ê •¥Ãfl¸‡ÊŸ ∑§⁄Uà „Ò¥– ÿ„ ’„ȬŒ •¥Ãfl¸‡ÊŸ ∑§ •ÁSÃàfl •ÊÒ⁄U ©‚∑§Ë •ÁmÃËÿÃÊ ¬⁄U ¬˝◊ÿ ∑§ •ÁmÃËÿÃÊ flÊ‹ ÷ʪ
∑§Ê •¥ÃÁfl¸⁄UÊœ ÄÿÊ¥ Ÿ„Ë¥ ∑§⁄UÃÊ „Ò?
29. If H is a subgroup of G and N is a normal subgroup of G, then H ∩ N is a normal subgroup of H.
ÿÁŒ G ∑§Ê H ∞∑§ ©¬‚◊Í„ „Ò ÃÕÊ G ∑§Ê N ∞∑§ ¬˝‚Ê◊Êãÿ ©¬‚◊Í„ „Ò, ÃÊ H ∑§Ê H ∩ N ∞∑§ ¬˝‚Ê◊Êãÿ ©¬‚◊Í„ „Ò–
30. Prove that a group G is abelian if the mapping f : G → G, given by f (x)=x2, is a homomorphism.
Á‚h ∑§ËÁ¡∞ Á∑§ ∑§Ê߸ ‚◊Í„ G •Ê’‹Ë „ÊÃÊ „Ò, ÿÁŒ f (x)=x2 mÊ⁄UÊ ÁŒÿÊ ¡ÊŸ flÊ‹Ê ¬˝ÁÃÁøòÊáÊ f : G → G ‚◊Ê∑§Ê⁄UË „ÊÃÊ „Ò–
31. Prove that the function f(z)=xy+iy is everywhere continuous but not analytic.
Á‚h ∑§ËÁ¡∞ Á∑§ »§‹Ÿ f(z)=xy+iy ¬˝àÿ∑§ SÕÊŸ ¬⁄U ‚¥Ãà „Ò ¬⁄¥UÃÈ flÒ‡‹Á·∑§ Ÿ„Ë¥ „Ò–
32. Consider the transformation w=2z, and determine the region R' of w-plane into which the
triangular region R enclosed by the lines x=0, y=0, x+y=1 in the z plane is mapped under the
map.
M§¬Ê¥Ã⁄UáÊ w=2z ¬⁄U ÁfløÊ⁄U ∑§ËÁ¡∞ ÃÕÊ w-‚◊Ë ∑§Ê fl„ ˇÊòÊ R' ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞, Á¡‚◊¥ ⁄UπÊ•Ê¥ x=0, y=0,
x+y=1 mÊ⁄UÊ z-‚◊Ë ◊¥ ¬Á⁄U’h ÁòÊ÷È¡Ëÿ ˇÊòÊ R ß‚ ¬˝ÁÃÁøòÊáÊ ∑§ •¥Ãª¸Ã ¬˝ÁÃÁøÁòÊà „ÊÃÊ „Ò–
z
33. Find the fixed points and the normal form of the bilinear transformation w = .
z +1
z
Ám⁄ÒUÁπ∑§ M§¬Ê¥Ã⁄UáÊ w = ∑§ ÁSÕ⁄U Á’¥ŒÈ ÃÕÊ ß‚∑§Ê ¬˝‚Ê◊Êãÿ M§¬Ê¥Ã⁄UáÊ ôÊÊà ∑§ËÁ¡∞–
z +1
RAS-1124 / ATS 6 Mathematics
Text of page 7
M A S T E R C O P Y
34. Show that a finite group G of composite order has a non-trivial subgroup.
Œ‡Êʸß∞ Á∑§ ‚¥ÿÈÄà ∑§ÊÁ≈U ∑§ ∞∑§ ¬Á⁄UÁ◊à ‚◊Í„ G ∑§Ê ∞∑§ •ÃÈë¿U ©¬‚◊Í„ „ÊÃÊ „Ò–
35. If F is the field of real numbers, prove that the vectors (a1, a2) and (b1, b2) in V2(F) are linearly
dependent if a1b2−a2b1=0.
ÿÁŒ F flÊSÃÁfl∑§ ‚¥ÅÿÊ•Ê¥ ∑§Ê ˇÊòÊ „Ò, ÃÊ Á‚h ∑§ËÁ¡∞ Á∑§ V2(F) ◊¥ ‚ÁŒ‡Ê (a1, a2) •ÊÒ⁄U (b1, b2) ⁄ÒUÁπ∑§Ã— •ÊÁüÊà „Ò¥, ÿÁŒ
a1b2−a2b1=0 „Ò–
36. If H ⊆ K be two subgroup of a finite group G, then show that [G : H]=[G : K] [K : H].
ÿÁŒ ∞∑§ ¬Á⁄UÁ◊à ‚◊Í„ G ∑§ ŒÊ ©¬‚◊Í„ H ⊆ K „Ò¥, ÃÊ Œ‡Êʸß∞ Á∑§ [G : H]=[G : K] [K : H] „Ò–
37. Let R be the ring of all real valued, continuous function on [0, 1]. Show that the set
1
S = f ∈R : f = 0 is an ideal of R.
2
◊ÊŸ ‹ËÁ¡∞ Á∑§ [0, 1] ¬⁄U R ‚÷Ë flÊSÃÁfl∑§ ◊ÊŸ flÊ‹ ‚Ãà »§‹ŸÊ ¥ ∑§Ê fl‹ÿ „Ò – Œ‡Êʸ ß ∞ Á∑§ ‚◊È ì Êÿ
1
S = f ∈R : f = 0 , R ∑§Ê ∞∑§ ªÈáÊ¡Êfl‹Ë „Ò–
2
38. A boat is moved uniformly along a canal by two horses pulling with forces
P=890 N and Q=1068 N acting under an angle α=608. Determine the magnitude of the resultant
pull on the boat and the angles β and v.
Á∑§‚Ë Ÿ„⁄U ◊¥ ∞∑§ ŸÊÒ∑§Ê ∑§Ê ŒÊ ÉÊÊ«∏Ê¥ mÊ⁄UÊ ’‹Ê¥ P=890 N •ÊÒ⁄U Q=1068 N mÊ⁄UÊ ∞∑§‚◊ʟ× πË¥øÊ ¡Ê ⁄U„Ê „Ò– ÿ ’‹
∑§ÊáÊ α=608 ∑§ •¥Ãª¸Ã ∑§Êÿ¸ ∑§⁄U ⁄U„ „Ò¥– ß‚ ŸÊÒ∑§Ê ∑§Ê πË¥øŸ ◊¥ ‹ª ¬Á⁄UáÊÊ◊Ë ’‹ ∑§Ê ¬Á⁄U◊ÊáÊ ÃÕÊ ∑§ÊáÊ β •ÊÒ⁄U v
ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞–
39. If the piston of the engine has a diameter of 101.6 mm and the gas pressure in the cylinder is
0.69 MPa. Calculate the turning moment M exerted on the crankshaft for the particular
configuration.
ÿÁŒ ߥ¡Ÿ ∑§ Á¬S≈UŸ ∑§Ê √ÿÊ‚ 101.6 mm „Ò ÃÕÊ Á‚Á‹¥«U⁄U ¬⁄U ªÒ‚ ∑§Ê ŒÊ’ 0.69 MPa „Ò, ÃÊ ß‚ ÁflÁ‡Êc≈U ÁflãÿÊ‚ ∑§ Á‹∞
∑˝Ò§ã∑§‡ÊÊç≈U ¬⁄U ‹ªŸ flÊ‹ fløŸ •ÊÉÊÍáʸ ∑§Ê ¬Á⁄U∑§Á‹Ã ∑§ËÁ¡∞–
40. Consider the triangle of base ‘b’ and height ‘h’. Determine the distance of centroid from the base.
•ÊœÊ⁄U ‘b’ •ÊÒ⁄U ™°§øÊ߸ ‘h’ ∑§ ÁòÊ÷È¡ ¬⁄U ÁfløÊ⁄U ∑§ËÁ¡∞– •ÊœÊ⁄U ‚ ∑§ãŒ˝∑§ ∑§Ë ŒÍ⁄UË ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞–
RAS-1124 / ATS 7 Mathematics
Text of page 8
M A S T E R C O P Y
Part ‘C’ / π¥«U “‚”
Part ‘C’ has 12 questions. Out of which any 10 questions are to be answered. Each question carries
10 marks. The question may be answered in maximum 100 words.
π¥«U “‚” ◊¥ 12 ¬˝‡Ÿ „Ò¥, Á¡Ÿ◊¥ ‚ Á∑§ã„Ë¥ 10 ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒŸ „Ò¥– ¬˝àÿ∑§ ¬˝‡Ÿ 10 •¥∑§ ∑§Ê „Ò– ¬˝‡Ÿ ∑§Ê ©ûÊ⁄U •Áœ∑§Ã◊
100 ‡ÊéŒÊ¥ ◊¥ ÁŒÿÊ ¡Ê ‚∑§ÃÊ „Ò–
41. Find the specific solution of the differential equation
2
( 1 − x2 ) ddxu2 − 2x du
dx
+ 12 u = 0 if u’(0)=4
and the function u(x) is well behaved at x=1.
d2 u du
•fl∑§‹ ‚◊Ë∑§⁄UáÊ ( 1 − x 2 ) 2
− 2x
dx
+ 12 u = 0 ∑§Ê ÁflÁ‡Êc≈U „‹ ôÊÊà ∑§ËÁ¡∞, ÿÁŒ u’(0)=4 „Ò ÃÕÊ x=1
dx
¬⁄U »§‹Ÿ u(x) ‚È•ÊøÁ⁄Uà „Ò–
42. If A3 be the subgroup of S3 consisting of all even permutations, show that A3 is normal subgroup
1
of S3 and O(A 3 ) = O(S 3 ) . Further show that S3/A3 is cyclic.
2
ÿÁŒ A3 ‚◊Í„ S3 ∑§Ê ©¬‚◊Í„ „Ò Á¡‚∑§ ‚÷Ë ∑˝§◊øÿ ‚◊ „Ò¥, ÃÊ Œ‡Êʸß∞ Á∑§ A3 ‚◊Í„ S3 ∑§Ê ‚Ê◊Êãÿ ©¬‚◊Í„ „Ò ÃÕÊ
1
O(A 3 ) = O(S 3 ) „Ò– •Êª, Œ‡Êʸß∞ Á∑§ S3/A3 ø∑˝§Ëÿ „Ò–
2
43. Gauss-Seidel Method is used to solve the system of equations
4 −1 1 x 7
4 −8 1 y = −21
−2 1 5 z 15
Determine the rate of convergence of the method and the number of iterations needed to make
(k )
max ε i ≤ 10−2 .
i
‚◊Ë∑§⁄UáÊ-ÁŸ∑§Êÿ
4 −1 1 x 7
4 −8 1 y = −21
∑§Ê „‹
−2 1 5 z 15
∑§⁄UŸ ∑§ Á‹∞ ªÊÚ‚-‚Ë«U‹ ÁflÁœ ∑§Ê ©¬ÿʪ Á∑§ÿÊ ¡ÊÃÊ „Ò–
(k )
ß‚ ÁflÁœ ∑§Ë •Á÷‚ÊÁ⁄UÃÊ ∑§Ë Œ⁄U ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞ ÃÕÊ ÿ„ ÷Ë ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞ Á∑§ max εi ≤ 10−2 ’ŸÊŸ ∑§ Á‹∞
i
Á∑§ÃŸË ¬ÈŸ⁄UÊflÎÁûÊÿÊ¥ ∑§Ë •Êfl‡ÿ∑§ÃÊ „Ò–
RAS-1124 / ATS 8 Mathematics
Text of page 9
M A S T E R C O P Y
44. A tight string 2 m long with a=30 m/s is initially at rest but is given an initial velocity of
1
300 sin4πx from its equilibrium position. Determine the maximum displacement at the x = m
8
location of the string.
2 m ‹¥’Ë ∞∑§ ÃŸË „È߸ «UÊ⁄UË a=30 m/s ∑§ ‚ÊÕ ¬˝Ê⁄¥U÷ ◊¥ ÁflüÊÊ◊ ∑§Ë ÁSÕÁà ◊¥ „Ò– ¬⁄¥UÃÈ ß‚∑§Ë ‚¥ÃÈ‹Ÿ ∑§Ë ÁSÕÁà ‚ ß‚◊¥
1
∞∑§ 300 sin4πx ∑§Ê ∞∑§ ¬˝Ê⁄¥UÁ÷∑§ flª ¬˝ŒÊŸ ∑§⁄U ÁŒÿÊ ¡ÊÃÊ „Ò– «UÊ⁄UË ∑§Ë •flÁSÕÁà x = m ¬⁄U •Áœ∑§Ã◊ ÁflSÕʬŸ
8
ÁŸœÊ¸Á⁄Uà ∑§ËÁ¡∞–
x
45. Use u = f ( ξ), ξ = to solve the parabolic system
4kt
ut=kuxx, −∞<x<∞, t>0
u(x, 0)=0, x<0
u(x, 0)=u0, x>0
where k and u0 are constant
¬⁄Ufl‹Áÿ∑§ ÁŸ∑§Êÿ
ut=kuxx, −∞<x<∞, t>0
u(x, 0)=0, x<0
u(x, 0)=u0, x>0
x
∑§Ê „‹ ∑§⁄UŸ ∑§ Á‹∞, u = f ( ξ), ξ = ∑§Ê ©¬ÿʪ ∑§ËÁ¡∞, ¡„Ê° k •ÊÒ⁄U u0 •ø⁄U „Ò¥–
4kt
i −w
46. In the transformation z = , show that half of w-plane given by v/0 corresponds to the
i +w
circle ?z?≤1 in z-plane.
i −w
M§¬Ê¥Ã⁄UáÊ z = ◊¥, Œ‡Êʸß∞ Á∑§ v/0 mÊ⁄UÊ ÁŒ∞ ¡ÊŸ flÊ‹Ê w-‚◊Ë ∑§Ê •œ¸Ã‹ z-‚◊Ë ◊¥ flÎûÊ ?z?≤1 ∑§ ‚¥ªÃ
i +w
„Ò–
47. Find the transformation which maps outside ?z?=1, on the half plane R(w)/0 so that the points
z=1, −i, −1 correspond to w=i, 0, −i respectively.
fl„ M§¬Ê¥Ã⁄UáÊ ôÊÊà ∑§ËÁ¡∞ ¡Ê •œ¸Ã‹ R(w)/0 ¬⁄U ?z?=1 ∑§ ’Ê„⁄U ß‚ ¬˝∑§Ê⁄U ¬˝ÁÃÁøÁòÊà „ÊÃÊ „Ò Á∑§ Á’¥ŒÈ z=1, −i, −1
∑˝§◊‡Ê— w=i, 0, −i ∑§ ‚¥ªÃ „Ò¥–
48. Find the matrix of the linear transformation T on V3(R) defined as T(a, b, c)=(2b+c, a−4b, 3a)
with respect to the basis B={(1, 1, 1) (1, 1, 0) (1, 0, 0)}.
V3(R) ¬⁄U ⁄ÒUÁπ∑§ M§¬Ê¥Ã⁄UáÊ T ∑§Ê •Ê√ÿÍ„ ôÊÊà ∑§ËÁ¡∞, ¡Ê •ÊœÊ⁄U B={(1, 1, 1) (1, 1, 0) (1, 0, 0)} ∑§ ‚ʬˇÊ
T(a, b, c)=(2b+c, a−4b, 3a) ∑§ M§¬ ◊¥ ¬Á⁄U÷ÊÁ·Ã „Ò–
RAS-1124 / ATS 9 Mathematics
Text of page 10
M A S T E R C O P Y
49. Prove that a continuous image of a connected metric space is connected.
Á‚h ∑§ËÁ¡∞ Á∑§ ∞∑§ ‚¥’h ŒÍ⁄UË∑§ ‚◊Ác≈U ∑§Ê ∞∑§ ‚¥Ãà ¬˝ÁÃÁ’¥’ ‚¥’h „ÊÃÊ „Ò–
50. Prove that metrices X1=(0, 1) and X2=[0, 1) are not complete metrices spaces with usual metrics.
Á‚h ∑§ËÁ¡∞ Á∑§ ŒÍ⁄UË∑§ X1=(0, 1) •ÊÒ⁄U X2=[0, 1) ‚Ê◊Êãÿ ŒÍ⁄UË∑§Ê¥ ∑§ ‚ÊÕ ‚¥¬Íáʸ ŒÍ⁄UË∑§ ‚◊Ác≈UÿÊ° Ÿ„Ë¥ „Ò¥–
x −y
51. Let X=R, and let d( x , y ) = ∀ x , y ∈ R . Show that the metric space (X, d) is not compact.
1+ x −y
x −y
◊ÊŸ ‹ËÁ¡∞ Á∑§ X=R „Ò ÃÕÊ d( x , y ) = ∀ x , y ∈ R „Ò– Œ‡Êʸß∞ Á∑§ ŒÍ⁄UË∑§ ‚◊Ác≈U (X, d) ‚¥„à Ÿ„Ë¥ „Ò–
1+ x −y
52. Let X be a non-empty set and let d1, d2 be two metrices on X. Show that d defined on X×X by
setting d(x, y)=max {d1(x, y), d2(x, y)}, x, y ∈ X is a metric on X.
◊ÊŸ ‹ËÁ¡∞ Á∑§ X ∞∑§ •Á⁄UÄà ‚◊ÈìÊÿ „Ò ÃÕÊ X ¬⁄U d1 •ÊÒ⁄U d2 ŒÊ ŒÍ⁄UË∑§ „Ò¥– Œ‡Êʸß∞ Á∑§ d(x, y)=max {d1(x, y),
d2(x, y)}, x, y ∈ X ‚ÈÁŸÁ‡øà ∑§⁄UŸ ¬⁄U, X×X ¬⁄U ¬Á⁄U÷ÁÊ·Ã d ‚◊ÈìÊÿ X ¬⁄U ∞∑§ ŒÍ⁄UË∑§ „Ò–
Part ‘D’ / π¥«U “Œ”
Part ‘D’ has 5 questions. Out of which any 3 questions are to be answered. Each question carries
30 marks. The question may be answered in maximum 300 words.
π¥«U “Œ” ◊¥ 5 ¬˝‡Ÿ „Ò¥, Á¡Ÿ◊¥ ‚ Á∑§ã„Ë¥ 3 ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒŸ „Ò¥– ¬˝àÿ∑§ ¬˝‡Ÿ 30 •¥∑§ ∑§Ê „Ò– ¬˝‡Ÿ ∑§Ê ©ûÊ⁄U •Áœ∑§Ã◊
300 ‡ÊéŒÊ¥ ◊¥ ÁŒÿÊ ¡Ê ‚∑§ÃÊ „Ò–
53. Two light rings can slide on rough horizontal rod. The rings are connected by a light inextensible
string of length a, to the mid-point of which is attached a weight W. Show that the greatest
µa
distance between rings, consistent with the equilibrium of the system, is , where µ is
1 + µ2
co-efficient of friction between either ring and rod.
ŒÊ „À∑§Ë •¥ªÍÁ∆UÿÊ° (Á⁄¥Uª) ¿UÑ ∞∑§ ⁄U»§ ˇÊÒÁá ¿U«∏ ¬⁄U Á»§‚‹ ‚∑§ÃË „Ò¥– ÿ ŒÊŸÊ¥ •¥ªÈÁ∆UÿÊ° ‹¥’Ê߸ a ∑§Ë ∞∑§ •ÁflÃÊãÿ „À∑§Ë
«UÊ⁄UË ‚ ¡È«∏Ë „È߸ „Ò¥, Á¡‚∑§ ◊äÿ-Á’¥ŒÈ ¬⁄U ∞∑§ ÷Ê⁄U W ‹ªÊ „È•Ê „Ò– Œ‡Êʸß∞ Á∑§ ß‚ ÁŸ∑§Êÿ ∑§ ‚¥ÃÈ‹Ÿ ∑§ ‚¥ªÃ •¥ªÈÁ∆UÿÊ¥
µa
∑§ ’Ëø ∑§Ë •Áœ∑§Ã◊ ŒÍ⁄UË 1 + µ2
„Ò, ¡„Ê° µ ¬˝àÿ∑§ •¥ªÍ∆UË •ÊÒ⁄U ¿U«∏ ∑§ ’Ëø ÉÊ·¸áÊ-ªÈáÊÊ¥∑§ „Ò–
54. Prove that J [ −3 ] is not a U.F.D., J being the ring of integers.
Á‚h ∑§ËÁ¡∞ Á∑§ J ∑§Ê ¬ÍáÊÊZ∑§Ê¥ ∑§Ê fl‹ÿ „ÊŸ ∑§ ∑§Ê⁄UáÊ, J [ −3 ] ∞∑§ U.F.D. Ÿ„Ë¥ „Ò–
RAS-1124 / ATS 10 Mathematics
Text of page 11
M A S T E R C O P Y
55. Find all the Mobius transformations which transform the half plane I(z)/0 into circles ?w?≤1.
fl ‚÷Ë ◊ÊÁ’ÿ‚ M§¬Ê¥Ã⁄UáÊ ôÊÊà ∑§ËÁ¡∞, ¡Ê •œ¸Ã‹ I(z)/0 ∑§Ê flÎûÊÊ¥ ?w?≤1 ◊¥ M§¬Ê¥ÃÁ⁄Uà ∑§⁄Uà „Ò¥–
56. Establish the validity of the formal solution of the vibrating string problem is given by
utt−c2uxx=0, 0<x<l, t>0
u(x, 0)=f(x), 0≤x≤l
ut(x, 0)=g(x), 0≤x≤l
u(0, t)=0, t/0
u(l, t)=0, t/0
ÁŸêŸÁ‹Áπà mÊ⁄UÊ ŒË ¡ÊŸ flÊ‹Ë ∑¥§Á¬Ã ∑§⁄UÃË «UÊ⁄UË ‚◊SÿÊ ∑§ •ÊÒ¬øÊÁ⁄U∑§ „‹ ∑§Ë flÒœÃÊ ∑§Ê SÕÊÁ¬Ã ∑§ËÁ¡∞ —
utt−c2uxx=0, 0<x<l, t>0
u(x, 0)=f(x), 0≤x≤l
ut(x, 0)=g(x), 0≤x≤l
u(0, t)=0, t/0
u(l, t)=0, t/0
57. A particle is projected with initial velocity V from a point O at angle of elevation α. Show that the
π
range on a plane through O inclined at β to the horizontal β < α < is
2
V2
[sin(2α−β)−sinβ] , and hence deduce that for a given velocity of propagation V, the
g cos 2 β
V2
maximum range is .
g(1 + sinβ)
Á∑§‚Ë ∑§áÊ ∑§Ê ∞∑§ Á’¥ŒÈ O ‚ ¬˝Ê⁄¥UÁ÷∑§ flª V ∑§ ‚ÊÕ ©ÛÊÿŸ ∑§ÊáÊ α ¬⁄U ¬˝ˇÊÁ¬Ã Á∑§ÿÊ ¡ÊÃÊ „Ò– Œ‡Êʸß∞ Á∑§ O ‚ „Ê∑§⁄U ¡ÊŸ
π V2
flÊ‹ ÃÕÊ ˇÊÒÁá β < α < 2 ‚ ∑§ÊáÊ β ¬⁄U Ÿ˜Ã ‚◊Ë ¬⁄U ¬⁄UÊ‚ g cos2 β [sin(2α−β)−sinβ] „Ò ÃÕÊ ß‚‚ ÁŸªÁ◊Ã
V2
∑§ËÁ¡∞ Á∑§ ÁŒ∞ „È∞ ¬˝ˇÊ¬ flª V ∑§ Á‹∞ •Áœ∑§Ã◊ ¬⁄UÊ‚ g(1 + sinβ) „Ò–
-oOo-
RAS-1124 / ATS 11 MathematicsOfficial key not published for this paper.
Page 1 of 11
Source: Central Board of Secondary Education. Question paper published by Central Board of Secondary Education.