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ŸÊ◊Ê¢∑§
Roll No.
No. of Questions — 24 No. of Printed Pages — 7
SS—15–1—Maths. I ©UìÊ ◊ÊäÿÁ◊∑§ ¬⁄UˡÊÊ, 2010 II
SENIOR SECONDARY EXAMINATION, 2010
flÒ∑§ÁÀ¬∑§ flª¸ I ÃÕÊ
ó ∑§‹Ê fl ÁflôÊÊŸ flª¸
( OPTIONAL GROUPS I & II — HUMANITIES AND SCIENCE )
ªÁáÊÃ ó ¬˝Õ◊ ¬òÊ
( MATHEMATICS — First Paper )
‚◊ÿ — 3 4 ÉÊá≈U ¬ÍáÊÊZ∑§ — 60 ¬⁄UˡÊÊÁÕ¸ÿÊ¥ ∑§ Á‹∞ •Êfl‡ÿ∑§ ÁŸŒ¸‡Ê — GENERAL INSTRUCTIONS FOR EXAMINEES : 1.
1
¬⁄UˡÊÊÕ˸ ‚fl¸¬˝Õ◊ •¬Ÿ ¬˝‡Ÿ ¬òÊ ¬⁄U ŸÊ◊Ê¢∑§ •ÁŸflÊÿ¸Ã— Á‹π¥ – Candidate must write first his / her Roll No. on the question paper compulsorily.
2.
¬˝‡Ÿ ¬òÊ ∑§ Á„UãŒË fl •¢ª˝¡Ë M§¬ÊãÃ⁄U ◊¥ Á∑§‚Ë ¬˝∑§Ê⁄U ∑§Ë òÊÈÁ≈U / •ãÃ⁄U / Áfl⁄UÙœÊ÷Ê‚ „UÙŸ ¬⁄U Á„UãŒË ÷Ê·Ê ∑§ ¬˝‡Ÿ ∑§Ù ‚„UË ◊ÊŸ¥ – If there is any error / difference / contradiction in Hindi and English versions of the Question paper, the question of Hindi version should be treated valid.
3.
‚÷Ë ¬˝‡Ÿ ∑§⁄UŸ •ÁŸflÊÿ¸ „Ò¥U – ¬˝‡Ÿ ∑˝§◊Ê¢∑§
21, 23
fl
24
◊¥ •ÊãÃÁ⁄U∑§ Áfl∑§À¬ „Ò¥U –
All questions are compulsory. Question Nos. 21, 23 and 24 have internal choice. 4. 5.
¬˝‡Ÿ ∑˝§◊Ê¢∑§
2
‚ 7 Ã∑§ •Áà ‹ÉÊÍûÊ⁄UÊà◊∑§ ¬˝‡Ÿ „Ò¥U –
Question Nos. 2 to 7 are Very Short Answer type.
¬˝àÿ∑§ ¬˝‡Ÿ ∑§Ê ©UûÊ⁄U ŒË ªß¸ ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ „UË Á‹π¥ – Write the answer of each question in answer-book only.
SS–528
SS—15–1—Maths. I
[ Turn over
2 6.
Á¡‚ ¬˝‡Ÿ ∑§ ∞∑§ ‚ •Áœ∑§ ‚◊ÊŸ •¢∑§ UflÊ‹ ÷ʪ „Ò¥U, ©UŸ ‚÷Ë ÷ʪ٥ ∑§Ê „U‹ ∞∑§ ‚ÊÕ ‚ÃØ Á‹π¥ – For questions having more than one part carrying similar marks, the answers of those parts are to be written together in continuity.
7.
•¬ŸË ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ∑§ ¬Îc∆UÊ¥ ∑§ ŒÊŸÊ¥ •Ê⁄U Á‹Áπ∞ – ÿÁŒ ∑§Ê߸ ⁄U$»§ ∑§Êÿ¸ ∑§⁄UŸÊ „UÊ, ÃÊ ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ∑§ •¢ÁÃ◊ ¬Îc∆UÊ¥ ¬⁄U ∑§⁄¥U •ÊÒ⁄U ßã„¥U ÁÃ⁄U¿UË ‹Êߟʥ ‚ ∑§Ê≈U∑§⁄U ©UŸ ¬⁄U “⁄U$»§ ∑§Êÿ¸” Á‹π Œ¥ – Write on both sides of the pages of your answer-book. If any rough work is to be done, do it on last pages of the answer-book and cross with slant lines and write ‘Rough Work’ on them. ¬˝‡Ÿ ∑˝§◊Ê¢∑§ 1 ∑§ øÊ⁄U ÷ʪ ( i, ii, iii ÃÕÊ iv ) „Ò¥U – ¬˝àÿ∑§ ÷ʪ ∑§ ©UûÊ⁄U ∑§ øÊ⁄U
8.
Áfl∑§À¬ ( ∑§, π, ª ∞fl¢ ÉÊ ) „Ò ¥ U – ‚„UË Áfl∑§À¬ ∑§Ê ©UûÊ⁄UÊˇÊ⁄U ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ ÁŸêŸÊŸÈ‚Ê⁄U ÃÊÁ‹∑§Ê ’ŸÊ∑§⁄U Á‹π¥ — There are four parts ( i, ii, iii and iv ) in Question No. 1. Each part has four alternatives A, B, C and D. Write the letter of the correct alternative in the answer-book at a place by making a table as mentioned below :
¬˝‡Ÿ ∑˝§◊Ê¢∑§
Question No. 1. (i) 1. (ii) 1. (iii) 1. (iv) 1. (i)
‚„Ë ©UûÊ⁄U ∑§Ê ∑˝§◊ÊˇÊ⁄U
Correct letter of the Answer
Á’ãŒÈ P ( 1, ÁŸŒ¸‡ÊÊ¢∑§ „Ò¥U (∑§) (ª)
2, 3 )
‚
xy
‚◊Ë ¬⁄U ‹ê’
PM
«UÊ‹Ê ªÿÊ – ‹ê’¬ÊŒ
M
∑§
( 1, 2, 3 ) ( 1, 0, 3 )
(π) (ÉÊ)
( 0, 2, 3 ) ( 1, 2, 0 ).
A perpendicular PM is drawn from the point P ( 1, 2, 3 ) on xy plane. The coordinates of the foot of perpendicular M are (A) (C) ( 1, 2, 3 ) ( 1, 0, 3 ) SS–528
(B) (D)
( 0, 2, 3 ) ( 1, 2, 0 ).
1 2
SS—15–1—Maths. I
3 (ii)
‚◊Ë x = 0 ∑§Ë Á’ãŒÈ ( 3, 4, 6 ) ‚ ŒÍ⁄UË „ÒU (∑§) 3 (π) (ª) 6 (ÉÊ) (A) (C) 3 6 (B) (D)
4 61 . 4 61 .
1 2
Distance of the plane x = 0 from the point ( 3, 4, 6 ) is
(iii)
∞∑§ ∑§áÊ ¬˝Ê⁄¢UÁ÷∑§ flª g ◊Ë/‚ ‚ ™§äflʸœ⁄U ÁŒ‡ÊÊ ◊¥ ™§¬⁄U ∑§Ë •Ê⁄U »¥§∑§Ê ¡ÊÃÊ „ÒU – ∑§áÊ mÊ⁄UÊ Ãÿ ∑§Ë ªß¸ ◊„UûÊ◊ ™°§øÊ߸ „ÒU (∑§) g ◊Ë≈U⁄U (π) g ◊Ë≈U⁄U 4 2 (ª) g
◊Ë≈U⁄U
(ÉÊ)
2g
◊Ë≈U⁄U –
A particle is projected vertically upwards with initial velocity g m/sec. The maximum height attained by the particle is g g (A) m (B) m 4 2 (C) g m (D) 2g m. (iv) 500
1 2
ª˝Ê◊ º˝√ÿ◊ÊŸ ∑§ ∞∑§ Á¬á«U ¬⁄U ∞∑§ ’‹ ‹ªÊŸ ¬⁄U 3 ◊Ë/‚ 2 ∑§Ê àfl⁄UáÊ ©Uà¬ÛÊ „UÊ ¡ÊÃÊ „ÒU, ÃÊ ‹ªŸ flÊ‹Ê ’‹ „ÒU (∑§) U 1 N (π) U1·5 N (ª) 2 N (ÉÊ) 3 N. A force acting on a body of mass 500 gms, produces an acceleration of 3 m/sec 2 . Then the impressed force is (A) (C) 1N 2N (B) (D) 1·5 N 3 N. 1 2
2.
Á‚h ∑§ËÁ¡∞ Á∑§ Á’ãŒÈ•Ê¥ P ∑§ËÁ¡∞ –
2 cos – 1 x = cos – 1 ( 2x 2 – 1 ) . 1
Prove that 2 cos – 1 x =...
Roll No.
No. of Questions — 24 No. of Printed Pages — 7
SS—15–1—Maths. I ©UìÊ ◊ÊäÿÁ◊∑§ ¬⁄UˡÊÊ, 2010 II
SENIOR SECONDARY EXAMINATION, 2010
flÒ∑§ÁÀ¬∑§ flª¸ I ÃÕÊ
ó ∑§‹Ê fl ÁflôÊÊŸ flª¸
( OPTIONAL GROUPS I & II — HUMANITIES AND SCIENCE )
ªÁáÊÃ ó ¬˝Õ◊ ¬òÊ
( MATHEMATICS — First Paper )
‚◊ÿ — 3 4 ÉÊá≈U ¬ÍáÊÊZ∑§ — 60 ¬⁄UˡÊÊÁÕ¸ÿÊ¥ ∑§ Á‹∞ •Êfl‡ÿ∑§ ÁŸŒ¸‡Ê — GENERAL INSTRUCTIONS FOR EXAMINEES : 1.
1
¬⁄UˡÊÊÕ˸ ‚fl¸¬˝Õ◊ •¬Ÿ ¬˝‡Ÿ ¬òÊ ¬⁄U ŸÊ◊Ê¢∑§ •ÁŸflÊÿ¸Ã— Á‹π¥ – Candidate must write first his / her Roll No. on the question paper compulsorily.
2.
¬˝‡Ÿ ¬òÊ ∑§ Á„UãŒË fl •¢ª˝¡Ë M§¬ÊãÃ⁄U ◊¥ Á∑§‚Ë ¬˝∑§Ê⁄U ∑§Ë òÊÈÁ≈U / •ãÃ⁄U / Áfl⁄UÙœÊ÷Ê‚ „UÙŸ ¬⁄U Á„UãŒË ÷Ê·Ê ∑§ ¬˝‡Ÿ ∑§Ù ‚„UË ◊ÊŸ¥ – If there is any error / difference / contradiction in Hindi and English versions of the Question paper, the question of Hindi version should be treated valid.
3.
‚÷Ë ¬˝‡Ÿ ∑§⁄UŸ •ÁŸflÊÿ¸ „Ò¥U – ¬˝‡Ÿ ∑˝§◊Ê¢∑§
21, 23
fl
24
◊¥ •ÊãÃÁ⁄U∑§ Áfl∑§À¬ „Ò¥U –
All questions are compulsory. Question Nos. 21, 23 and 24 have internal choice. 4. 5.
¬˝‡Ÿ ∑˝§◊Ê¢∑§
2
‚ 7 Ã∑§ •Áà ‹ÉÊÍûÊ⁄UÊà◊∑§ ¬˝‡Ÿ „Ò¥U –
Question Nos. 2 to 7 are Very Short Answer type.
¬˝àÿ∑§ ¬˝‡Ÿ ∑§Ê ©UûÊ⁄U ŒË ªß¸ ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ „UË Á‹π¥ – Write the answer of each question in answer-book only.
SS–528
SS—15–1—Maths. I
[ Turn over
2 6.
Á¡‚ ¬˝‡Ÿ ∑§ ∞∑§ ‚ •Áœ∑§ ‚◊ÊŸ •¢∑§ UflÊ‹ ÷ʪ „Ò¥U, ©UŸ ‚÷Ë ÷ʪ٥ ∑§Ê „U‹ ∞∑§ ‚ÊÕ ‚ÃØ Á‹π¥ – For questions having more than one part carrying similar marks, the answers of those parts are to be written together in continuity.
7.
•¬ŸË ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ∑§ ¬Îc∆UÊ¥ ∑§ ŒÊŸÊ¥ •Ê⁄U Á‹Áπ∞ – ÿÁŒ ∑§Ê߸ ⁄U$»§ ∑§Êÿ¸ ∑§⁄UŸÊ „UÊ, ÃÊ ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ∑§ •¢ÁÃ◊ ¬Îc∆UÊ¥ ¬⁄U ∑§⁄¥U •ÊÒ⁄U ßã„¥U ÁÃ⁄U¿UË ‹Êߟʥ ‚ ∑§Ê≈U∑§⁄U ©UŸ ¬⁄U “⁄U$»§ ∑§Êÿ¸” Á‹π Œ¥ – Write on both sides of the pages of your answer-book. If any rough work is to be done, do it on last pages of the answer-book and cross with slant lines and write ‘Rough Work’ on them. ¬˝‡Ÿ ∑˝§◊Ê¢∑§ 1 ∑§ øÊ⁄U ÷ʪ ( i, ii, iii ÃÕÊ iv ) „Ò¥U – ¬˝àÿ∑§ ÷ʪ ∑§ ©UûÊ⁄U ∑§ øÊ⁄U
8.
Áfl∑§À¬ ( ∑§, π, ª ∞fl¢ ÉÊ ) „Ò ¥ U – ‚„UË Áfl∑§À¬ ∑§Ê ©UûÊ⁄UÊˇÊ⁄U ©UûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ ÁŸêŸÊŸÈ‚Ê⁄U ÃÊÁ‹∑§Ê ’ŸÊ∑§⁄U Á‹π¥ — There are four parts ( i, ii, iii and iv ) in Question No. 1. Each part has four alternatives A, B, C and D. Write the letter of the correct alternative in the answer-book at a place by making a table as mentioned below :
¬˝‡Ÿ ∑˝§◊Ê¢∑§
Question No. 1. (i) 1. (ii) 1. (iii) 1. (iv) 1. (i)
‚„Ë ©UûÊ⁄U ∑§Ê ∑˝§◊ÊˇÊ⁄U
Correct letter of the Answer
Á’ãŒÈ P ( 1, ÁŸŒ¸‡ÊÊ¢∑§ „Ò¥U (∑§) (ª)
2, 3 )
‚
xy
‚◊Ë ¬⁄U ‹ê’
PM
«UÊ‹Ê ªÿÊ – ‹ê’¬ÊŒ
M
∑§
( 1, 2, 3 ) ( 1, 0, 3 )
(π) (ÉÊ)
( 0, 2, 3 ) ( 1, 2, 0 ).
A perpendicular PM is drawn from the point P ( 1, 2, 3 ) on xy plane. The coordinates of the foot of perpendicular M are (A) (C) ( 1, 2, 3 ) ( 1, 0, 3 ) SS–528
(B) (D)
( 0, 2, 3 ) ( 1, 2, 0 ).
1 2
SS—15–1—Maths. I
3 (ii)
‚◊Ë x = 0 ∑§Ë Á’ãŒÈ ( 3, 4, 6 ) ‚ ŒÍ⁄UË „ÒU (∑§) 3 (π) (ª) 6 (ÉÊ) (A) (C) 3 6 (B) (D)
4 61 . 4 61 .
1 2
Distance of the plane x = 0 from the point ( 3, 4, 6 ) is
(iii)
∞∑§ ∑§áÊ ¬˝Ê⁄¢UÁ÷∑§ flª g ◊Ë/‚ ‚ ™§äflʸœ⁄U ÁŒ‡ÊÊ ◊¥ ™§¬⁄U ∑§Ë •Ê⁄U »¥§∑§Ê ¡ÊÃÊ „ÒU – ∑§áÊ mÊ⁄UÊ Ãÿ ∑§Ë ªß¸ ◊„UûÊ◊ ™°§øÊ߸ „ÒU (∑§) g ◊Ë≈U⁄U (π) g ◊Ë≈U⁄U 4 2 (ª) g
◊Ë≈U⁄U
(ÉÊ)
2g
◊Ë≈U⁄U –
A particle is projected vertically upwards with initial velocity g m/sec. The maximum height attained by the particle is g g (A) m (B) m 4 2 (C) g m (D) 2g m. (iv) 500
1 2
ª˝Ê◊ º˝√ÿ◊ÊŸ ∑§ ∞∑§ Á¬á«U ¬⁄U ∞∑§ ’‹ ‹ªÊŸ ¬⁄U 3 ◊Ë/‚ 2 ∑§Ê àfl⁄UáÊ ©Uà¬ÛÊ „UÊ ¡ÊÃÊ „ÒU, ÃÊ ‹ªŸ flÊ‹Ê ’‹ „ÒU (∑§) U 1 N (π) U1·5 N (ª) 2 N (ÉÊ) 3 N. A force acting on a body of mass 500 gms, produces an acceleration of 3 m/sec 2 . Then the impressed force is (A) (C) 1N 2N (B) (D) 1·5 N 3 N. 1 2
2.
Á‚h ∑§ËÁ¡∞ Á∑§ Á’ãŒÈ•Ê¥ P ∑§ËÁ¡∞ –
2 cos – 1 x = cos – 1 ( 2x 2 – 1 ) . 1
Prove that 2 cos – 1 x =...
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