NIOS Class 12 October 2024 Question Paper Mathematics

This Question Paper consists of 45 questions and 15 printed pages + Graph Sheet.
ß‚ ¬˝‡Ÿ-¬òÊ ◊¥ 45 ¬˝‡Ÿ ÃÕÊ 15 ◊ÈÁŒ˝Ã ¬Îc∆U + ª˝Ê»§ ‡ÊË≈U „Ò¥–

Roll No. Code No.
•ŸÈ∑˝§◊Ê¥∑§ ∑§Ê«U Ÿ¥.
68/ESS/1
MATHEMATICS SET/‚ ≈ A
ªÁáÊÃ
(311)
Day and Date of Examination :
(¬⁄UˡÊÊ ∑§Ê ÁŒŸ fl ÁŒŸÊ¥∑§)
Signature of Invigilators :
(ÁŸ⁄UˡÊ∑§Ê¥ ∑§ „SÃÊˇÊ⁄U) 1.

2.

General Instructions :
1. Candidate must write his/her Roll Number on the first page of the Question Paper.
2. Please check the Question Paper to verify that the total pages and total number of questions contained in the
Question Paper are the same as those printed on the top of the first page. Also check to see that the questions are
in sequential order.
3. Making any identification mark in the Answer-Book or writing Roll Number anywhere other than the specified
places will lead to disqualification of the candidate.
4. Write your Question Paper code No. 68/ESS/1-A on the Answer-Book.
5. (a) The Question Paper is in English/Hindi medium only. However, if you wish, you can answer in any one
of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada, Telugu, Marathi, Oriya, Gujarati,
Konkani, Manipuri, Assamese, Nepali, Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in the box provided in the
Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and English, the responsibility for any
errors/mistakes in understanding the question will be yours only.
6. In case of any doubt or confusion in the question paper, the English Version will prevail.
‚Ê◊Êãÿ •ŸÈŒ‡Ê —
1. ¬⁄UˡÊÊÕ˸ ¬˝‡Ÿ-¬òÊ ∑§ §¬„‹ ¬Îc∆U ¬⁄U •¬ŸÊ •ŸÈ∑˝§◊Ê¥∑§ •fl‡ÿU Á‹π¥–
2. ∑Χ¬ÿÊ ¬˝‡Ÿ-¬òÊ ∑§Ê ¡ÊÚ°ø ‹¥ Á∑§ ¬˝‡Ÿ-¬òÊ ∑§ ∑ȧ‹ ¬Îc∆UÊ¥ ÃÕÊ ¬˝‡ŸÊ¥ ∑§Ë ©ÃŸË „Ë ‚¥ÅUÿÊ „Ò Á¡ÃŸË ¬˝Õ◊ ¬Îc∆ ∑ §‚’‚ ™§¬⁄U ¿U¬Ë „Ò– ß‚
’Êà ∑§Ë ¡ÊÚ°ø ÷Ë ∑§⁄U ‹¥ Á∑§ ¬˝‡Ÿ ∑˝ Á◊∑§ UM§¬ ◊¥ „Ò¥–
3. § ©ûÊ⁄U-¬ÈÁSUÃ∑§Ê ◊¥ ¬„øÊŸ-Áøq ’ŸÊŸ •ÕflÊ ÁŸÁŒ¸c≈U SÕÊŸÊ¥ ∑§§•ÁÃÁ⁄UÄà ∑§„Ë¥ ÷Ë •ŸÈ∑§˝ ◊Ê¥∑§ Á‹πŸ ¬⁄U ¬⁄UˡÊÊÕ˸ ∑§Ê •ÿÊÇÿ ∆U„⁄UÊÿÊ ¡ÊÿªÊ–
4. •¬ŸË ©ûÊ⁄U-¬ÈÁSUÃ∑§Ê ¬⁄U ¬˝‡Ÿ-¬òÊ ∑§Ë ∑§Ê«U ‚¥ÅÿÊ 68/ESS/1-A Á‹π¥–
5. (∑§) ¬˝‡Ÿ-¬òÊ ∑§fl‹ Á„¥ŒË/•¥ª˝¡Ë ◊Êäÿ◊ ◊¥ „Ò– Á»§⁄U ÷Ë, ÿÁŒ •Ê¬ øÊ„¥ ÃÊ ŸËø ŒË ªß¸ Á∑§‚Ë ∞∑§ ÷Ê·Ê ◊¥ ©ûÊ⁄ Œ ‚∑§Ã „Ò¥ —
•¥ª˝¡Ë, Á„¥ŒË, ©ŒÍ¸, ¬¥¡Ê’Ë, ’°ª‹Ê, ÃÁ◊‹, ◊‹ÿÊ‹◊, ∑§ãŸ«∏, Ã‹ÈªÈ, ◊⁄UÊ∆UË, ©Á«∏ÿÊ, ªÈ¡⁄UÊÃË, ∑§Ê¥∑§áÊË, ◊ÁáʬÈ⁄UË, •‚Á◊ÿÊ,
Ÿ¬Ê‹Ë, ∑§‡◊Ë⁄UË, ‚¥S∑Χç•ÊÒ⁄U Á‚¥œË–
∑Χ¬ÿÊ ©ûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ ÁŒ∞ ª∞ ’ÊÚÄ‚ ◊¥ Á‹π¥ Á∑§ •Ê¬ Á∑§‚ ÷Ê·Ê ◊¥ ©ûÊ⁄U Á‹π ⁄U„ „Ò¥–
(π) ÿÁŒ •Ê¬ Á„¥ŒË ∞fl¥ •¥ª˝¡Ë ∑§ •ÁÃÁ⁄UÄà Á∑§‚Ë •ãÿ ÷Ê·Ê ◊¥ ©ûÊ⁄U Á‹πÃ „Ò¥, ÃÊ ¬˝‡Ÿ ∑§Ê ‚◊¤ÊŸ ◊¥ „ÊŸ flÊ‹Ë
òÊÈÁ≈UÿÊ¥/ª‹ÁÃÿÊ¥ ∑§Ë Á¡ê◊UŒÊ⁄Ë ∑ fl‹ •Ê¬∑§Ë „ÊªË–
6. ¬˝‡Ÿ¬òÊ ◊¥ Á∑§‚Ë ÷Ë ¬˝∑§Ê⁄U ∑§ ‚¥Œ„ •ÕflÊ ŒÈÁflœÊ ∑§Ë ÁSÕÁà ◊¥ •¥ª˝¡Ë •ŸÈflÊŒ „Ë ◊Êãÿ „ÊªÊ–
68/ESS/1-311-A ] 1 [ Contd...

MATHEMATICS
ªÁáÊÃ
(311)
Time : 3 Hours ] [ Maximum Marks : 100
‚◊ÿ — 3 ÉÊá≈U ] [ ¬ÍáÊÊZ∑§ — 100
Note :
(i) This question paper consists of 45 questions in all.
(ii) All questions are compulsory.
(iii) Marks are given against each question.
(iv) Section - A consists of
(a) Q.No. 1 to 20 - Multiple Choice Type Questions (MCQs) carrying 1 mark each. Select and
write the most appropriate option out of the four options given in each of these questions.
(b) Q.No. 21 to 29 - Objective Type Questions. Q.No. 21 to 24 carry 2 marks each (with two
sub-parts of 1 mark each). Q.No. 25 to 28 carries 4 marks each (with 4 sub-parts of
1 mark each) and Q.No. 29 carries 6 marks (with 6 sub-parts of 1 mark each). Attempt these
questions as per the instructions given for each of the questions 21 to 29.
(v) Section - B consists of
(a) Q.No. 30 to 38 - Very Short Answer Type Questions carrying 2 marks each.
(b) Q.No. 39 to 43 - Short Answer Type Questions carrying 4 marks each.
(c) Q.No. 44 and 45 - Long Answer Type Questions carrying 6 marks each.
ÁŸŒ¸‡Ê —
(i) ß‚ ¬˝‡Ÿ-¬òÊ ◊¥ ∑ȧ‹ 45 ¬˝‡Ÿ „Ò¥–
(ii) ‚÷Ë ¬˝‡Ÿ •ÁŸflÊÿ¸ „Ò¥–
(iii) ¬˝àÿ∑§ ¬˝‡Ÿ ∑§ ‚Ê◊Ÿ ©‚∑§ •¥∑§ ÁŒ∞ ª∞ „Ò¥–
(iv) πá«U - ∑§ ◊¥ ‡ÊÊÁ◊‹ „Ò¥ —
(a) ¬˝‡Ÿ ‚¥ÅÿÊ 1 ‚ 20 - ’„ÈÁfl∑§À¬Ëÿ ¬˝∑§Ê⁄U ∑§ ¬˝‡Ÿ „Ò¥ ¡Ê ¬˝àÿ∑§ 1 •¥∑§ ∑§Ê „Ò– ߟ ¬˝‡ŸÊ¥ ◊¥ ¬˝àÿ∑§ ∑§ øÊ⁄U Áfl∑§À¬Ê¥
◊¥ ‚ ‚’‚ ©¬ÿÈÄà Áfl∑§À¬ ∑§Ê øÈÁŸ∞–
(b) ¬˝‡Ÿ ‚¥ÅÿÊ 21 ‚ 29 - flSÃÈÁŸc∆U ¬˝‡Ÿ „Ò¥– ¬˝‡Ÿ ‚¥ÅÿÊ 21 ‚ 24 - ¬˝àÿ∑§ 2 •¥∑§Ê¥ ∑§Ê „Ò¥ (ŒÊ ©¬÷ʪ ∑§ ‚ÊÕ
¬˝àÿ∑§ 1 •¥∑§ ∑§Ê „Ò) ¬˝‡Ÿ ‚¥ÅÿÊ 25 ‚ 28 ¬˝àÿ∑§ 4 •¥∑§Ê¥ ∑§Ê „Ò (øÊ⁄U ©¬÷ʪ ¬˝àÿ∑§ ∑§Ê 1 •¥∑§) •ÊÒ⁄U ¬˝‡Ÿ
‚¥ÅÿÊ 29 ∑§ 6 •¥∑§ „Ò¥ (¿U— ©¬÷ʪ ¬˝àÿ∑§ ∑§Ê 1 •¥∑§) ¬˝‡Ÿ ‚¥ÅÿÊ 21 ‚ 29 ◊¥ ¬˝àÿ∑§ ¬˝‡Ÿ ◊¥ ŒË ªß¸ ‚ÍøŸÊ
•ŸÈ‚Ê⁄U ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒËÁ¡∞–
(v) πá«U - π§ ◊¥ ‡ÊÊÁ◊‹ „Ò¥ —
(a) ¬˝‡Ÿ ‚¥ÅÿÊ 30 ‚ 38 - •Áà ‹ÉÊÈ-©ûÊ⁄UËÿ ¬˝∑§Ê⁄U ∑§ ¬˝‡Ÿ „Ò¥ •ÊÒ⁄U ¬˝àÿ∑§ ∑§ 2 •¥∑§ „Ò¥–
(b) ¬˝‡Ÿ ‚¥ÅÿÊ 39 ‚ 43 - ‹ÉÊÈ-©ûÊ⁄UËÿ ¬˝∑§Ê⁄U ∑§ ¬˝‡Ÿ „Ò¥ •ÊÒ⁄U ¬˝àÿ∑§ ∑§ 4 •¥∑§ „Ò¥–
(c) ¬˝‡Ÿ ‚¥ÅÿÊ 44 •ÊÒ⁄U 45 - ŒËÉʸ ©ûÊ⁄UËÿ ¬˝∑§Ê⁄U ∑§ ¬˝‡Ÿ „Ò¥ •ÊÒ⁄U ¬˝àÿ∑§ ∑§ 6 •¥∑§ „Ò¥–

68/ESS/1-311-A ] 2 [ Contd...

Note / ÁŸŒ¸‡Ê —
(1) Answers of all questions are to be given in the Answer-Book given to you.
‚÷Ë ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U •Ê¬∑§Ê ŒË ªß¸ ©ûÊ⁄U-¬ÈÁSÃ∑§Ê ◊¥ „Ë Á‹π¥–
(2) 15 minutes time has been allotted to read this Question Paper. The Question Paper will be
distributed at 2 : 15 p.m. From 2 : 15 p.m. to 2 : 30 p.m., the students will read the Question
Paper only and will not write any answer on the Answer-Book during this period.
ß‚ ¬˝‡Ÿ-¬òÊ ∑§Ê ¬…∏Ÿ ∑§ Á‹∞ 15 Á◊Ÿ≈U ∑§Ê ‚◊ÿ ÁŒÿÊ ªÿÊ „Ò– ¬˝‡Ÿ-¬òÊ ∑§Ê ÁflÃ⁄UáÊ ŒÊ¬„⁄U ◊¥ 2 : 15 ’¡ Á∑§ÿÊ
¡Ê∞ªÊ– 2 : 15 ’¡ ‚ 2 : 30 ’¡ Ã∑§ ¿UÊòÊ ∑§fl‹ ¬˝‡Ÿ-¬òÊ ∑§Ê ¬…∏¥ª •ÊÒ⁄U ß‚ •flÁœ ∑§ ŒÊÒ⁄UÊŸ fl ©ûÊ⁄U-¬ÈÁSÃ∑§Ê ¬⁄U
∑§Ê߸ ©ûÊ⁄U Ÿ„Ë¥ Á‹π¥ª–

SECTION - A / πá«U - ∑§
1. The points A(−1, −1), B(2, 3) and C(−2, 6) are the vertices of : 1
(A) an equilateral triangle (B) an isosceles triangle
(C) a scalene triangle (D) an isosceles right triangle
Á’ãŒÈ A(−1, −1), B(2, 3) •ÊÒ⁄U C(−2, 6) ‡ÊË·¸ „Ò —
(A) ∞∑§ ‚◊’Ê„È ÁòÊ÷È¡ ∑§ (B) ∞∑§ ‚◊Ám’Ê„È ÁòÊ÷È¡ ∑§
(C) ∞∑§ Áfl·◊’Ê„È ÁòÊ÷È¡ ∑§ (D) ∞∑§ ‚◊Ám’Ê„È ‚◊∑§ÊáÊ ÁòÊ÷È¡ ∑§

2. The slope of a line which makes an angle of 608 with the negative direction of x-axis, is : 1
©‚ ⁄UπÊ ∑§Ë ¬˝fláÊÃÊ, ¡Ê ´§áÊÊà◊∑§ x-•ˇÊ ∑§Ë ÁŒ‡ÊÊ ∑§ ‚ÊÕ 608 ∑§Ê ∑§ÊáÊ ’ŸÊÃË „Ò, „Ò —
1 1
(A) (B) − (C) 3 (D) − 3
3 3

x2 y2
3. For the hyperbola − = 1 , the length of the latus rectum is : 1
16 9
9 8
(A) units (B) units (C) 3 units (D) 4 units
2 3

x2 y2
•Áà ¬⁄Ufl‹ÿ − = 1 ∑§ Á‹∞ ŸÊÁ÷‹¥’ ¡ËflÊ ∑§Ë ‹¥’Ê߸ „Ò —
16 9
9 8
(A)
2
ß∑§Ê߸ (B)
3
ß∑§Ê߸ (C) 3 ß∑§Ê߸ (D) 4 ß∑§Ê߸

4. The radius of the circle 4x2+4y2−2x+3y−6=0 is : 1

(A) 37 units (B) 109 units (C) 71 units (D) 107 units
4 8 6 8
flÎûÊ 4x2+4y2−2x+3y−6=0 ∑§Ë ÁòÊíÿÊ „Ò —
(A) 37 ß∑§Ê߸ (B) 109 ß∑§Ê߸ (C) 71 ß∑§Ê߸ (D) 107 ß∑§Ê߸
4 8 6 8

68/ESS/1-311-A ] 3 [ Contd...

5. The equation x+7y−4=0 in the slope-intercept form is : 1
⁄UπÊ x+7y−4=0 ∑§Ê •¥Ãπá«U ¬˝fláÊÃÊ M§¬ „Ò —
x 7y 1 4 1 4
(A) + =1 (B) x=−7y+4 (C) y =− x + (D) y= x+
4 4 7 7 7 7

 2a b   3 −2 
6. The two matrices −4 6  and d 3c  are equal for the values a, b, c, d : 1
   

 2a b  3 −2 
ŒÊ •Ê√ÿÍ„ −4 
6 ÃÕÊU d 3c  ’⁄UÊ’⁄U „Ê¥ª, ¡’ a, b, c, d ∑§ ◊ÊŸ „Ò¥ —
  

3 3
(A) a= , b=−2, c=−2, d=4 (B) a =− , b=−2, c=2, d=4
2 2

3 3
(C) a= , b=−2, c=2, d=−4 (D) a= , b=−2, c=2, d=4
2 2

0
 2 3  
7. For the matrix A =   and B = −1 : 1
 0 1   3 

(A) AB exists (B) BA exists
(C) AB and BA both exists (D) Neither AB nor BA exists

0
 2 3
•Ê√ÿÍ„ A =   •ÊÒ⁄U B = −1 ∑§ Á‹∞ —
 0 1  
 3 

(A) AB ∑§Ê •ÁSÃàfl „Ò (B) BA ∑§Ê •ÁSÃàfl „Ò

(C) AB •ÊÒ⁄U BA ŒÊŸÊ¥ ∑§Ê •ÁSÃàfl „Ò¥ (D) AB •ÊÒ⁄U BA ŒÊŸÊ¥ ∑§ „Ë •ÁSÃàfl Ÿ„Ë¥ „Ò¥

1 6 
8. If A =   , then |A'|, where A' denotes the transpose of the matrix A, is : 1
3 2 

ÿÁŒ A = 
1 6
 „Ò, ÃÊ |A'|, ¡„Ê° A' •Ê√ÿÍ„ A ∑§ ¬Á⁄Uflø ∑§Ê Œ‡ÊʸÃÊ „Ò, „Ò —
3 2 
(A) −16 (B) 16 (C) −8 (D) 8

68/ESS/1-311-A ] 4 [ Contd...

9. The relation R defined in the set A={x ∈ I, 0 ≤ x ≤ 12} as R={(a, b); a=b a, b ∈ A} is : 1
(A) reflexive. (B) transitive.
(C) symmetrical. (D) an equivalence relation.
‚◊Èëøÿ A={x ∈ I, 0 ≤ x ≤ 12} ◊¥ ‚¥’œ¥ R ¡Ê ÁŸêŸ ¬˝∑§Ê⁄U ‚ ¬Á⁄U÷ÊÁ·Ã „Ò R={(a, b); a=b a, b ∈ A} ∞∑§ —
(A) Sfl¥ÃÈÀÿ „Ò– (B) ‚¥∑˝§Ê◊∑§ „Ò–
(C) ‚◊Á◊à „Ò– (D) ‚◊ÃÈÀÿ ‚¥’¥œ „Ò–

10. The domain of the function y=sec−1 x is : 1
(A) [−1, 1] (B) R
(C) x / 1 or x ≤−1 (D) R−{0}
»§‹Ÿ y=sec−1 x ∑§Ê ¬˝Ê¥Ã „Ò —
(A) [−1, 1] (B) R
(C) x / 1 ÿÊ x ≤−1 (D) R−{0}

11. Let * be a binary operation on the set N of natural numbers defined by the rule a*b=ab, for 1
all, a, b ∈ N, then :
(A) * is commutative.
(B) * is associative.
(C) * is both commutive and associative.
(D) * is neither commutive nor associative.
∞∑§ Ám•ÊœÊ⁄UË ‚¥Á∑˝§ÿÊ * ¬˝Ê∑ΧÁÃ∑§ ‚¥ÅÿÊ•Ê¥ ∑§ ‚◊Èëøÿ N ¬⁄U ß‚ ¬˝∑§Ê⁄U ¬Á⁄U÷ÊÁ·Ã „Ò Á∑§ a*b=ab ¡„Ê° ‚÷Ë
a, b ∈ N – Ã’ —
(A) * ∑˝§◊ÁflÁŸ◊ÿ „Ò–
(B) * ‚„øÊ⁄UË „Ò–
(C) * ∑˝§◊ÁflÁŸ◊ÿ •ÊÒ⁄U ‚„øÊ⁄UË „Ò–
(D) * Ÿ ÃÊ ∑˝§◊ÁflÁŸ◊ÿ •ÊÒ⁄U Ÿ „Ë ‚„øÊ⁄UË „Ò–

12. The inverse of the function y=x2, for all x ∈ R is : 1
(A) f−1(x)=−x (B) f −1(x)=x
(C) f−1(x)=|x| (D) does not exist
»§‹Ÿ y=x2, ¡„Ê° ‚÷Ë x ∈ R, ∑§Ê ¬˝ÁËÊ◊ »§‹Ÿ —
(A) f−1(x)=−x „Ò– (B) f −1(x)=x „Ò –
(C) f−1(x)=|x|„Ò– (D) ∑§Ê •ÁSÃàfl Ÿ„Ë¥ „Ò–

dy
13. If y=(1−x2)5, then is : 1
dx

ÿÁŒ y=(1−x2)5 „Ò, ÃÊ dy „Ò —
dx
(A) 5(1−x 2)4 (B) (1−2x)5
(C) 5(1−x2)4 (−2x) (D) 5(1−2x) 4

68/ESS/1-311-A ] 5 [ Contd...

dy
14. If y=sin2x cos3x, then is : 1
dx

ÿÁŒ y=sin2x cos3x „Ò, ÃÊ dy „Ò —
dx
(A) 2 cos2x cos3x−3 sin2x sin3x (B) 2 cos2x cos3x+3 sin2x sin3x
(C) 6 cos2x sin3x−6 cos2x sin3x (D) 6 cos2x sin3x+6 cos2x sin3x

15. Which of the following is a vector quantity ? 1
(A) Mass (B) Density (C) Force (D) Time
ÁŸêŸ ◊¥ ∑§ÊÒŸ‚Ë ‚ÁŒ‡Ê ⁄UÊÁ‡Ê „Ò?
(A) Œ˝√ÿ◊ÊŸ (B) ÉÊŸàfl (C) ’‹ (D) ‚◊ÿ

→ ∧ ∧ ∧ ∧ ∧ ∧ ∧
16. The unit vector parallel to the resultant of vector a = 3 i − 2 j + k and b=−2 i + 4 j + k is : 1

→ ∧ ∧ ∧ ∧ ∧ ∧ ∧
‚ÁŒ‡ÊÊ¥ a = 3 i − 2 j + k •ÊÒ⁄U b=−2 i + 4 j + k ∑§ ¬Á⁄UáÊÊ◊Ë ‚ÁŒ‡Ê ∑§ ‚◊Ê¥Ã⁄ ∞∑§∑§ ‚ÁŒ‡Ê „Ò —U

(A)
∧ ∧
i + 2 j + 2k
∧
(B)
5
(
1 ∧ ∧ ∧
i + 2 j + 2k )
(C)
1 ∧
3
( ∧ ∧
i + 2 j + 2k ) (D) ±
3
(
1 ∧ ∧ ∧
i + 2 j + 2k )
17. The distance of the point (3, 4, −5) from the plane 2x−3y+3z+27=0 is : 1

6
(A) 6 units (B) 22 units (C) units (D) 132 units
22
‚◊Ë 2x−3y+3z+27=0 ∑§Ë Á’¥ŒÈ (3, 4, −5) ‚ ŒÍ⁄UË „Ò —
6
(A) 6 ß∑§Ê߸ (B) 22 ß∑§Ê߸ (C) ß∑§Ê߸ (D) 132 ß∑§Ê߸
22

18. The equation of the line passing through the points (1, 4, 7) and (3, −2, 5) is : 1
Á’ãŒÍ•Ê¥ (1, 4, 7) •ÊÒ⁄U (3, −2, 5) ‚ ªÈ¡⁄UŸ flÊ‹Ë ⁄UπÊ ∑§Ê ‚◊Ë∑§⁄UáÊ „Ò —
x− 1 y− 4 z− 7 x− 1 y− 4 z
(A) = = (B) = =
2 −6 −2 2 2 12

x− 1 y− 4 z −7 x− 1 y− 4 z −7
(C) = = (D) = =
2 6 −2 2 −2 2

68/ESS/1-311-A ] 6 [ Contd...

19. Converse of the statement “If x is divisible by 4, then x is even” is : 1
(A) If x is not divisible by 4, then x is not even.
(B) If x is even, then x is divisible by 4.
(C) If x is even, then x is not divisible by 4.
(D) If x is not divisible by 4, then x is even.
∑§ÕŸ ““ÿÁŒ x, 4 ‚ Áfl÷ÊÁ¡Ã „Ò ÃÊ x ∞∑§ ‚◊ ‚¥ÅÿÊ „Ò”” ∑§Ê Áfl‹Ê◊ ∑§ÕŸ „Ò —
(A) ÿÁŒ x, 4 ‚ Áfl÷ÊÁ¡Ã Ÿ„Ë¥ „Ò, ÃÊ x ‚◊ ‚¥ÅÿÊ Ÿ„Ë¥ „Ò–
(B) ÿÁŒ x ‚◊ ‚¥ÅÿÊ „Ò, ÃÊ x, 4 ‚ Áfl÷ÊÁ¡Ã „Ò–
(C) ÿÁŒ x ‚◊ ‚¥ÅÿÊ „Ò, ÃÊ x, 4 ‚ Áfl÷ÊÁ¡Ã Ÿ„Ë¥ „Ò–
(D) ÿÁŒ x, 4 ‚ Áfl÷ÊÁ¡Ã Ÿ„Ë¥ „Ò, ÃÊ x ‚◊ ‚¥ÅÿÊ „Ò–

2
 d2 y  6  dy 
4
20. The degree of the differential equation   +x   = 0 is : 1
 dx 2   dx 

2
 2  4
•fl∑§‹ ‚◊Ë∑§⁄UáÊ  d y  + x6  dy  = 0 ∑§Ë ÉÊÊà „Ò —
 dx 2   dx 
(A) 2 (B) 4 (C) 6 (D) 10

21. Match Column - I statement with the correct option of Column - II. 1x2=2
Column - I Column - II

−1 3 6
(a) The cofactor of the element −2 in the matrix  2 5 −2  is (P) 13
 
 4 1 3 

−4 5
(b) If A =   , then ?adj A? is (Q) −2
 2 −3 
(R) −13
(S) 2
∑§ÊÚ‹◊ - I ∑§ ∑§ÕŸÊ¥ ∑§Ê ∑§ÊÚ‹◊ - II ∑§ ‚„Ë Áfl∑§À¬ ‚ Á◊‹Ê∞°–
∑§ÊÚ‹◊ - I ∑§ÊÚ‹◊ - II
−1 3 6
(a) •Ê√ÿÍ„ 2 5 −2  ∑§ •flÿfl −2 ∑§Ê ‚„πá«U „Ò
 (P) 13
 
 4 1 3 

−4 5
(b) ÿÁŒ A =   „Ò, ÃÊ ?adj A? „Ò — (Q) −2
 2 −3 
(R) −13
(S) 2

68/ESS/1-311-A ] 7 [ Contd...

22. Fill in the blanks : 1x2=2

dy
(i) If y = sin 3 x , then is __________.
dx
(ii) The order of the differential equation xdx+ydy=0 is __________.
Á⁄UÄà SÕÊŸÊ¥ ∑§Ê ÷Á⁄U∞ —
dy
(i) ÿÁŒ y = sin 3 x „Ò, ÃÊ „Ò __________–
dx
(ii) •fl∑§‹ ‚◊Ë∑§⁄UáÊ xdx+ydy=0 ∑§Ë ∑§ÊÁ≈U „Ò __________–

23. Write True for correct statement and False for incorrect statements. 1x2=2
(i) The relation “is a factor of” from R to R is reflexive and Transitive but not symmetric.
(ii) The function f : R → R defined by f (x)=x2+3 is one-one and onto.
‚„Ë ∑§ÕŸ ∑§ Á‹∞ ‚àÿ •ÊÒ⁄U ª‹Ã ∑§ÕŸ ∑§ Á‹∞ •‚àÿ Á‹Áπ∞ —
(i) R ‚ R ◊¥ ¬Á⁄U÷ÊÁ·Ã ‚¥’¥œ ““∑§Ê ªÈáÊŸπ¥«U „Ò”” SflÃÈÀÿ ÃÕÊ ‚¥∑˝§Ê◊∑§ „Ò ¬⁄UãÃÈ ‚◊Á◊à Ÿ„Ë¥ „Ò–
(ii) »§‹Ÿ f : R → R ¡Ê f (x)=x2+3 mÊ⁄UÊ ¬Á⁄U÷ÊÁ·Ã „Ò ∞∑§ ∞∑Ò§∑§Ë •ÊÒ⁄U •Êë¿UÊŒ∑§ »§‹Ÿ „Ò–

24. Write the negation of each of the following statements : 1x2=2
(i) The number is less than 5.
(ii) All prime numbers are odd.
ÁŸêŸÁ‹Áπà ∑§ÕŸÊ¥ ∑§Ê ÁŸ·œŸ Á‹Áπ∞ —
(i) ‚¥ÅÿÊ 5 ‚ ¿UÊ≈UË „Ò–
(ii) ‚÷Ë •÷Êíÿ ‚¥ÅÿÊ∞° Áfl·◊ ‚¥ÅÿÊ∞° „ÊÃË „Ò¥–

25. Fill in the blanks : 1x4=4
(i) It is not possible to add two matrices of __________ orders.
(ii) If A is an invertible matrix, then (A−1)−1=__________.
(iii) Three points are collinear if the area of the triangle formed by these three points is
_________.
(iv) A square matrix A is said to be a _________ matrix, if A’=−A
Á⁄UÄà SÕÊŸÊ¥ ∑§Ê ÷Á⁄U∞ —
(i) __________ ∑˝§◊Ê¥ ∑§ ŒÊ •Ê√ÿÍ„Ê¥ ∑§Ê ÿÊª ‚¥÷fl Ÿ„Ë¥ „Ò–
(ii) ÿÁŒ •Ê√ÿÍ„ A ∑§ Á‹∞ A−1 ∑§Ê •ÁSÃàfl „Ò, ÃÊ (A−1)−1=__________–
(iii) ÃËŸ Á’¥ŒÈ ‚⁄Uπ „Ê¥ª ÿÁŒ ߟ Á’ãŒÈ•Ê¥ ‚ ’ŸŸ flÊ‹Ë ÁòÊ÷È¡ ∑§Ê ˇÊòÊ»§‹ _________ „Ò–
(iv) ∞∑§ flª¸ •Ê√ÿÍ„ A ∞∑§ _________ •Ê√ÿÍ„ ∑§„‹Ê∞ªÊ ÿÁŒ A’=−A „Ò–

68/ESS/1-311-A ] 8 [ Contd...

26. Fill in the blanks : 1x4=4
dy
(i) If y=sin−1(x2), then is __________.
dx
(ii) ∫3 x5 −x dx=__________.
dy
(iii) The general solution of the first order linear differential equation + Py = Q is
dx
__________.
2
a xe x
(iv) ∫−a 1 + x 2 dx = __________.
Á⁄UÄà SÕÊŸÊ¥ ∑§Ê ÷Á⁄U∞ —

(i) ÿÁŒ y=sin−1(x2) „Ò, ÃÊ dy „Ò __________–
dx
(ii) ∫3 x 5 −xdx=__________ –

dy
(iii) ∑§ÊÁ≈U ∞∑§ ∑§ ⁄ÒUÁπ∑§ •fl∑§‹ ‚◊Ë∑§⁄UáÊ + Py = Q ∑§Ê •Á÷c≈ „‹ „ÊÃÊ „ÒU __________–
dx
2
a xe x
(iv) ∫−a 1 + x2 dx = __________–

27. Write True for correct statements and False for incorrect statements : 1x4=4
b b
(i) ∫a f (x) dx = ∫a f (a + b − x) dx.
d2 y
(ii) If x=at2 and y=2at, then =t.
dx 2

The slope of normal to curve y=f(x) at (x1, y1) is given by 
dy 
(iii)  at (x1, y1)
 dx 
(iv) The degree of a differential equation is the degree of the highest differential coefficient.
‚„Ë ∑§ÕŸ ∑§ Á‹∞ ‚àÿ •ÊÒ⁄U ª‹Ã ∑§ÕŸ ∑§ Á‹∞ •‚àÿ Á‹Áπ∞ —
b b
(i) ∫a f (x) dx = ∫a f (a + b − x) dx –
2
(ii) ÿÁŒ x=at2 •ÊÒ⁄U y=2at „Ò, ÃÊ d y2 =t–
dx

(iii) fl∑˝§ y=f(x) ∑§Ê Á’ãŒÈ (x1, y1) ¬⁄U «UÊ‹ ª∞ •Á÷‹¥’ ∑§Ë ¬˝fláÊÃÊ Œ‡Êʸ߸ ¡ÊÃË „Ò  dy  (x1, y1) ¬⁄U mÊ⁄UÊ–
 dx 
(iv) •fl∑§‹ ‚◊Ë∑§⁄UáÊ ∑§Ë ÉÊÊà ‚’‚ ’«∏Ë ∑§ÊÁ≈U flÊ‹ •fl∑§‹¡ ∑§Ë ÉÊÊà „ÊÃË „Ò–

68/ESS/1-311-A ] 9 [ Contd...

28. Carefully study the figure given below and answer the following : 1x4=4

∧ ∧ ∧ →
(i) What are x i , y j and z k called for the vector r ?
→
(ii) If OA = 4 , ?OB?=5 and ?OC?=6, then express OP in terms of its component vectors.
→ → →
(iii) If a , b and c are the position vectors of vertices A, B and C of ∆ABC, then write the
position vector of the centroid of ∆ABC.
→ →
(iv) If a and b are the position vectors of A and B respectively, then find the position
vector of the point which divides the join of A and B in the ratio 2 : 3 internally.
‚¥‹ÇŸ ÁøòÊ ∑§Ê äÿÊŸ¬Ífl¸∑§ ¬Á…∏∞ •ÊÒ⁄U ÁŸêŸ ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒËÁ¡∞ —
→ ∧ ∧ ∧
(i) ‚ÁŒ‡Ê r ∑§ ‚¥Œèʸ ◊¥ x i , y j •ÊÒ⁄U z k ∑§Ê ÄÿÊ ∑§„Ê ¡ÊÃÊ „Ò?
→
(ii) ÿÁŒ OA = 4 , ?OB?=5 ÃÕÊ ?OC?=6 „Ò, ÃÊ OP ∑§Ê ÉÊ≈U∑§ ‚ÁŒ‡ÊÊ¥ ∑§ M§¬ ◊¥ √ÿÄà ∑§ËÁ¡∞–
→ → →
(iii) ÿÁŒ ÁòÊ÷È¡ ABC ∑§ ‡ÊË·ÊZ A, B •ÊÒ⁄U C ∑§ ÁSâÊÁà ‚ÁŒ‡ÊÊ¥ ∑§Ê a , b •ÊÒ⁄U c ‚ Œ‡ÊʸÿÊ ¡Ê∞, ÃÊ ÁòÊ÷È¡
ABC ∑§ ∑§ãŒ˝∑§ ∑§Ê ÁSÕÁÃU ‚ÁŒ‡Ê Á‹Áπ∞–
→ →
(iv) ÿÁŒ a •ÊÒ⁄U b , ŒÊ Á’ãŒÈ•Ê¥ A •ÊÒ⁄U B ∑§ ÁSÕÁà ‚ÁŒ‡Ê „Ò, ÃÊ ©‚ Á’¥ŒÈ ∑§Ê ÁSÕÁà ‚ÁŒ‡Ê ôÊÊà ∑§ËÁ¡∞ ¡Ê A
•ÊÒ⁄U B ∑§Ê Á◊‹ÊŸ flÊ‹ ⁄UπÊ-π¥«U ∑§Ê 2 : 3 ∑§ •ŸÈ¬Êà ◊¥ •ã× Áfl÷ÊÁ¡Ã ∑§⁄UÃÊ „Ò–
29. Read the passage and answer the question that follow it. 1x6=6
Let f be a real function and let c be any point in the domain of f. Then,
(i) c is called the point of local maxima if there exists h > 0 such that f(c)/f(x), for all
x e (c−h, c+h). The number f (c) is called the local maximum value of f.
(ii) c is called the point of local minima. If there exists h > 0 such that f(c)≤f(x), for all
x e (c−h, c+h).
The number f(c) is called the local minimum value of f.
The values of x for which f '(x)=0 are called stationary points or turning points.
The value of x for which f '(x)=0 or f '(x) does not exist are called critical points.
Further, the end of points of domain or f cannot be the points of local maxima or local minima.
 1 − x + x2
Consider the function f (x ) =  on R.
 1 + x + x2
(i) How many critical points does the function f(x) have ?
(a) 0 (b) 1 (c) 2 (d) 3

68/ESS/1-311-A ] 10 [ Contd...

(ii) Which statement about the critical points of the function f (x) is correct ?
(a) All the critical points of f (x) are positive.
(b) All the critical points of f (x) are negative.
(c) Some critical points of f (x) are positive while others are negative.
(d) None of above as function f (x) does not have any critical point.
(iii) At positive value of critical point of f (x), the function f (x) has :
(a) Local maxima.
(b) Local minima.
(c) Neither local maxima nor local minima.
(d) None of the above as function f (x) have any critical point.
(iv) At negative value of critical point of f (x), the function has :
(a) Local maxima.
(b) Local minima.
(c) Neither local maxima nor local minima
(d) None of the above as function f (x) have any critical point.
(v) Local minimum value of f (x) is :
1
(a) 3 (b) (c) −1 (d) None of these
3
(vi) Local maximum value of f (x) is :
(a) 7 (b) 3 (c) 5 (d) None of these
ÁŸêŸ ªlÊ¥‡Ê ∑§Ê ¬…∏¥ •ÊÒ⁄U ŸËø Á‹π ¬˝‡ŸÊ¥ ∑§ ©ûÊ⁄U ŒËÁ¡∞ —
◊ÊŸ ‹ËÁ¡∞ Á∑§ f ∞∑§ flÊSÃÁfl∑§ »§‹Ÿ „Ò •ÊÒ⁄U c, f ∑§ ¬˝Ê¥Ã ∑§Ê ∑§Ê߸ Á’¥ŒÈ „Ò Ã’,
(i) c ∑§Ê SÕÊŸËÿ ©Áëøc∆U ∑§Ê Á’¥ŒÈ ∑§„Ê ¡ÊÃÊ „Ò ÿÁŒU h > 0 ∑§Ê •ÁSÃàfl ß‚ ¬˝∑§Ê⁄U „Ò Á∑§ ‚÷Ë x e (c−h, c+h)
∑§ Á‹∞ f(c)/f(x) ‚¥ÅÿÊ f (c) ∑§Ê f ∑§Ê SÕÊŸËÿ ©Áëøc≈◊ÊŸ ∑§„Ê ¡ÊÃÊ „Ò–U
(ii) c ∑§Ê SÕÊŸËÿ ÁŸêŸc∆U U ∑ §Ê Á’¥ Œ È ∑§„Ê ¡ÊÃÊ „Ò , ÿÁŒ h > 0 ∑§Ê •ÁSÃàfl ß‚ ¬˝ ∑ §Ê⁄U „Ò Á∑§ ‚÷Ë
x e (c−h, c+h) ∑§ Á‹∞ f(c)≤ f(x).
‚¥ÅÿÊ f(c) ∑§Ê f ∑§Ê SÕÊŸËÿ ÁŸÁêŸc∆U ◊ÊŸ ∑§„Ê ¡ÊÃÊ „Ò–U
x ∑§ fl ◊ÊŸ Á¡Ÿ∑§ Á‹∞ f '(x)=0 SÃéœ Á’¥ŒÈ ÿÊ ◊Ê«∏ Á’¥ŒÈ ∑§„‹ÊÃ „Ò–
x ∑§ fl ◊ÊŸ Á¡Ÿ∑§ Á‹∞ f '(x)=0 ÿÊ f '(x) ∑§Ê ∑§Ê߸ •ÁSÃàfl Ÿ„Ë¥ „Ò, ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ ∑§„‹ÊÃ „Ò¥–
ß‚∑§ •‹ÊflÊ, f ∑§ ¬˝Ê¥Ã ∑§ •¥ÁÃ◊ Á’¥ŒÈ, SÕÊŸËÿ ©Áëøc∆U •ÕflÊ SÕÊŸËÿ ÁŸêŸc∆ Á’¥ŒÈ Ÿ„Ë¥ „Ê ‚∑§Ã „Ò¥–
 2
◊ÊŸ ‹¥ Á∑§ »§‹Ÿ R ‚ ‚¥’h „Ò, ÃÊ f (x ) =  1 − x + x 2 –
1 + x + x
(i) f(x) ∑§ Á∑§ÃŸ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ „Ò¥?
(a) 0 (b) 1 (c) 2 (d) 3
(ii) ÁŸêŸ ∑§ÕŸ »§‹Ÿ f (x) ∑§ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ ∑§ Á‹∞ ‚àÿ „Ò?
(a) f (x) ∑§ ‚÷Ë ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ œŸÊà◊∑§ „ÊÃ „Ò¥–
(b) f (x) ∑§ ‚÷Ë ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ ´§áÊÊà◊∑§ „ÊÃ „Ò¥–
(c) f (x) ∑§ ∑ȧ¿U ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ œŸÊà◊∑§ •ÊÒ⁄U •ãÿ ´§áÊÊà◊∑§ „ÊÃ „Ò¥–
(d) ™§¬⁄ËU ∑§ÕŸÊ¥ ◊¥ ∑§Ê߸ ÷Ë ‚àÿ Ÿ„Ë¥ „Ò, ÄÿÊ¥Á∑§ f (x) ∑§Ê ∑§Ê߸ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ Ÿ„Ë¥ „Ò–

68/ESS/1-311-A ] 11 [ Contd...

(iii) œŸÊà◊∑§ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ ¬⁄U, »§‹Ÿ f (x) ∑§Ê —
(a) SÕÊŸËÿ ©Áëøc∆U „ÊªÊU–
(b) SÕÊŸËÿ ÁŸêŸc∆U „ÊªÊUU–
(c) Ÿ ÃÊ SÕÊŸËÿ ©Áëøc∆U •ÊÒ⁄U Ÿ „Ë SÕÊŸËÿ ÁŸêŸc∆U „ÊªÊUU–
(d) ™§¬⁄U ◊¥ ÁŒ∞ ª∞ ◊¥ ‚ ∑§Ê߸ ÷Ë Ÿ„Ë¥ „Ò ÄÿÊ¥Á∑§ f (x) ∑§Ê ∑§Ê߸ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ Ÿ„Ë¥ „Ò–
(iv) ´§áÊÊà◊∑§ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ ¬⁄U, »§‹Ÿ f (x) ∑§Ê —
(a) SÕÊŸËÿ ©Áëøc∆U „ÊªÊU
(b) SÕÊŸËÿ ÁŸêŸc∆U „ÊªÊUU
(c) Ÿ ÃÊ SÕÊŸËÿ ©Áëøc∆U •ÊÒ⁄U Ÿ „Ë SÕÊŸËÿ ÁŸêŸc∆U „ÊªÊUU
(d) ™§¬⁄U ◊¥ ÁŒ∞ ª∞ ◊¥ ‚ ∑§Ê߸ ÷Ë Ÿ„Ë¥ „Ò ÄÿÊ¥Á∑§ f (x) ∑§Ê ∑§Ê߸ ∑˝§Ê¥ÁÃ∑§ Á’¥ŒÈ Ÿ„Ë¥ „Ò–
(v) f (x) ∑§Ê SÕÊŸËÿ ÁŸêŸc∆U ◊ÊŸ „Ò —U

1
(a) 3 (b)
3
(c) −1 (d) ߟ◊¥ ‚ ∑§Ê߸ Ÿ„Ë¥
(vi) f (x) ∑§Ê SÕÊŸËÿ ©Áëøc∆U ◊ÊŸ „Ò —U
(a) 7 (b) 3 (c) 5 (d) ߟ◊¥ ‚ ∑§Ê߸ Ÿ„Ë¥

SECTION - B / πá«U - π

5
30. Find the equation of the hyperbola with vertices at (0, ±6) and e = . 2
3

©‚ •Áà ¬⁄Ufl‹ÿ ∑§Ê ‚◊Ë∑§⁄UáÊ ôÊÊà ∑§ËÁ¡∞ Á¡‚∑§ ‡ÊË·¸ (0, ±6) ÃÕÊ ©à∑§ãŒ˝ÃÊ e = 5 „Ò–
3
OR / •ÕflÊ

1
Find the equation of the ellipse, when focus is (−1, 1), directrix is x−y+3=0 and e = .
2

1
ŒËÉʸflÎûÊ ∑§Ê ‚◊Ë∑§⁄UáÊ ôÊÊà ∑§ËÁ¡∞ Á¡‚∑§Ë ŸÊÁ÷ (−1, 1), ©à∑§ãŒ˝ÃÊ e = 2 ÃÕÊ ÁŸÿÃÊ x−y+3=0 „Ò–

 2 −3   x   1
31. Solve the matrix equation :     =  2
1 1  y  3

•Ê√ÿÍ„ ‚◊Ë∑§⁄UáÊ „‹ ∑§ËÁ¡∞ — 
2 −3   x   1
   = 
1 1  y  3
OR / •ÕflÊ

68/ESS/1-311-A ] 12 [ Contd...

0 1 0
Solve for x : x 2 x = 0
1 3 x

0 1 0
x ∑§ Á‹∞ „‹ ∑§ËÁ¡∞ — x 2 x = 0
1 3 x

−a2 ab ac
32. Prove that ab −b 2 bc = 4 a 2 b 2c 2 2

ac bc −c 2

−a2 ab ac
Á‚h ∑§ËÁ¡∞ — ab −b 2 bc = 4 a 2 b 2c 2
ac bc −c 2

Simplify : tan sin ( 1 − x ) 
−1
33. 2

‚⁄U‹ ∑§ËÁ¡∞ — tan sin−1 ( 1 − x )

x− 3
34. Evaluate : lim 2
x → 3 x− 3

◊ÊŸ ôÊÊà ∑§ËÁ¡∞ — lim x − 3
x → 3 x− 3
OR / •ÕflÊ
sin 3x
Evaluate : lim
x→0 x
sin 3x
◊ÊŸ ôÊÊà ∑§ËÁ¡∞ — xlim
→0 x

x+ 1 d2 y
35. If y = , (x ≠ 1) find 2
x− 1 dx 2
x+ 1 d2 y
ÿÁŒ y = x − 1 , (x ≠ 1) „Ò, ÃÊ ôÊÊà ∑§ËÁ¡∞–
dx 2
OR / •ÕflÊ
1 x dy
If y = e , Find .
x dx
1 dy
ÿÁŒ y = x e x „Ò, ÃÊ ôÊÊà ∑§ËÁ¡∞–
dx

68/ESS/1-311-A ] 13 [ Contd...

dy cos 2 (a + y )
36. If cosy=xcos(a+y), then prove that = 2
dx sin a
dy cos 2 (a + y )
ÿÁŒ cosy=xcos(a+y) „Ò, ÃÊ Á‚h ∑§ËÁ¡∞ Á∑§ =
dx sin a

37. Find the distance of the point (3, 4, −5) from the plane 2x−3y+3z+27=0. 2
‚◊Ë 2x−3y+3z+27=0 ‚ Á’¥ŒÈ (3, 4, −5) ∑§Ë ŒÍ⁄UË ôÊÊà ∑§ËÁ¡∞–

38. Reduce the equation of the plane 4x−5y+6z−60=0 to the intercept form. Find its intercepts 2
on the coordinate axes.
‚◊Ë ∑§ ‚◊Ë∑§⁄UáÊ 4x−5y+6z−60=0 ∑§Ê •ã× πá«U SflM§¬ ◊¥ ’ŒÁ‹ÿ– ß‚∑§ ÁŸŒ¸‡ÊÊ¥∑§-•ˇÊÊ¥ ¬⁄U •ã×
πá«U ôÊÊà ∑§ËÁ¡∞–

39. For the ellipse 3x2+2y2=6, find the lengths of minor and major axes, coordinate of foci, 4
vertices and the eccentricity.
ŒËÉʸflÎûÊ 3x2+2y2=6 ∑§ Á‹∞ ©‚∑§ •ˇÊÊ¥ ∑§Ë ‹¥’ÊßÿÊ°, ŸÊÁ÷ÿÊ¥ ∑§ ÁŸŒ¸‡ÊÊ¥∑§, ‡ÊË·¸ •ÊÒ⁄U ©à∑§ãŒ˝ÃÊ ôÊÊà ∑§ËÁ¡∞–

 2 −6  4
40. Using elementary column transformation, fluid the inverse of the matrix A =  .
 1 −2 

 2 −6 
¬˝Ê⁄UÁê÷∑§ SÃ¥÷ ‚¥Á∑˝§ÿÊ•Ê¥ mÊ⁄UÊ •Ê√ÿÍ„ A ∑§Ê √ÿÍà∑˝§◊ ôÊÊà ∑§ËÁ¡∞ ¡„Ê° A =  –
 1 −2 

−1  1  −1  1  −1  2 
41. Prove that : tan   + tan   = tan   4
7  13  9

Á‚h ∑§ËÁ¡∞ — tan−1 
1 −1  1  −1  2 
 + tan   = tan  
7  13  9

π/2 sin x − cos x
42. Find the value of ∫ dx . 4
0 1 + sin x cos x

◊ÊŸ ôÊÊà ∑§ËÁ¡∞ — ∫ π/2 sin x − cos x dx –
0 1 + sin x cos x
OR / •ÕflÊ
3
Evaluate :
∫−3
x + 1
dx.
◊ÊŸ ôÊÊà ∑§ËÁ¡∞ — ∫ 3
x + 1
dx –
−3

68/ESS/1-311-A ] 14 [ Contd...

43. Prove that the lines 4
x+ 1 y+ 3 z+ 5 x− 2 y− 4 z− 6
= = and = = are coplanar. Also, find the
3 5 7 1 4 7
equation of the plane containing these lines.
x+ 1 y+ 3 z+ 5 x− 2 y− 4 z− 6
Á‚h ∑§ËÁ¡∞ Á∑§ ⁄UπÊ∞° = = •ÊÒ⁄U = = ‚◊ËËÿ „Ò– ©‚
3 5 7 1 4 7
‚◊Ë ∑§Ê ‚◊Ë∑§⁄UáÊ ÷Ë ôÊÊà ∑§ËÁ¡∞ Á¡‚◊¥ ÿ ⁄UπÊ∞° ÁSÕà „Ò¥–

44. A machine producing either products A or B can produce one unit of A by using 2 units of 6
chemicals and 1 unit of a compound and can produce one unit of B by using 1 unit of
chemicals and 2 units of the compound. Only 800 units of chemicals and 1000 units of the
compound are available. The profit available per unit of A and B are respectively ` 30 and
` 20. Find the optimum allocation of units between A and B to maximise, the total profit.
Find the maximum profit.
©à¬ÊŒ A ÿÊ B ∑§Ê ©à¬ÊŒŸ ∑§⁄UŸ flÊ‹Ë ∞∑§ ◊‡ÊËŸ, A ∑§Ë 1 ß∑§Ê߸ ∑§ ©à¬ÊŒŸ ∑§ Á‹∞ 2 ß∑§Ê߸ ⁄U‚ÊÿŸ •ÊÒ⁄U 1 ß∑§Ê߸
ÿÊÒÁª∑§ ∑§Ê ©¬ÿÊª ∑§⁄UÃË „Ò; •ÊÒ⁄U B ∑§Ë 1 ß∑§Ê߸ ∑§ ©à¬ÊŒŸ ∑§ Á‹∞ 1 ß∑§Ê߸ ⁄U‚ÊÿŸ •ÊÒ⁄U 2 ß∑§Ê߸ ÿÊÒÁª∑§ ∑§Ê ©¬ÿÊª
∑§⁄UÃË „Ò– ∑§fl‹ 800 ß∑§Ê߸ ⁄U‚ÊÿŸ •ÊÒ⁄U 1000 ß∑§Ê߸ ÿÊÒÁª∑§ ∑§Ë ◊ÊòÊÊ ©¬‹éœ „Ò– A •ÊÒ⁄U B ∑§Ë ¬˝àÿ∑§ ß∑§Ê߸ ¬⁄U
∑˝§◊‡Ê— ` 30 •ÊÒ⁄U ` 20 ∑§Ê ‹Ê÷ •Á¡¸Ã ∑§⁄UÃË „Ò– •Áœ∑§Ã◊ ‹Ê÷ •Á¡¸Ã ∑§⁄UŸ „ÃÈ ß∑§ÊßÿÊ¥ A •ÊÒ⁄U B ∑§Ê ßc≈Ã◊
•Ê’¥≈UŸ πÊÁ¡∞– •Áœ∑§Ã◊ ‹Ê÷ ÷Ë ôÊÊà ∑§ËÁ¡∞–
OR / •ÕflÊ
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours
on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on
machine B to produce a package of bolts. He earns a profit of ` 17.50 per package on nuts
and ` 7 per package on botts. How many packages of each should be produced each day so
as to maximise his profits if he operates his machine for at the most 12 hours a day ? Form
the above as a L.P.P. and then solve it graphically.
∞∑§ ÁŸ◊ʸÃÊ “Ÿ≈U” •ÊÒ⁄U “’ÊÀ≈U” ∑§Ê ©à¬ÊŒŸ ∑§⁄UÃÊ „Ò– Ÿ≈˜‚ ∑§Ê ∞∑§ ¬Ò∑§≈U ÃÒÿÊ⁄U ∑§⁄UŸ ◊¥ ◊‡ÊËŸ A ¬⁄U 1 ÉÊ¥≈UÊ ∑§Ê◊
∑§⁄UŸÊ ¬«∏ÃÊ „Ò •ÊÒ⁄U ◊‡ÊËŸ B ¬⁄U 3 ÉÊ¥≈U ∑§Ê◊ ∑§⁄UŸÊ ¬«∏ÃÊ „Ò– ’ÊÀ≈U ∑§Ê ∞∑§ ¬Ò∑§≈U ÃÒÿÊ⁄U ∑§⁄UŸ ◊¥ ◊‡ÊËŸ A ¬⁄U 3 ÉÊ¥≈U
∑§Ê◊ ∑§⁄UŸÊ ¬«∏ÃÊ „Ò •ÊÒ⁄U ◊‡ÊËŸ B ¬⁄U 1 ÉÊ¥≈UÊ ∑§Ê◊ ∑§⁄UŸÊ ¬«∏ÃÊ „Ò– ÁŸ◊ʸÃÊ ∑§Ê Ÿ≈˜‚ ∑§ ∞∑§ ¬Ò∑§≈U •ÊÒ⁄U ’ÊÀ≈U ∑§ ∞∑§
¬Ò∑§≈U ¬⁄U ∑˝§◊‡Ê— ` 17.50 •ÊÒ⁄U ` 7 ∑§Ê ‹Ê÷ ¬˝Êåà „ÊÃÊ „Ò– ÿÁŒ fl„ ߟ ŒÊŸÊ¥ ◊‡ÊËŸÊ¥ ∑§Ê ¬˝ÁÃÁŒŸ 12 ÉÊ¥≈U ©¬ÿÊª
∑§⁄UÃÊ „Ê, ÃÊ •Áœ∑§Ã◊ ‹Ê÷ •Á¡¸Ã ∑§⁄UŸ ∑§ Á‹∞, ©‚∑§Ê ¬˝ÁÃÁŒŸ ߟ ŒÊŸÊ¥ ©à¬ÊŒ∑§Ê¥ ∑§ Á∑§ÃŸ-Á∑§ÃŸ ¬Ò∑§≈U ÃÒÿÊ⁄U
∑§⁄UŸ „Ê¥ª? ∞∑§ ⁄ÒUÁπ∑§ ¬˝Êª˝Ê◊Ÿ ‚◊SÿÊ ’ŸÊß∞ •ÊÒ⁄U •Ê‹π ÁflÁœ ‚ „‹ ∑§ËÁ¡∞–

45. Show that of all the rectangles of given perimeter, the square has the greatest area. 6
Œ‡Êʸß∞ ∑§Ë ÁŒÿ ª∞ ¬Á⁄U◊ʬ ∑§ •ÊÿÃÊ¥ ◊¥ ‚ flª¸ ∑§Ê ˇÊòÊ»§‹ •Áœ∑§Ã◊ „ÊÃÊ „Ò–
OR / •ÕflÊ
Find the point on the curve y2=4x which is nearest to the point (z, 1).
fl∑˝§ y2=4x ∑§Ê fl„ Á’¥ŒÈ ôÊÊà ∑§ËÁ¡∞ ¡Ê Á’¥ŒÈ (z, 1) ‚ ÁŸ∑§≈UÃ◊ ŒÍ⁄UË ¬⁄U „Ò–

-oOo-

68/ESS/1-311-A ] 15 [ Contd...