This document contains a mathematics sample paper with 29 questions across 4 sections - A, B, C and D. The questions cover a wide range of mathematics topics including calculus, algebra, trigonometry, matrices, probability, linear programming, and geometry. The paper is designed to test students' mathematical skills and knowledge over 3 hours with the maximum marks being 100.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
This is a 3 hour sample paper for cbse class 12 board exam. This covers all chapters of ncert 12th math book. For more such papers visit clay6.com/papers/.
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
This is a 3 hour sample paper for cbse class 12 board exam. This covers all chapters of ncert 12th math book. For more such papers visit clay6.com/papers/.
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
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Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Biological screening of herbal drugs: Introduction and Need for
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. MATHEMATICS (Sample Paper)
Time allowed: 3 hours M.M.- 100
SECTION - A
1. Evaluate : 𝑙𝑜𝑔
2+𝑥
2−𝑥
1
−1
dx
2. If ‘*’ be a binary operation on ‘R’ defined by a*b =
𝑎𝑏
5
then find identity element for the binary operation ‘*’ if
exists.
3. If ∆=
2 −3 5
6 0 1
1 5 −7
𝑡𝑒𝑛 𝑓𝑖𝑛𝑑 ∶ 𝑎12. 𝐴32 + 𝑎11. 𝐴21
4. If 𝑎 + 𝑏 = 𝑎 − 𝑏 then what is the angle between 𝑎 and 𝑏
SECTION B
5. Show : tan−1 1+𝑥+ 1−𝑥
1+𝑥− 1−𝑥
=
𝜋
4
+
1
2
cos−1
𝑥 , −1 < 𝑥 < 1
6. If y = 𝑙𝑜𝑔 (𝑥 + 𝑥2 + 1) 2
then show : (1 + x2
)
𝑑2 𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
− 2 = 0
7. Simplify: tan-1 a cos x−b sin x
b cos x+a sin x
8. Find 2 x 2 matrix B : B
1 −2
1 4
=
6 0
0 6
.
9. The radius of a spherical diamond is measured as 7 cm with an error of 0.04cm . find the approximate error in
calculating its volume.
10. The mean and variance of a binomial distribution are 12 and 3 respectively. Find the probability distribution.
11. Find the angle between the line
𝑥+1
2
=
3𝑦+5
9
=
3−𝑧
−6
and the plane 10x + 2y – 11z = 3.
12. Evaluate:
𝑑𝑥
7−6𝑥−𝑥2
SECTION -C
13. By using the properties of determinants show :
𝑎 𝑏 𝑐
𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎
𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏
= 𝑎3
+ 𝑏3
+ 𝑐3
− 3𝑎𝑏𝑐
14. Find the intervals in which f(x) is strictly increasing or strictly decreasing where f(x) = -2x3
-9x2
-12x+1
OR
Find the equation of tangent to the curve y = (3𝑥 − 2) which is parallel to line 4x – 2y + 5 = 0
15. If x =a Sin2t(1+Cos2t), y = bcos2t(1-Cos2t) find
𝑑𝑦
𝑑𝑥
16. Roger Federer wins thrice as often as he loses. If he plays two matches then find the probability distribution of
number of wins.
17. Evaluate :
𝑥2+𝑥+1
(𝑥+2)(𝑥2+1)
dx
18. Find the equation of plane which contains the point (1, -1, 2) and perpendicular to each of the planes 2x+3y-2z =
5 and x+2y-3z = 11
2. 19. f(x) =
𝑠𝑖𝑛 3𝑥
𝑡𝑎𝑛 2𝑥
, 𝑥 < 0
𝑘, 𝑥 = 0
𝑒3𝑥2
−1
1−𝑐𝑜𝑠 2𝑥
, 𝑥 > 0
, find k if f(x) is continuous at x = 0
20. Prove that volume of largest cone that can be inscribed in a sphere of radius ‘R’ is
8
27
of the volume of sphere.
Also find the height of cone.
OR
Given the sum of perimeter of a square and circle. Show that sum of their areas is least when the side of the
square is equal to the diameter of circle.
21. Three bags contain balls as shown in the table below :
Bag Number of white balls Number of green balls Number of red balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag chosen at random and two balls are drawn from it. These balls happen to be white and green.
What is the probability that these balls came from bag III?
22. If 𝑎 , 𝑏, 𝑐 are mutually perpendicular vectors of equal magnitudes then show that the vector
𝑎 + 𝑏 + 𝑐 is equally inclined to 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐.
OR
If𝑎 = 𝑖 + 𝑗 + 𝑘 and 𝑏 = 𝑗 − 𝑘 then find a vector 𝑐 such that 𝑎 × 𝑐 = 𝑏 𝑎𝑛𝑑 𝑎. 𝑐 = 3
23. Solve the following differential equation : x dy – y dx = 𝑥2 + 𝑦2 dx
SECTION D
24. Find the area bounded by the circles x2
+ y2
= 4 and x2
+ (y-2)2
= 4.
25. Show that relation ‘R’ defined on the set N x N given by : (a.b) R (c.d) ⇒a+d = b+c is a equivalence relation.
26. Evaluate :
𝑥 𝑡𝑎𝑛 𝑥
𝑠𝑒𝑐 𝑥+𝑡𝑎𝑛 𝑥
𝑑𝑥
𝜋
0
OR
𝑎−𝑥
𝑎+𝑥
𝑎
−𝑎
dx
27. Find the foot of perpendicular drawn from the point (0, 2, 3) to the line
𝑥+3
5
=
𝑦−1
2
=
𝑧+4
3
. Also find the image of
the point w.r.t. the line.
28. If A =
3 2 1
4 −1 2
7 3 −3
find A-1
and by using A-1
solve :
3𝑥 + 4𝑦 + 7𝑧 = 14
2𝑥 − 𝑦 + 3𝑧 = 4
𝑥 + 2𝑦 − 3𝑧 = 0
29. Prerna Singhal has Rs. 1500 for purchase the bags of chocolates and sweets. A bag of chocolates and a bag of
sweets cost Rs. 180 and Rs. 120 respectively. She has the storage capacity of 10 bags only. She earns a profit of
Rs. 11 and Rs. 9 respectively per bag of chocolates and sweets. Formulate it as a LPP and solve it graphically for
the maximum profit she can get for her birthday party.