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Msbshse class 12th maths question paper with solutions 2019.

MSBSHSE Class 12th Maths Question Paper With Solutions 2019 QUESTION PAPER CODE J – 60

Select and write the most appropriate answer from the given alternatives for each question. [6]

Question 1: The principal solution of cot x = -тИЪ3 are

(a) ЁЭЫС / 6, 5ЁЭЫС / 6 (b) 5ЁЭЫС / 6, 7ЁЭЫС / 6

(c) 5ЁЭЫС / 6, 11ЁЭЫС / 6 (d) ЁЭЫС / 6, 11ЁЭЫС / 6

Question 2: The acute angle between the two planes x + y + 2z = 3 and 3x – 2y + 2z = 7 is _____.

(a) sin -1 (5 / тИЪ102) (b) cos -1 (5 / тИЪ102)

(c) sin -1 (15 / тИЪ102) (d) cos -1 (15 / тИЪ102)

The acute angle between the two planes x + y + 2z = 3 and 3x – 2y + 2z = 7.

n 1 = i + j + 2k and n 2 = 3i – 2j + 2k

|n 1 | = тИЪ1 2 + 1 2 + 2 2 = тИЪ6

|n 2 | = тИЪ3 2 + (-2) 2 + 2 2 = тИЪ17

cos ╬╕ = | (i + j + k) . (3i – 2j + 2k) / тИЪ6 . тИЪ17 |

cos ╬╕ = | (1 x 3) + (1 x -2) + (2 x 2) / тИЪ102 |

(a) -2, -1, -2 (b) 2,1, 2 (c) 2, -1, -2 (d) -2, 1, -2

Question 4: If f(x) = (1 + 2x) 1/x , for x тЙа 0 is continuous at x = 0, then f(0) = _____.

f (0) = lim xтЖТ0 [(1 + 2x) 1/2x ] 2 = e 2

Question 5: тИлdx / 9x 2 + 1 = _____.

(a) (1 / 3) tan -1 (2x) + c (b) (1 / 3) tan -1 (x) + c

(c) (1 / 3) tan -1 (3x) + c (d) (1 / 3) tan -1 (6x) + c

= (1 / 9) тИлdx / x 2 + (1 / 3) 2

Question 6: If y = ae 5x + be -5x , then the differential equation is _____.

(a) d 2 y / dx 2 = 25y (b) d 2 y / dx 2 = -25y

(c) d 2 y / dx 2 = -5y (d) d 2 y / dx 2 = 5y

Write the truth values of the following statement:

(i) 2 is a rational number and тИЪ2 is an irrational number.

(ii) 2 + 3 = 5 or тИЪ2 + тИЪ3 = тИЪ5

The truth value of p and q is T.

The truth value of p ^ q is T.

q = тИЪ2 + тИЪ3 = тИЪ5 which is F.

The truth value of the given statement is T v F = T.

[b] Find the value of p if the vectors i – 2j + k, 2i – 5j + pk and 5i – 9j + 4k are coplanar.

= |2 (- 14 – 25) – 5 (- 10 – 20) – 4 (25 – 28)|

Since a, b and c are coplanar vectors, their scalar triple product is 0.

1 (-20 + 9p) + 2 (8 – 5p) + 1 (-18 + 25) = 0

– 20 + 9p + 16 – 10p – 18 + 25 = 0

Question 9: Show that the points A (- 7, 4, -2), B ( – 2, 1, 0) and C (3, – 2, 2) are collinear.

A (- 7, 4, -2), B ( – 2, 1, 0) and C (3, – 2, 2).

If the area of the triangle formed by these points is zero, then, the three points are collinear.

If the value of the determinant of the three vertices is zero, then, the points are collinear.

= – 7 (2 – 0) – 4 ( – 4 – 0) – 2 (4 – 3)

The three points A (- 7, 4, -2), B ( – 2, 1, 0) and C (3, – 2, 2) are collinear.

Question 10: Write the equation of the plane 3x + 4y – 2z = 5 in the vector form.

The equation of the given plane is 3x + 4y – 2z = 5.

(xi + yj + zk) . (3i + 4j – 2k) = 5

r . (3i + 4j – 2k) = 5 which is the required form.

Question 11: If y = x x , then find dy / dx.

On differentiating both sides,

(1 / y) * dy / dx = [x * 1 / x] + log x * 1

Question 12: Find the equation of a tangent to the curve y = x 2 + 4x + 1 at point (- 1, – 2).

(dy / dx) at (- 1, – 2) = 2 (-1) + 4 = 2

Slope of the tangent at the point (-1, -2) = 2

Equation of the tangent at the point (-1, -2) is

Question 13: Evaluate: тИлe x (1 + x) / cos 2 [xe x ] dx.

Let I = тИлe x (1 + x) / cos 2 [xe x ] dx

Differentiate with respect to x

тИлe x (1 + x) / cos 2 [xe x ] = tan (xe x ) + c

Question 14: Evaluate: тИл 0 ЁЭЫС/2 sin 2 x dx.

Let I = тИл 0 ЁЭЫС/2 sin 2 x dx

= тИл 0 ЁЭЫС/2 [1 – cos 2x] / [2] dx

= [(x / 2) – (sin 2x / 4)] 0 ЁЭЫС/2

[a] In тЦ│ABC, prove that sin ([B – C] / 2) = ([b – c] / a) cos (A / 2)

[b] Show that sin -1 [5 / 13] + cos -1 [3 / 5] = tan -1 [63 / 16].

= [(k sin B – k sin C)] / [(k sin A)] * cos (A / 2)

= k (sin B – sin C) / k sin A * cos (A / 2)

= 2 [cos [B + C] / 2] [sin [(B – C)] / 2] / 2 sin (A / 2) cos (A / 2) * cos (A / 2)

= 2 cos ([ЁЭЫС / 2] – [A / 2]) sin ([B – C] / 2) / 2 sin (A / 2)

тЗТ sin -1 5 / 13 + cos -1 3 / 5

тЗТ tan -1 5 / 12 + tan -1 4 / 3

R is a midpoint of the line segment AB and A-R-B which divides the AB in the ratio m:n.

n( \(\begin{array}{l}(\bar {AR})\end{array} \) ) = m( \(\begin{array}{l}(\bar {RB})\end{array} \) )

The cartesian equation of the line is 2x – 2 = 3y + 1 = 6z – 2.

2 (x – 1) = 3 (y + 1 / 3) = 6 (z – 1 / 3)

(x – 1) / 3 = y + (1 / 3) / 2 = z – (1 / 3) / 1

The direction ratios of the line are 3, 2, 1.

The line passes through the point (1, (-1 / 3), (1 / 3)).

Let A = (1, (-1 / 3), (1 / 3))

The line passes through the point having the position vector.

The line is parallel to the vector b.

The vector equation of the line is

Question 18: Discuss the continuity of the function

f (x) = [log (2 + x) – log (2 – x) / tan x], for x тЙа 0

= lim xтЖТ0 f (x) = lim xтЖТ0 log (2 + x) – log (2 – x) / tan x

= lim xтЖТ0 log ([(1 + x / 2) / (1 – x / 2)] / x) / (tan x) / x

= lim xтЖТ0 log [(1 + x / 2) – log (1 – x / 2) / x] / lim xтЖТ0 (tan x / x)

From (i) and (ii), lim xтЖТ0 f (x) = f (0).

The given function is continuous at x = 0.

[b] For the following probability density function of X, find

if f (x) = x 2 / 18, -3 < x < 3

E (X) = ╬╝ = тИСp i x i = 0.82

Question 20: Given X ~ B (n, p). If E (X) = 6, Var (X) = 4.2, then find n and p.

X ~ B (n, p); E (X) = 6, Var (X) = 4.2

Var (X) / E (X) = npq / np = q = 4.2 / 6 = 0.7

Let p,q denote the statements that the switches S 1 , S 2 are closed respectively.

The switching table for the given switching circuit is as follows:

Let the three numbers be x, y and z.

The matrix form of these equations are

On applying the row operations,

The required numbers are 1, -2, 3.

Question 23: [a] In triangle ABC, with usual notations prove that b 2 = c 2 + a 2 – 2a cos B.

Using the cartesian co-ordinate system in the above figure,

we get B тЙг (0,0) ; A тЙг (c cos B , c cos B) and C тЙг (a,0)

b 2 = (a – c cos B) 2 + (0 – c sin B) 2

b 2 = a 2 – 2 ac cos B + c 2 cos 2 B + c 2 sin 2 B

b 2 = a 2 – 2 ac cos B + c 2 (sin 2 B + cos 2 B)

= (a – b) 2 cos 2 C / 2 + (a + b) 2 sin 2 C / 2

= (a 2 + b 2 – 2ab) cos 2 C / 2 + (a 2 + b 2 + 2ab) . sin C / 2

= (a 2 + b 2 ) cos C / 2 – 2ab cos 2 C / 2 + (a 2 + b 2 ) . sin 2 C / 2 + 2ab sin 2 C / 2

= (a 2 + b 2 ) (cos 2 C / 2 + sin 2 C / 2) – 2ab (cos 2 C / 2 – sin 2 C / 2)

= (a 2 + b 2 ) – 2ab cos C {By cosine Rule}

Given general equation is px 2 – 8xy + 3y 2 + 14x + 2y + q = 0.

a = p , h = -4 , b = 3 , g = 7, f = 1 , c = q

Since the equation represents perpendicular lines;

Since the equation represents a pair of lines.

-3 (3q – 1) + 4 (- 4q – 7) + 7 (- 4 – 21) = 0

Question 25: Maximise: z = 3x + 5y subject to

x + 4y тЙд 24, 3x + y тЙд 21, x + y тЙд 9, x тЙе 0, y тЙе 0.

Z is maximum at the point (4, 5).

dy / dx = (dy / dt) / (dx / dt), where dx / dt тЙа 0.

Hence, find dy / dt if x = acos 2 t and y = asin 2 t.

Since x and y are differentiable functions of t,

dx / dt = lim (╬┤tтЖТ0) (╬┤x) / (╬┤x) and dy / dt = lim (╬┤tтЖТ0) (╬┤y) / (╬┤t)…(1)

Also, as (╬┤tтЖТ0), (╬┤xтЖТ0) … (2)

Now , (╬┤y) / (╬┤x) = (╬┤y) / (╬┤t) / (╬┤x) / (╬┤t) …. (╬┤t тЙа 0)

lim (╬┤tтЖТ0) (╬┤y) / (╬┤x) = lim (╬┤tтЖТ0) (╬┤y) / (╬┤t) / (╬┤x) / (╬┤t) as (╬┤t тЙа 0)

lim (╬┤tтЖТ0) (╬┤y) / (╬┤x) = lim (╬┤tтЖТ0) (╬┤y) / (╬┤t) / (╬┤x) / (╬┤t)

lim (╬┤xтЖТ0) (╬┤y) / (╬┤x) = (lim (╬┤tтЖТ0) (╬┤y) / (╬┤t) / (lim (╬┤xтЖТ0) (╬┤x) / (╬┤t)

lim (╬┤xтЖТ0) (╬┤y) / (╬┤x) exists and is equal to dy / dx.

dy / dx = (dy / dt) / (dx / dt) if dx / dt тЙа 0

dy / dx = (dy / dt) / (dx / dt)

= 2a sint cost / -2a cost sint

Question 27: [a] f (x) = (x – 1) (x – 2) (x – 3), x те║ [0, 4], find тАШcтАЩ if LMVT can be applied.

As f (x) is a polynomial in x,

(i) f (x) is continuous on [0, 4].

(ii) f (x) is differentiable on (0, 4).

Thus, all the conditions of LMVT are satisfied.

To verify LMVT we have to find c & c те║ (0, 4), such that

fтАЩ (c) = f (4) – f (0) / 4 – 0 ……..(1)

f (4) = (4 – 1) (4 – 2) (4 – 3) = 6

f (0) = (0 – 1) (0 – 2) (0 – 3) = -6 and

f (4) – f (0) / 4 – 0 = 3c 2 – 12c + 11

(6 – (-6)) / 4 = 3c 2 – 12c + 11

Both the value of c lie between 0 and 4.

Now, area of rectangle = xy = x (54 – x)

For extreme values f ‘( x) = 0

Area is maximum when x =27 , y = 27

The dimensions of rectangle are 27m * 27m.

Question 28: Prove that: тИлdx / тИЪx 2 + a 2 = log |x + тИЪx 2 + a 2 | + c.

Let I = тИл 1 / тИЪx 2 + a 2 dx

I = тИл 1/ тИЪ(a 2 tan 2 ЁЭЫЙ + a 2 ) asec 2 ЁЭЫЙ d ЁЭЫЙ

= тИл (a . sec 2 ЁЭЫЙ) / (a тИЪ(1 + tan 2 ЁЭЫЙ)) dЁЭЫЙ

= тИл (sec 2 ЁЭЫЙ) / (sec ЁЭЫЙ) dЁЭЫЙ

= log |sec ЁЭЫЙ + tan ЁЭЫЙ| + c 1

= log |x / a + тИЪ(sec 2 ЁЭЫЙ)| + c 1

= log |x / a + тИЪ 1 + tan 2 ЁЭЫЙ | + c 1

= log |x / a + тИЪ(1 + x 2 / a 2 )| + c 1

= log |x / a + тИЪ(a 2 + x 2 ) / a| + c 1

= log |x + тИЪ(x 2 + a 2 )| – log a + c 1

тИлdx / тИЪx 2 + a 2 = log |x + тИЪx 2 + a 2 | + c, where c = – log a +c

Question 29: Show that: тИл 0 ЁЭЫС/4 log (1 + tan x) dx = ЁЭЫС / 8 log 2.

Question 30: [a] Solve the differential equation (dy / dx) + y sec x = tan x.

[b] Solve the differential equation (x + y) dy / dx = 1.

IF = e тИлsec xdx = e log(sec x + tan x) = secx + tanx

y (sec + tan x) = тИл(sec x + tan x) tan x.dx

= тИлsec x tan x + тИл tan 2 x dx

y(sec x + tan x) = sec x + tan x – x + c

Comparing with dx / dy + P * x = Q,

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## Maharashtra Board (MSBSHSE) Question Paper for Class 12 Maths 2019

## Maharashtra Board HSC Maths Class 12 Question Paper 2019 with Solutions - Free PDF Download

## Download PDF of Maharashtra Board (MSBSHSE) Question Paper for Class 12 Maths 2019

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## Maharashtra HSC (Class 12th) 2023 Exam Pattern & Marking Scheme - Check Details Here

## Maharashtra HSC Exam Pattern 2023 - Highilghts

## Maharashtra HSC Exam 2023:┬аPreparation Tips

Confused about your exam or college applications, be the first to know.

Refer to the table below about the important highlights of┬аMaharashtra Board exam 2023:

- Each subject has a 100 marks┬аexam, with 80 marks for theory and 20 marks for practicals, orals, and projects.
- For the areas of agriculture science and technology, animal science and technology, biology, chemistry, geology, and physics, the theory test is worth 70 marks and the practical/oral/project is for 30 marks.
- To pass the┬аexam, each student must have a score of 35% on each topic.

Also Read:┬а Maharashtra B.Tech Admissions - All You Need to Know!

These 20 marks will be bifurcated in the following way-

- 5 marks will be for┬аlistening skills
- 5 marks for speech skills
- 10 marks for the respective subject connected activity/experiment

In all of Class 10th's topics, the same criterion will be applied.

You may also find helpful: How to Prepare - Maharashtra Board and MHT CET 2022 Simultaneously?

Given in the table below is the subject wise┬аMaharashtra HSC (Class 12)┬аMarking Scheme 2023:

## Maharashtra HSC (Class 12) Paper Pattern & Marking Scheme 2023┬аPDF

- Make a proper study schedule┬атАУ┬а ┬аAfter going through the Maharashtra HSC time table 2023, candidates should make an effective study plan where they must┬аessentially devote a proper amount of time to all subjects.┬а
- Objective-type questions┬атАУ ┬аTo prepare for objective-type questions, students should have a┬аclear┬аconcepts and learn all definitions, formulae and important terms/facts by heart. While writing the exam, read the question and options carefully, calculate the answer first in the mind and choose the answer.
- Practice Everyday тАУ As we know practice makes man perfect. Make an habit to practice Mathematics,┬аgrammar, and writing comprehension, composition on a regular basis. Also make┬аnotes on the important formulae, laws, definitions.┬а
- Cover the entire Syllabus┬а тАУ ┬аIt is utmost important to cover┬аthe entire syllabus. Do not leave out any topic for the exam or do not do suggestive study. Consult teachers and expert faculty in case of any doubts.
- Health тАУ┬а Students are advised to have a proper diet┬аand sleep well apart from studying. Ensuring physical soundness is integral part for regular preparation of the exam.
- Solve Previous YearsтАЩ┬аPapers and Sample Papers-┬а Once you are done with the Maharashtra HSC syllabus you just start solving previous┬аyearтАЩs question papers. Solving sample papers and previous years' question papers will definitely ace your preparation. This will help them know about the┬аtype of questions that are asked in the exams. It will also help them work on their time management.┬а

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## HSC Maths Paper 'leak': Five people from Buldhana arrested

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Find and download HSC past exam papers, with marking guidelines and notes from the marking centre (HSC marking feedback) , are available for each course. Get our latest COVID-19 advice. ... Mathematics K-10 Introduction Students with disability ...

We've collated HSC Maths past papers for Standard Maths, Advanced Maths, Extension 1 Maths and Extension 2 Maths from the past decade to ensure you get enough practice and feel prepared when exam time comes around!

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See the exam paper, plus marking guidelines and feedback from markers, for the 2020 NSW Mathematics Standard Higher School Certificate (HSC) exam.

HSC Science (General) Mathematics and Statistics question paper solved pdf is key to increase score in the final exam. H.S.C Mathematics and Statistics question paper pdf download with the answer available on this page. Students can get solutions for the 12th Board Exam Mathematics and Statistics question paper.

Mathematics Advanced 2020 HSC exam pack | NSW Education Standards Attention: NSW Caretaker Period has commenced. The caretaker period for the NSW Election commenced on 3 March 2023. Accordingly, no ministerial press releases or related information issued by the Government from this date will be available on this website.

See the exam paper, plus marking guidelines and feedback from markers, for the 2019 NSW Mathematics Standard Higher School Certificate (HSC) exam. Attention: NSW Caretaker Period has commenced. The caretaker period for the NSW Election commenced on 3 March 2023. Accordingly, no ministerial press releases or related information issued by the ...

Download Free Previous Years HSC Science Question Papers from 2013-2020. Physics, Chemistry, Maths, Biology, English, Hindi and Marathi Subjects. Target Publications is a long-standing educational content publishing enterprise that has remained an indispensable part of the education industry for over 15 years. ...

The three-hour Maharashtra HSC Mathematics Exam was scheduled to take place from 11 am to 2 pm. Due to rules prohibiting entry after 10.30 a.m., the pupils were present within the exam room. Students were there when the pictures were posted on social media. READ | UP Board paper evaluation to begin on March 18, results expected by this date

By India Today Education Desk: The Maharashtra HSC mathematics board exam paper went viral on social media at 10 am today, right before the exam began. The Class 12 paper leaked in Maharashtra's Buldana district. The news comes just days after the Maharashtra Class 12 English paper leak in which a case was filed against six teachers in Parbhani ...

Maths Ext 2 HSC Papers General / Standard Maths Trial Papers 101 papers online, 95 w. sol Maths (2U) Trial Papers 537 papers online, 515 w. sol Maths Ext 1 Trial Papers 660 papers online, 611 w. sol Maths Ext 2 Trial Papers 596 papers online, 493 w. sol General / Standard Maths Internal Assessment Tasks 35 papers online, 23 w. sol Maths (2U ...

We have collected the HSC Science (General) previous year Question Papers by Year and Subject. Solving these Maharashtra State Board 12th Board Exam question papers is a very good approach to prepare for the examinations. Download 12th Board Exam Chapter-wise Solved Papers PDF.

You've come to the right place! Not only will we give you past papers for various subjects studied in the HSC, we'll also give you the best tips to help you ace those subjects. The marking guidelines are also included just to make things even more convenient for you. These are the subjects we've listed HSC past papers for: English. Maths ...

HSC maths paper may have leaked in Buldhana zone NCP leader Ajit Pawar claimed on Friday that the HSC maths paper had been leaked. "It is on television. The leak will cause huge damage to students," he said. Congress MLA Varsha Gaikwad, who was the education minister in the MVA government, said, "This is the third case of something going wrong.

Online library of HSC resources. Resources including past trial papers, study notes, past papers, assessments, essays & many more. Updated to new Syllabus

Maharashtra State Board Class 12th maths 2019 question paper with solutions are available on this page, by BYJU'S, in downloadable pdf format and also in the text for the students to prepare well for their exams. The page also consists of Maths Question Paper 2019 Class 12, which acts as an extra resource.

The board has introduced the right chapters for the basic concept development of the students. To study better and to check the efficiency level of your preparation, you can use these mathematics papers of the previous years' board exams to practice. Find these HSC Maths Previous Year Question Papers here and download them.

Maharashtra HSC Mathematics Question Paper Analysis 2022: First Reaction. The first reaction of students for the Maharashtra HSC Mathematics Exam 2022 is provided below: The Maths paper had an overall difficulty level of Easy to Moderate unlike the previous papers of Physics, Chemistry, and English which inclined more on the difficult side.

Browse Mathematics 2 Unit trial papers with the choice of School or Year. We would like to wish you good luck hope these resources help you prepare & ace your upcoming exams. Note: Some trial papers will not have solutions. So we have named the trial papers "with solutions" to save your time.

That's why we have provided Maharashtra Board (MSBSHSE) Maths Class 12 Previous Year papers 2019 PDF here so that you become thorough with the question paper pattern. Practising Maharashtra Board HSC Class 12 Maths previous year question papers has many advantages. One of the most important factors these papers address is the management of time.

HSC Past Exam Paper: 2021 Higher School Certificate (HSC) maths paper is a great revision resource for students sitting for General Mathematics Examination, GM Paper 1 and Paper 2. The HSC past exam paper is available in PDF format - see below how to download the exam paper and what you are downloading. How to download 2021 HSC Past Exam Paper

The Board has revised the Maharashtra HSC Exam Pattern 2022-23. The following is a list of changes made to the Maharashtra Board HSC exam pattern. 1. A 25% increase in objective questions: The board agreed to include 25% objective type questions in the HSC examination question paper as part of the updated evaluation process. These questions ...

Hello Friends ЁЯЩПWelcome To Our Channel P N SirYou can support this channel on upi id : [email protected]ЁЯУМ HSC 2023 MIQ Mathshttps://bit.ly/3F5ZEmEЁЯУМ HSC 2022 MIQ M...

Find all the solutions to the 2018 - 2022 HSC Maths Adv Exams, here. 1. The given equation has a gradient of -2. This corresponds to a downward sloping graph, so the solution must be either (A) or (B). тИ┤ (A) is the correct solution - (B) has a negative y-intercept. 2.

In yet another case of 'paper leak', the Mathematics paper of the ongoing Class 12 Higher Secondary Certificate (HSC) exams purportedly went viral on social media around 30 minutes ahead of ...

The purported leaked images of HSC Maths paper |. Mumbai: The Buldhana Police on Saturday arrested five people, including two teachers at a private school, for allegedly leaking parts of the Maths ...

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