Sample paper for class 12 Mathematics

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Sample Paper for class 12 Mathematics
Session 2017-2018

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General Instructions:

• All questions are compulsory.
• This question paper contains 29 questions.
• Question 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
• Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
• Questions 13-23 in Section C are long-answer-I type questions carrying 4 marks each.
• Questions 24-29 in Section D are long-answer-II type questions carrying 6 marks each.

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Section A

Questions 1 to 4 carry 1 mark each.

1. Let A= {1,2,3,4}. Let R be the equivalence relation on A{tex} \times {/tex}A defined by

(a,b) R(c,d) iff a + d= b+c  . Find the equivalence class [(1,3)].

2. If A [a_ij] is a matrix of order 2 {tex} \times {/tex}2, such that {tex}\left| A \right|{/tex}=-15 and C^ij represents the cofactor of a_ij, then find a₂₁C₂₁ a₂₂C₂₂

3. Give an example of vectors {tex}\vec a{/tex} and {tex}\vec b{/tex}such that {tex}| {\vec a} | = | {\vec b} |{/tex} but {tex}\vec a{/tex} ≠ {tex}\vec b{/tex}.

4. Determine whether the binary operation * on the set N of natural numbers defined by a*b 2^ab is associative or not.

Section B

Questions 5 to 12 carry 2 marks each

5. If 4sin^-1 cos^-1 x=π, then find the value of x.

6. Find the inverse of the matrix {tex}\left[ {\frac{{ – 3}}{5}\frac{2}{{ – 3}}} \right]{/tex}Hence, find the matrix P satisfying the matrix equation P{tex}\left[ {\frac{{ – 3}}{5}\frac{2}{{ – 3}}} \right]{/tex}={tex}\left[ {\frac{1}{2}\frac{2}{{ – 1}}} \right]{/tex}.

7. Prove that if {tex}\frac{1}{2}{/tex}≤ x ≤ then cos^-1 x+cos^-1 {tex}\left[ {\frac{x}{2} + \frac{{\sqrt {3 – 3{x^2}} }}{2}} \right] = \frac{\pi }{3}{/tex}

8. Find the approximate change in the value of  {tex}\frac{1}{{{x^2}}}{/tex}, when x changes from x = 2 to x = 2.002

9. Find {tex}\int {{e^x}\frac{{\sqrt {1 + \sin 2x} }}{{1 + \cos 2x}}dx} {/tex}

10. Verify that ax² + by² =1 is a solution of the differential equation x(yy₂+y₁²)=yy₁

11. Find the Projection (vector) of {tex}2\hat i – \hat j + \hat k\;on\;\hat i – 2\hat j + \hat k.{/tex}

12. If A and B are two events such that P(A)=0.6, then find P (A/B).

Section C

Questions 13 to 23 carry 4 marks each.

13.If  {tex}\Delta =\left| \begin{matrix} 1 & a & {{a}^{2}} \\ a & {{a}^{2}} & 1 \\ {{a}^{2}} & 1 & a \\ \end{matrix} \right|=-4{/tex}

Then find the value of

{tex}\left[ \begin{matrix} {{a}^{3}}-1 & 0 & a-{{a}^{4}} \\ 0 & a-{{a}^{4}} & {{a}^{3}}-1 \\ a-{{a}^{4}} & {{a}^{3}}-1 & 0 \\ \end{matrix} \right]{/tex}
14. Find ‘a’ and ‘b’, if the function given by f(x)={tex}\left[ \begin{matrix} {{a}^{3}}-1 & 0 & a-{{a}^{4}} \\ 0 & a-{{a}^{4}} & {{a}^{3}}-1 \\ a-{{a}^{4}} & {{a}^{3}}-1 & 0 \\ \end{matrix} \right]{/tex}

OR

Determine the values of ‘a’ and ‘b’ such that the following function is continuous at x = 0:{tex}f(x) = \left\{ \begin{gathered} \frac{{x + \sin x}}{{\sin (a + 1)x}},if – \pi < x < 0 \hfill \\ 2,if\;x = 0 \hfill \\ 2\frac{{{e^{\sin bx}} – 1}}{{bx}},if\;x > 0 \hfill \\ \end{gathered} \right.{/tex}

15. If y=log{tex}{(\sqrt x+ \frac{1}{{\sqrt x }})^2},{/tex} then prove that x(x+1)² y₂+(x+1)² y₁=2.

16. Find the equation(s) of the tangent(s) to the curve y=(x³-1)(x-2) 3 at the points where the curve intersects the x –axis.

OR

Find the intervals in which the function f(x)=-3log(1+x )+4log(2+x)-{tex}\frac{4}{{2 + x}}{/tex}  is strictly increasing or strictly decreasing.

17. A person wants to plant some trees in his community park. The local nursery has to perform this task. It charges the cost of planting trees by the following formula:

C(x)=x³-45x² + 600x, Where x is the number of trees and C(x) is the cost of planting x trees in rupees. The local authority has imposed a restriction that it can plant 10 to 20 trees in one community park for a fair distribution. For how many trees should the person place the order so that he has to spend the least amount?

How much is the least amount? Use calculus to answer these questions. Which value is being exhibited by the person?

18. Find {tex}\int {\frac{{\sec x}}{{1 + \cos ecx}}dx} {/tex}

19. Find the particular solution of the differential equation:

{tex}y{e^y}dx = ({y^3} + 2x{e^y})dy,y(0) = 1{/tex}

OR

Show that (x-y) dy=(x+2y)dx is a homogenous differential equation. Also, find the general solution of the given differential equation.

20. If  {tex}\vec a,\vec b,\vec c{/tex}are three vectors such that, {tex}\vec a,\vec b,\vec c = \vec 0{/tex}then prove that {tex}\vec a \times \vec b,\vec c = \vec 0{/tex}and hence show that {tex}\left[ {\vec a \times \vec b,\vec c} \right]{/tex}=0.

21. Find the equation of the line which intersects the lines and passes through the point (1, 1,

22. Bag I contains 1 white, 2 black and 3 red balls; Bag II contains 2 white, 1 black and 1 red balls; Bag III contains 4 white, 3 black and 2 red balls. A bag is chosen at random and two balls are drawn from it with replacement. They happen to be one white and one red. What is the probability that they came from Bag III.

23. Four bad oranges are accidentally mixed with 16 good ones. Find the probability distribution of the number of bad oranges when two oranges are drawn at random from this lot. Find the mean and variance of the distribution.

Section D

Questions 24 to 29 carry 6 marks each.

24. If the function f:{tex}R \to R{/tex} be defined by f (x)= 2x-3and g:{tex}R \to R{/tex} by g (x)=x³+5, then find fog and show that fog is invertible. Also, find (fog)-1, hence find (fog)^-1(9).

OR

A binary operation * is defined on the set {tex}R{/tex} of real numbers bya*b={tex}\left\{ \begin{gathered} a,\;if\;b = 0 \hfill \\ \left| a \right|\; + b,if\;b \ne 0 \hfill \\ \end{gathered} \right.{/tex}

If at least one of a and b is 0, then prove that a* b=b*a. Check whether * is commutative. Find the identity element for*, if it exists.
25. If {tex}\left[ {\begin{array}{*{20}{c}} 3&2&1 \\ 4&{ – 1}&2 \\ 7&3&{ – 3} \end{array}} \right.{/tex} then find A^-1and hence solve the following system of equations:3x+4y+7z=14,2x-y+3z=4,x+2y-3z=0

OR
If A={tex}\left[ {\begin{array}{*{20}{c}} 2&1&1 \\ 1&0&1 \\ 0&2&{ – 1} \end{array}} \right]{/tex}, find the inverse of A using elementary row transformations and hence solve the following matrix equation XA=[1 0 1] .

26. Using integration, find the area in the first quadrant bounded by the curve {tex}y = x\left| x \right|,{/tex} the circle {tex}{x^2} + {y^2} = 2{/tex} and the y-axis

27. Evaluate the following: {tex}\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{x + \frac{\pi }{4}}}{{2 – \cos 2x}}dx} {/tex}

OR

Evaluate {tex}\int\limits_{ – 2}^2 {(3{x^2} – 2x + 4)} dx{/tex} as the limit of a sum.

28. from the line {tex}\hat r = \hat i + 2\hat j – \hat k + \lambda (\hat i + 3\hat j – 9\hat k){/tex}

measured parallel to the plane: x – y + 2z – 3 = 0.

29. A company produces two different products. One of them needs 1/4 of an hour of assembly work per unit, 1/8 of an hour in quality control work and Rs1.2 in raw   materials. The other product requires 1/3 of an hour of assembly work per unit, 1/3 of an hour in quality control work and Rs 0.9 in raw materials. Given the current availability of staff in the company, each day there is at most a total of 90 hours available for assembly and 80 hours for quality control. The first product described has a market value (sale price) of Rs 9 per unit and the second product described has a market value (sale price) of Rs 8 per unit. In addition, the maximum amount of daily sales for the first product is estimated to be 200 units, without there being a maximum limit of daily sales for the second product.

Formulate and solve graphically the LPP and find the maximum profit.

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Sample Paper for class 12

It is Sample paper for class 12 Mathematics. However, myCBSEguide provides the best sample papers for all the subjects. There are number of sample papers which you can download from myCBSEguide website. Sample paper for class 12 all subjects

• CBSE Sample Papers for Class 12 Physics
• CBSE Sample Papers for Class 12 Chemistry
• CBSE Sample Papers for Class 12 Mathematics
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• CBSE Sample Papers for Class 12 Computer Science
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• CBSE Sample Papers for Class 12 Hindi Core
• CBSE Sample Papers for Class 12 Hindi Elective
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• CBSE Sample Papers for Class 12 Physical Education
• CBSE Sample Papers for Class 12 Other Subjects

These are also Sample Paper for class 12 Mathematics available for download through myCBSEguide app. These are the latest Sample Paper for class 12 Physics as per the new board exam pattern. Download the app today to get the latest and up-to-date study material.

Marking Scheme for Class 12 Board exam

Subject	Board Marks	Practical or internal Marks
English	100 Marks	ZERO Marks
Hindi	100 Marks	ZERO Marks
Mathematics	100 Marks	ZERO Marks
Chemistry	70 Marks	30 Marks
Physics	70 Marks	30 Marks
Biology	70 Marks	30 Marks
Computer Science	70 Marks	30 Marks
Informatics Practices	70 Marks	30 Marks
Accountancy	80 Marks	20 Marks
Business Studies	80 Marks	30 Marks
Economics	80 Marks	20 Marks
History	80 Marks	20 Marks
Political Science	100 Marks	ZERO Marks
Geography	70 Marks	30 Marks
Sociology	80 Marks	20 Marks
Physical Education	70 Marks	30 Marks
Home Science	70 Marks	30 Marks

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