CBSE Class-12 Mathematics Sample Paper (By CBSE) 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. 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 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 2, such that =-15 and C ij represents the cofactor of a ij, then find a 21 C 21 a 22 C 22 3. Give an example of vectors and such that but. 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 Hence, find the matrix P satisfying the matrix Material downloaded from mycbseguide.com. 1 / 18
equation P =. 7. Prove that if x then cos -1 x+cos -1 8. Find the approximate change in the value of, when x changes from x = 2 to x = 2.002 9. Find 10. Verify that ax 2 + by 2 =1 is a solution of the differential equation x(yy 2 +y 1 2 )=yy1 11. Find the Projection (vector) of 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 Then find the value of 14. Find a and b, if the function given by f(x)= OR Determine the values of a and b such that the following function is continuous at x = 0: Material downloaded from mycbseguide.com. 2 / 18
15. If y=log then prove that x(x+1) 2 y 2 +(x+1) 2 y 1 =2. 16. Find the equation(s) of the tangent(s) to the curve y=(x 3-1)(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)- 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)=x 3-45x 2 + 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 19. Find the particular solution of the differential equation: OR Show that (x-y) dy=(x+2y)dx is a homogenous differential equation. Also, find the general solution of the given differential equation. Material downloaded from mycbseguide.com. 3 / 18
20. If are three vectors such that, then prove that and hence show that =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: be defined by f (x)= 2x-3and g: by g (x)=x 3 +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 of real numbers bya*b= 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 then find A -1 and hence solve the following system of equations:3x+4y+7z=14,2x-y+3z=4,x+2y-3z=0 OR If A=, find the inverse of A using elementary row transformations and hence Material downloaded from mycbseguide.com. 4 / 18
solve the following matrix equation XA=[1 0 1]. 26. Using integration, find the area in the first quadrant bounded by the curve the circle and the y-axis 27. Evaluate the following: OR Evaluate as the limit of a sum. 28. from the line 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. Material downloaded from mycbseguide.com. 5 / 18
CBSE Class-12 Mathematics Sample Paper (By CBSE) Answer 1. 2. -15 3. (or any other correct answer) 4. (1*2)*3=2 2 *3=2 12, 1*(2*3)=1*2 6 =2 64. Hence, * is not associative. Section B 5. 6. 7. Lets cos -1 x=θ.then The given expression on LHS 8. Material downloaded from mycbseguide.com. 6 / 18
9. 10. Substituting this value, for a in the equation (1), we get, 11. The required Projection (vector) of 12. 13. Where Cij=the cofactor of a ij and a ij =the (i,j) the element of determinant D. Material downloaded from mycbseguide.com. 7 / 18
We know that 14. Since, f is differentiable at 1, f is continuous at 1 Hence, f(x)= (2x+1)=3 As f is continuous at 1, we have a+b=3 (1) =2a As f is differentiable at 1, we have 2a=2, i.e., a=1 and b=2. OR f(0)=2. For the function to be continuous at 0, we must have i.e., we must b may be any real number other than 0. 15. = Material downloaded from mycbseguide.com. 8 / 18
(using (1)) Hence, proved. 16. When y=0, we have (x-1)(x 2 +x+1)(x-2)=0, i.e.,x=1 or 2. The required equations of the tangents are y-0=-3(x-1) or, y=-3x+3 and y-0=7(x-2)or, y=7x-14. OR Domain f= In(-1,0), f (x) Therefore, f is strictly decreasing in (-1,0]. In Therefore, f is strictly increasing in. 17. We have C(x)=x 3-45x 2 +600x,10 x 20. For the time being we may assume that the function C(x) is continuous in [10,20]. C (x)=3x 2-90x+600=3(x-10)(x-20) C (x)=0 if x=10 or x=20. But,10,20 (10,20). Therefore, the maximum or the minimum value will occur at the points. C(10)=2500,C(20)=2000. Hence, the person must place the order for 20 trees and the least amount to be spent=rs2000. Value: The person cares for a healthy environment despite being economically constrained. 18. Material downloaded from mycbseguide.com. 9 / 18
Put t=-1,-1=2b,i.e., B=-1/2. Put, t=1,1=4c, i.e., C=1/4. Put t=0,0=a+b+c, which gives. Therefore the required intergral 19. The given differential equation is ye y dx=(y3+2xe y )dy,y(0)=1 Or, which is linear in x. I.F.= Multiplying both sides by the I.F. and integrating, we get, (the general solution). When x=0, y=1.0=-e -1 +C Hence, the required particular solution is The given differential equation is (x-y)dy=(x+2y)dx or, Hence, homogeneous. Material downloaded from mycbseguide.com. 10 / 18
Put y=v x Then equation becomes Or, Integrating, we get Or, log (v 2 +v+1)- OR, log (y 2 +xy+x 2 )-2 (the general solution). 20. =0 [As the scalar triple product of three vectors is zero if any two of them are equal.] 21. General point on the first line is General point on the first line is Direction ratios of the required Therefore, the direction ratios of the required line are (6,20,34) or, (3,10,17). Hence, the required equation of the line is 22. Let E1=Bag I is chosen, E 2 =Bag II is chosen, E 2 =Bag III is chosen, A=The two balls drawn Material downloaded from mycbseguide.com. 11 / 18
from the chosen bag are white and red. By Bayes s Theorem, the required probability = 23. Let X denotes the random variable. Then X=0,1,2. X i P i Xipi Xi 2 pi 0 60/95 0 0 1 32/95 32/95 32/95 Variance= 24. defined by fog(x) = f(g(x)) = f(x 3 +5) = 2(x 3 +5)-3 = 2x 3 +7 Let Material downloaded from mycbseguide.com. 12 / 18
Hence, fog is one-one. Let y (Codomain fog ). Hence, for every y (Codomain fog ), such that fog Hence, Fog is onto. Since fog is both one-one and onto, it is invertible. (fog) -1 : defined by (fog) -1 (x) = (fog) -1 (9)= OR Let a,b such that a=0, b 0. Then a*b= Let a,b such that a=0,b 0. Then a*b=a,b*a= Let a,b such that a=0, b=0. Then a*b=a=0, b*a=b=0, a*b=b*a. Now we need to check whether * is communicative. One more case is needed to be examined. Let a,b such that a 0,b 0. Then a*b= and a*b may not be equal to b*a, e.g., (-1) * 2=3,2 * (-1)=1, hence, (-1)*2 2*(-1). Thus * is not communicative. Material downloaded from mycbseguide.com. 13 / 18
The element will be the identify element for * if a * e=e*a=a for all a. a*e=a provided e=0 and e*a=a provided e=0 (As 0*0=0 and 0*a= +a=a for a 0). Hence, 0 is the identify element for *. 25.. Hence, A -1 exist. Let C ij represent the cofactor of (I,j) th element of A. Then, C 11 =-3, C 12 =26, The given system of equations is equivalent to the matrix equation Hence, x=1,y=1, z=1 Material downloaded from mycbseguide.com. 14 / 18
26. Solving y= simultaneously, (y=x 2 lies in quadrant I). Material downloaded from mycbseguide.com. 15 / 18
The required area= the shaded area 27. The give definite integral=i = Hence, f is odd. Therefore, g(x)= Hence, g is even. Thus Hence, Material downloaded from mycbseguide.com. 16 / 18
Let f(x)=3x 2-2x+4. Then the given definite integral = where nh=4. f(-2+rh)=3(-2+rh) 2-2 (-2+rh)+4=3r 2 h 2-14rh+20 28. The general point on the given line is The direction ratios of the line parallel to the plane x-y+2z-3=0 intersecting the given line and passing through the point (-2,3,-4) are and + (-1)+. The point of intersection is The required distance Material downloaded from mycbseguide.com. 17 / 18
= unit. 28. Let x= the number of units of product 1 to be produced daily Y= the number of units of Products 2 to be produced daily To maximize P=(9-1.2) x +(8-0.9)y=7.8x+7.1y subject to the constraints: At the point P (0,0) 0 (200, 120) 2412 (0,240) 1704 (200,0) 1560 (80,210) 2115 The maximum profit=rs.2412. Material downloaded from mycbseguide.com. 18 / 18