Please refer to Class 12 Mathematics Sample Paper With Solutions Set E below. These Class 12 Mathematics Sample Papers will help you to get more understanding of the type of questions expected in the upcoming exams. All sample guess papers for Mathematics Class 12 have been designed as per the latest examination pattern issued by CBSE. Please practice all CBSE Sample Papers for Mathematics in Standard 12.

**Sample Paper Class 12 Mathematics With Solutions Set E**

**SECTION – A**

**1. If **

**Solution:** We have

**2. Find the differential equation representing the curve y = cx + c ^{2} .**

**Solution:**We’ve y = cx + c

^{2}… (i) ⇒ y1 = c

Replacing value of c in (i), we get : y = (dy/dx)x + (dy/dx)

^{2}.

**3. Write the integrating factor of the following differential equation :****(1+ y ^{2} )dx – (tan^{-1} y – x)dy = 0 .**

**Solution:**We have (1+ y2 )dx – (tan-1 y – x)dy = 0

**4. Write the value of a̅.(b̅xa̅) .****Solution:** We’ve a̅.(b̅xa̅) = [a̅ b̅ a̅] = 0.

**5. If **

**Solution:** We have

**6. Write the direction ratios of the following line : **

**Solution:** Given line x = -3,y-4/3 = 2-z/1 can be re-written as x+3/0 = y-4/3 = z-2/1

So its direction ratios are 0, 3, –1.

**SECTION – B**

**7. To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of ₹ 20, ₹15 and ₹ 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.****By such exhibition, which values are generated in the students?****Solution:** Let the amounts collected by schools A, B and C be x, y and z (in ₹) respectively.

By equality of matrices, we get : x = 850, y = 805, z = 970 .

So the amounts collected by school A is ₹ 850, by school B is ₹ 805 and by school C is ₹970.

**8. Prove that : 2 tan ^{-1} **

**Solution:**

**OR**

**Solve for x : tan ^{-1}**

**Solution: **

**9. If **

**Solution:**

**10. Using properties of determinants, prove the following : **

**Solution:**

**11. If x = α sin 2t(1+ cos2t) and y = β cos 2t(1- cos 2t) , show that dy/dx = β/α tan t .****Solution:** We have x = αsin 2t(1+ cos2t) and y = βcos 2t(1- cos 2t)

**12. **

**Solution:** Let

**13. Find the derivative of the following function f (x) w. r. t. x, at x =1 : **

**Solution:**

**14. **

**Solution: **

**OR**

**Evaluate : **

**Solution: **

**15. Evaluate **

**Solution:**

**16. Find **

**Solution: **

**17. Show that four points A, B, C and D whose position vectors are **

**Solution:**

**18. Show that the following two lines are coplanar : **

**Solution: **

Hence the lines 1 2 L and L are coplanar.

**OR**

**Find the acute angle between the plane 5x – 4y + 7z – 13 = 0 and the y-axis.****Solution: **Given equation of plane is 5x – 4y + 7z -13 = 0…(i)

**19. A and B throw a die alternatively till one of them gets a number greater than four and wins the game. If A starts the game, what is the probability of B winning? ****Solution:** Let E : getting a no. greater than four on the die.

**OR **

**die is thrown three times. Events A and B are defined as below :****A : 5 on the first and 6 on the second throw.****B : 3 or 4 on the third throw.****Solution: **Given that A : 5 on the first and 6 on the second throw and B : 3 or 4 on the third throw.

Here A = {(5, 6, 1), (5, 6, 2), (5, 6, 3), (5, 6, 4), (5, 6, 5), (5, 6, 6)}

**SECTION – C**

**20. If the function f : R → R be defined by f (x) = 2x – 3 and g : R → R by g (x) = x3 + 5, then find the value of (fog) –1 (x).****Solution:**

**OR**

**Let A = Q x Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b)*(c,d) = (ac, b + ad) for all (a, b),(c,d)∈A. Then find**

(i) the identity element in A.

(ii) the invertible element of A.**Solution:** Given A = QxQ, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a, b)*(c,d) = (ac, b + ad) for (a, b),(c,d)∈A.

(i) Let (e,e’) be the identity element of * in A. Then (a, b)*(e,e’) = (a,b) = (e,e’)*(a, b)

**21. If the function f (x) = 2x ^{3} – 9m x^{2} + 12m^{2} x + 1, where m > 0 attains its maximum and minimum at p and q respectively such that p^{2} = q, then find the value of m.**

**Solution:**Given f (x) = 2x

^{3}– 9mx

^{2}+ 12m

^{2}x +1, m > 0

**22. Using integration, find the area of the region bounded by the lines y = 2 + x , y = 2 – x and x = 2.****Solution:** We have y = 2 + x…(i), y = 2 – x…(ii) and x – 2

**23. Find the differential equation for all the straight lines, which are at a unit distance from the origin.****Solution:** Let the equation of the line be y = mx + c…(i)

**OR**

**Show that the differential equation 2xy , dy/dx = x2 + 3y2 is homogeneous and solve it.****Solution: **

**24. Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle π/4 with the plane x + y = 3. Also find the equation of the plane.****Solution:** Let the d.r.’s of the normal to the plane through the points (1, 0, 0) and (0, 1, 0) be A, B and C.

**25. 40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?****Solution:** Let E_{1}, E_{2} and E be the events that students residing in hostel, outside hotel and students getting grade A respectively.

**26. The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10.****According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively.****A man receives ₹ 225 a day and a woman receives ₹ 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum?****Formulate an LPP and solve it graphically.****Solution:** Let x men and y women number of helpers be hired.

Here the feasible region (in fact, feasible point) is B(6, 4).

Hence the minimum value of Z is ₹ 2150.

Therefore 6 men and 4 women helpers should be hired.