free download cbse solved paper class xii mathematics 2007
Sample Paper – 2008
Class – XII
Subject - Mathematics
____________________________________________________________
Application of Derivatives
Q1 The volume of a cube is increasing at a constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.
Q2 Use differentials to find the approximate value of 
Q3 It is given that for the function f(x) = x3 – 6x2 + ax + b on [1, 3], Rolle’s theorem holds with
c = 2+ 
. Find the values of a and b if f(1)= f(3) = 0
Q4 Find a point on the curve y = (x – 3)2, where the tangent is parallel to the line joining (4, 1)
and (3, 0).
Q5 Find the intervals in which the function f(x) = x4 – 8x3 + 22x2 – 24x + 21 is decreasing or increasing.
Q6 Find the local maximum or local minimum of the function.
f(x) = sin4x + cos4x, 0
Q7 Find the point on the curve y2 = 4x which is nearest to the point (2, 1).
Q8 A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.
Q9 A balloon which always remain spherical has a variable diameter 
. Find the rate of change of its volume with respect to x.
Q10 Find the intervals in which f(x) = (x+1)3 (x – 3)3 is strictly increasing or decreasing.
Q11 Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1
Q12 Using differentials, find the approximate value of (26.57)1/3
Q13 Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Q14 Find the equation of the tangent and normal to the hyperbola 
at the point (x0,y0)
Q15 Find the intervals of the function f(x) = sinx + cosx, 0£x£2p is strictly increasing or strictly decreasing.
Q16 An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.
Inverse Trigonometric Functions
Class – XII
Q.1. Find the value of : tan-1 (1) + cos -1 (-1/2) + sin-1 (-1/2).
Q.2. Prove : tan-1x + tan-1 
= tan-1 
, çxê<
Q.3. If tan-1 
then find the value of x.
Q.4. Find the value of sin 
.
Q.5. Prove : sin-1
Q.6. Solve : tan-12x + tan-13x = 
Q.7. Prove : 
Q.8. Solve : sin-1 ( 1 –x) – 2sin-1x = 
.
Q.9. Evaluate: tan-1
- sec-1 (-2) + cosec-1 
.
Q.10. Prove : tan-1 
= 
Q.11. Simplify: sin-1 
, 
Q.12. Prove: sec2 (tan-12) + cosec2 ( cot-13) = 15.
Q.13. Simplify : tan-1 
Q.14. Prove : tan-1 
=
Q.15. If sin(sin-1 
, then find the value of x.
Q16.. Prove that :
2tan-1
= cos-1
MATRICES
Class - XII
Q.1. Construct a 3´4 matrix, whose elements are given by aij= 
Q.2. If A = 
and B = 
, then find the matrix X, such that 2A + 3X = 5B.
Q.3. If A = 
and I = 
, then show that I+A= (I – A) 
Q.4. Express the matrix A= 
as the sum of symmetric and skew-symmetric matrix
Q.5. Obtain the inverse of the matrix A = 
using elementary transformations.
Q.6. If f(x)= 
Prove that f(x). f(y) = f(x + y)
Q.7. Show that the matrix
Q.8. If A and B are invertible matrices of the same order, then prove that (AB)-1 = B-1A-1
Q.9. Let f(x) = x2 – 5x + 6. Find f(A) If A = 
Q.10. If A = 
Show that A2 -5A + 7I = 0, Use this to find A4.
Q.11. Express the matrix A = 
as the sum of a symmetric and a skew-symmetric matrix.
Q.12. Find the values of x, y, z if the matrix A = 
satisfy the equation A¢A = I3.
Q.13. Show that : 
= 
Q.14. Show that the following system of equations is consistent 2x – y + 3z = 5, 3x + 2y – z = 7, 4x + 5y – 5z = 9, Also, find the solution.
DETERMINANTS
Class - XII
Q.1. Prove that : 
= (1- x3)2
Q.2. Find the equation of the line joining A(1,3) and B(0,0) using determinants and find if D (K, 0) is a point such that area of a triangle ABD is 3 square units.
Q.3. If A = 
Verify that A3 – 6A2 + 9A – 4I = 0 and hence find A-1
Q.4. Prove that : 
= (1 – x2) 
Q.5. Solve by matrix method:
2x + y + z = 1
x - 2y – z = 3/2
3y - 5z = 9
Q.6. Prove that :
= a3
Q.7. Prove that : 
= abc + bc + ca + ab.
Q.8. Solve : 
= 0
Q.9. Using determinants, find the area of the triangle whose vertices are (1, 4), (2, 3), (-5, 3). Are the given points collinear.
Q.10. If the points (a1, b1), (a2, b2) and (a1 + a2, b1 + b2) are collinear, Show that a1b2 = a2b1.
Q.11. If a, b, c are all positive and are pth , qth and rth terms of G.P., then show that
D = 
= 0
Q.12. If 
= 0, then Prove that
a, b, c are in G.P or x, y, z are in G.P
Integrals
Class – XII
Q.1. 
Q.2. 
dx
Q.3. 
Q.4. 
Q.5. 
Q.6. 
Q.7. 
Q.8. 
Q.9. 
Q.10. Evaluate: 
.
Q.11. 
Q12. 
Q.13. 
Q.14 
Q.15. 
Q.16. 
Q.17. 
Q.18. Evaluate 
Differential Equations
Class – XII
Q.1. sec2x. tany dx + sec2y. tanx dy = 0.
Q.2. xdy – ydx = 
Q.3. 2xy + y2 – 2x2 dy/dx = 0, y = 2 when x = 1.
Q.4. x logx. dy/dx + y = (2/x) logx
Q.5. y dx + (x – y2) dy = 0
Q.6. (tan-1y - x)dy = (1 + y2) dx
Q.7. (1 + x2)dy/dx + 2xy = 
; y = 0 when x = 1
Q.8. x2dy + (xy + y2)dx = 0, y = 1 when x = 1
Q.9. dx/dy + ycotx = 2x + x2cotx, y = 0 when x =
Q.10. (x + y)dy = dx
Q.11. x (xdy – ydx) = ydx, y(1) = 1.
Q.12. dy/dx = cos(x + y) + sin(x + y)
Q.13. x dy/dx = y – x tan(y/x).
Q.14. (x2 + 1) dy/dx + 2xy = 
Q.15. (x2 + y2)dx + xy.dy = 0, y(1) = 1
Q.16. (x + y + 1)2 dy = dx, y(-1) = 0
Q.17. (xy2 + 2x)dx + (x2y + 2y)dy = 0
Q.18. dy/dx + 
= cosx + 
Q.19. Find the differential equation of all circles in the first quadrant which touch the co-ordinate axis.
Q.20. Form the differential equation corresponding to y2 = m(a2 – x2) by eliminating parameters m and a.
Application of Integrals
Class – XII
Q.1. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle
Q.2. Find the area of the region bounded by the ellipse 
.
Q.3. Find the area of the region bounded by the parabola y = x2 and y = 
.
Q.4. Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the linex=
.
Q.5. Using integration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2,2) and (3, 1).
Q.6. Prove that the curves y2 = 4x and x2 = 4y divide the area of the square bounded by x=0, x=4, y=4 and y=0 into three equal parts.
Q.7. Sketch the graph of y= 
Q.8. Using the method of integration, find the area bounded by the curve 
.
Q.9. Find the area of the smaller region bounded by the ellipse 
.
Q.10. Using integration, find the area of the triangular region, the equations of whose sides are y=2x + 1, y=3x +1 and x = 4.
Q.11. Find the area of the region 
Q.12. Find the area of the region between the circles x2 + y2 = 4 and (x – 2)2 + y2 = 4.
Q.13. Find the area bounded by the ellipse 
and the co-ordinates x = ae and x = 0, where b2=a2(1 – e2) and e<1.
Q.14. Find the area bounded by the curve y2 = 4a2(x – 1) and the lines x = 1and y = 4a.
Q.15. Using integration, find the area of the region bounded by the following curves, after making a rough sketch:
y = 1 + 
Q16. Draw a rough sketch of the curves y = sinx and y = cosx as x varies from o to and find the area of the region enclosed by them and x-axis.
Continuity & Differentiation
Class – XII
Q.1. Find the values of a and b such that the function defined by
f(x) = ( 5, if x £ 2
ax + b if 2
21, if x 
10 ) is a continuous function
Q.2. Find 
of sin2y + cos (xy) = p
Q.3. Differentiate w.r.t. x (x cosx)x + (x sinx)1/x
Q.4. If x = 
, y = 
, show that 
Q.5. If y = (tan-1x)2, show that
(x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2.
Q.6. Differentiate sin-1 
w.r.t. x
Q.7. If x 
for -1
Q.8. Find 
if y = a t + 1/t , x = ( t + 1/t)a
Q9. Discuss the continuity of the function given by:-
f(x) = êx-1ç + çx-2ê at x = 1, and x = 2.
Q10 If the function f(x) is given by f(x) = { 3ax + b) if x>1
11 if x = 1
(5ax – 2b if x<1}
is continuous at x = 1, find the values of a and b.
Q11 If y = [x + 
]n, then prove that 
Q12 Prove : 
Q13 Find when y = sec-1
Q14 If ex + ey = ex+y, prove that 
Q15 Given that cos
-------------------= 
prove that 
Q16 If x=a(q + sinq), y= a(1+ cosq), prove that 
VECTOR ALGEBRA
Class – XII
Q.1. Find a vector in the direction of vector 
that has magnitude 7 units.
Q.2. Show that the points A, B and C with position vectors, 
respectively, form the vertices of a right angled triangle.
Q.3. Find 
, if two vectors 
are such that 
.
Q.4. Find the area of the parallelogram whose adjacent sides are determined by the vectors 
Q.5. If a unit vector 
makes angles 
with 
, 
and acute angle θ with 
, then find θ and hence
the components of .
Q.6. Let,
and 
be three vectors such that 
and each one of them being perpendicular to the sum of other two, Find 
Q.7. Find the value of 
Q.8. The scalar product of the vector 
with a unit vector along the sum of vectors 
Find the value of l.
Q.9. If the sum of two unit vectors is a unit vector, Prove that the magnitude of their difference is 
.
Q.10. If are position vectors of points A and B respectively, then find the position vector of points of trisection of AB.
Q.11. Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.
Q.12. ABCD is a parallelogram. If the coordinates of A, B, C are (-2, -1), (3, 0) and (1, -2) respectively, Find the co-ordinate of D.
Q.13. Show that the points A, B, C with position vectors 
Q.14. If a vector makes a, b, 
with OX, OY and OZ respectively, prove that sin2a+sin2b+sin2γ=2.
Q.15. If 
inclined at an angle 
, then prove that sin 
= 
.
Q.16 If 
.
Q.17. If 
.
Three Dimensional Geometry
Class – XII
Q.1. Find the direction cosines of X, Y and Z-axis.
Q.2. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 
.
Q.3. Find the value of p so that the lines,
Q.4. Find the shortest distance between the lines whose vector equations are:-
Q.5. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z -6 = 0.
Q.6. Find the vector equation of the plane passing through the intersection of planes 
Q.7. Find the angle between the line 
Q.8. Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then
Q.9. Show that the angles between the diagonals of a cube is cos-1
.
Q.10. Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines 
Q.11. Find the foot of the perpendicular drawn from the point (0, 2, 3) on the line 
Also, find the length of the perpendicular.
Q.12. Find the shortest distance between the following pairs of lines whose cartesian equations are :
Q.13. A plane meets the coordinate axis in A, B, C such that the centroid of triangle ABC is the point (p,q,r). Show that the equation of the plane is 
.
Q.14. Find the equation of the plane passing through the point (1, 1, -1) and perpendicular to the planes x + 2y + 3z – 7=0 and 2x – 3y + 4z = 0.
Q.15. Find the distance between parallel planes,
Q.16. Show that the lines :
are coplanar.
Also, find the plane containing these two lines.
Class-XII
PROBABILITY
Q.1. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once.
Q.2. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that at least one is a girl.
Q.3. If A and B are two independent events, show that the probability of occurrence of at least one of A and B is given by :
1 – P(A').P(B')
Q.4. Probability of solving specific problem independently by A and B are ½ and 1/3 respectively. If both try to solve the problem independently, find the probability that the problem is solved.
Q.5. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red.
Q.6. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings.
Q.7. In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’, if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.
Q.8. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
Q.9. If P(A) = 3/8, P(B) = ½ and P (AÇB) = ¼, find P
and P
.
Q.10. The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3 and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.
Q.11. A can solve 90% of the problem given in a book and B can solve 70%. What is the probability that at least one of them will solve the problem, selected at random from the book.
Q.12. Three persons A, B, C throw a die in succession till one gets a ‘six’ and wins the game. Find their respective probabilities of winning, if A begins.
Q.13. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Q.14. An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in a random draw of three balls.
Q.15. In a meeting 70% of the members favour a certain proposal,30% being opposed. A member is selected at random and let X=0 if he opposed and X =1 if he is in favour. Find E(x) and Var(x).
Q.16. Find the probability distribution of the number of doublets in 4 throws of a pair of dice.
LINEAR PROGRAMMING
Q.1. One kind of cake requires 200gm of flour and 25gm of fat, and another kind of cake requires 100gm of flour and 50gm of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
Q.2. A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The Vitamin contents of one Kg food is given below :-
| Food | Vitamin A | Vitamin B | Vitamin C |
| X | 1 | 2 | 3 |
| Y | 2 | 2 | 1 |
One Kg of food X costs Rs. 16 and one Kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet.
Q.3. Maximise and Minimise:
Z = x + 2y
Subject to constraints x + 2y ³100, 2x – y £ 0, 2x + y £ 200, x, y ³ 0
Q.4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit.
Q.5. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in following table :
Transportation Cost Per Quintal (in Rs.)
| From / To | A | B |
| D | 6 | 4 |
| E | 3 | 2 |
| F | 2.50 | 3 |
How should the supplies be transported in order that the transportation cost is minimum. What is the minimum cost.
Q.6. Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.
Q.7. Solve the following L.P.P graphically:
Maximise: Z = 60x + 15y
Subject to constraints
x + y £ 50
3x + y £ 90, x, y ³ 0
Q.8. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit.
Q.9. If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2/km on petrol. If he drives it at a faster speed of 40km/hr, the petrol cost increases to Rs. 5/km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an L.P.P. and solve.
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