# Class 12 Mathematics Sample Paper With Solutions Set F

Please refer to Class 12 Mathematics Sample Paper With Solutions Set F 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 F

SECTION A

Question numbers 01 to 20 carry 1 mark each.

1. If A is a square matrix of order 3 and |A| = 5 , then the value of |2A’| is
(a) – 10
(b) 10
(c) – 40
(d) 40

D

2. If A is a square matrix such that A2 = A, then (I – A)3 + A is equal to
(a) I
(b) 0
(c) I – A
(d) I +A

A

3. The principal value of tan −1 (tan 3π/5) is
(a) 2π/5
(b) −2π/5
(c) 3π/5
(d) −2π/5

B

4. If the projection of a̅ = î – 2ĵ+ 3k̂ on b̅ = 2î + λk̂ is zero, then the value of λ is
(a) 0
(b) 1
(c) – 2/3
(d) – 3/2

C

5. The vector equation of the line passing through the point (– 1, 5, 4) and perpendicular to the plane z = 0 is
(a) r̅ = – î + 5ĵ+ 4k̂ + λ(î + ĵ)
(b) r̅ = – î + 5ĵ+ (4 + λ)k̂
(c) r̅ = î – 5ĵ – 4k̂ + λk̂
(d) r̅ = λk̂

B

6. The number of arbitrary constants in the particular solution of a differential equation of second order is (are)
(a) 0
(b) 1
(c) 2
(d) 3

A

7.

(a) – 1
(b) 0
(c) 1
(d) 2

D

8. The length of the perpendicular drawn from the point (4, – 7, 3) on the y-axis is
(a) 3 units
(b) 4 units
(c) 5 units
(d) 7 units

C

9. If A and B are two independent events with P(A) = 1/3 and P(B) = 1/4 , then P(B’ |A) is equal to
(a) 1/4
(b) 1/3
(c) 3/4
(d) 1

C

10. The corner points of the feasible region determined by the system of linear inequalities are (0, 0), (4,0), (2, 4) and (0,5). If the maximum value of z = ax + by, where a, b > 0 occurs at both (2,4) and (4,0), then
(a) a = 2b
(b) 2a = b
(c) a = b
(d) 3a = b

A

Fill in the blanks

11. A relation R in a set A is called _________, if (a1 , a2 ) ∈ R implies 2 1 (a , a ) ∈ R, for all a1 , a∈ A .
Sol. symmetric

12. The greatest integer function defined by f (x) = [x], 0 < x < 2 is not differentiable at x = ___.
Sol. As the greatest Integer function is not differentiable at integral points.
So, here f (x) will be non-differentiable at x =1.

13. If A is a matrix of order 3 × 2, then the order of the matrix A’ is _________.
Sol. 2×3

OR

A square matrix A is said to be skew-symmetric, if ____________.
Sol. A = – A’ or, A’ = – A .

14. The equation of the normal to the curve y2 = 8x at the origin is __________.
Sol.
For y2 = 8x , we have 2y × dy/dx = 8 i.e., dy/dx = 4/y
Slope of normal at (0, 0) is −0/4 = 0
Equation of normal at origin is : y − 0 = 0(x − 0) i.e., y = 0 .

OR

The radius of a circle is increasing at the uniform rate of 3 cm/sec . At the instant when the
radius of the circle is 2 cm, its area increases at the rate of _______ cm2 / s .
Sol. Area of the circle is, A = πr2
⇒ dA/dt = 2πr × dr/dt
When r = 2 cm, dA/dt = 2π×2×3 = 12π cm2 /sec

15. The position vectors of two points A and B are O̅A̅ = 2î – ĵ – k̂ and O̅B̅ = 2î – ĵ+ 2k̂ ,
respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 :1 is ____________.
Sol.

Question numbers 16 to 20 are of very short answer type questions.

16.

Sol.

17. Find : ∫ x4 log x dx .
Sol.

OR

Find :

Sol.

18. Evaluate :

Sol.

19. Two cards are drawn at random and one-by-one without replacement from a well-shuffled pack of 52 playing cards. Find the probability that one card is red and the other is black.
Sol.

20. Find :

Sol.

SECTION B

Question numbers 21 to 26 carry 2 marks each.

21. Prove that

Sol.

= 2cos−1 x
= R.H.S.

OR

Consider a bijective function f : R+ (7, ∞) given by f (x) =16x2 + 24x + 7, where R+ is the set of all positive real numbers. Find the inverse function of f.
Sol.

22. If x = at2 , y = 2at, then find d2y/dx2.
Sol. Given that x = at2 , y = 2at,

23. Find the points on the curve y = x3 – 3x2 – 4x at which the tangent lines are parallel to the line 4x + y – 3 = 0 .
Sol. For y = x3 – 3x2 – 4x , we have
dy/dx = 3x2 − 6x– 4
Slope of tangent is ‘–4’ as tangent lines are parallel to 4x + y – 3 = 0 , whose slope is ‘–4’.
Note that the parallel lines have same slope.
That is, 3x2 – 6x – 4 = –4
⇒ 3x(x – 2) = 0
∴ x = 0 ; x = 2
Therefore, the required points on the curve are (0, 0); (2, – 12) .

24. Find a unit vector perpendicular to each of the vectors a̅ and b̅  where a̅ = 5î + 6ĵ – 2k̂ and b̅ = 7î + 6ĵ+ 2k̂
Sol.

OR

Find the volume of the parallelopiped whose adjacent edges are represented by 2a̅, – b̅ and 3c̅, where
a̅ = î – ĵ+ 2k̂,
b̅ = 3î + 4ĵ – 5k̂, and
c̅ = 2î – ĵ+ 3k̂ .
Sol.

25. Find the value of k, so that the lines x = – y = kz and x – 2 = 2y +1= – z +1 are perpendicular  to each other.
Sol.

26. The probability of finding a green signal on a busy crossing X is 30%. What is the probability of finding a green signal on X on two consecutive days out of three?
Sol.

SECTION C

Question numbers 27 to 32 carry 4 marks each.

27. Let N be the set of natural numbers and R be the relation on N × N defined by (a, b) R (c, d) iff ad = bc for all a, b, c, d∈N. Show that R is an equivalence relation.
Sol. Reflexive: For any (a, b)∈N× N , we always have a·b = b·a
∴ (a, b) R (a, b) .
Thus, R is reflexive.
Symmetric: For (a, b), (c, d) ∈ N × N
Let (a, b) R (c, d)
⇒ a · d = b · c
⇒ c · b = d · a
⇒ (c, d) R (a, b)
∴ R is symmetric .
Transitive: For any (a, b), (c, d), (e, f )∈N × N
Let (a, b) R (c, d) and (c, d) R (e, f )
⇒ a .d = b.c and c.f = d.e
⇒ a · d · c · f = b · c · d · e
⇒ a · f = b · e
∴ (a, b) R (e, f )
∴ R is transitive
Therefore, R is an equivalence relation .

28. If y = ex² cos x + (cos x)x , then find dy/dx.
Sol. Let u = (cos x)x
⇒ y = ex² cos x + u
∴ dy/dx = ex² cos x(2x·cos x – x2 ·sin x) + du/dx    …(i)
Now u = (cos x)x
⇒ log u = log (cos x)x
⇒ log u = x . log (cos x)
Differentiate w.r.t. x both sides,
1/u × du/dx = log (cos x) – x tan x
⇒ du/dx = (cos x)x{log (cos x) – x tan x}
Therefore by (i), we have
dy/dx = ex² cos x (2x · cos x – x2 sin x) + (cos x)x{log (cos x) – x tan x}

29. Find : ∫sec3 x dx .
Sol.

30. Find the general solution of the differential equation y eydx = (y3 + 2xey ) dy .
Sol.

OR

Find the particular solution of the differential equation

Sol.

31. A furniture trader deals in only two items – chairs and tables. He has Rs 50,000 to invest and a space to store at most 35 items. A chair costs him Rs 1,000 and a table costs him Rs 2,000. The trader earns a profit of Rs 150 and Rs 250 on a chair and table, respectively. Formulate the above problem as an LPP to maximize the profit and solve it graphically.
Sol.

∴ Max. (Z) = Rs6750
Number of chairs = 20, no. of tables =15 .

32. There are two bags, I and II. Bag I contains 3 red and 5 black balls and Bag II contains 4 red and 3 black balls. One ball is transferred randomly from Bag I to Bag II and then a ball is drawn randomly from Bag II. If the ball so drawn is found to be black in colour, then find the probability that the transferred ball is also black.
Sol. Let E1 : the ball transfered from Bag I is Black ,
E2 : the ball transfered from Bag I is Red ,
A: the ball drawn from Bag II is Black.

OR

An urn contains 5 red, 2 white and 3 black balls. Three balls are drawn, one-by-one, at random without replacement. Find the probability distribution of the number of white balls. Also, find the mean and the variance of the number of white balls drawn.
Sol. Let X = Number of white balls.
So, values of X = 0, 1, 2.
Table for probability distribution is :

SECTION D

Question numbers 33 to 36 carry 6 marks each.

33.

then find A–1 and use it to solve the following system of the equations:
x + 2y – 3z = 6,
3x + 2y – 2z = 3,
2x – y + z = 2.
Sol.

OR

Using properties of determinants, prove that

Sol.

34. Using integration, find the area of the region bounded by the triangle whose vertices are (2, –2), (4, 5) and (6, 2) .
Sol. Let A(2, – 2), B(4, 5), C(6, 2).
Equations of the sides of triangle ABC are

35. Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height h and radius r is one-third of the height of the cone, and the greatest volume of the cylinder is 4/9 times the volume of the cone.
Sol.

36. Find the equation of the plane that contains the point A(2,1, – 1) and is perpendicular to the line of intersection of the planes 2x + y – z = 3 and x + 2y + z = 2 . Also find the angle between the plane thus obtained and the y-axis.
Sol. Let equation of the required plane be
a(x – 2) + b(y –1) + c(z +1) = 0…
Also it is given that (i) is perpendicular to the planes 2x + y – z = 3 and x + 2y + z = 2 .
So, 2a + b – c = 0
a + 2b + c = 0

OR

Find the distance of the point P(–2, – 4, 7) from the point of intersection Q of the line r̅ = (3î – 2ĵ+ 6k̂ ) + λ(2î – ĵ+ 2k̂) and the plane r̅ ·(î – ĵ+ k̂ ) = 6 . Also write the vector equation of the line PQ.
Sol. General point on line is r̅ = (3+ 2λ)î + (–2 – λ)ĵ + (6 + 2λ)k̂
For the point of intersection of line and plane,
[(3 + 2λ)î + (–2 – λ)ĵ+ (6 + 2λ)k̂ ] ·(î – ĵ+ k̂) = 6
⇒ 3+ 2λ + 2 + λ + 6 + 2λ = 6
⇒ λ = – 1
∴ Q(î – ĵ+ 4k̂) = Q(1, – 1, 4) .
Now the required distance of P(–2, – 4, 7) from the point of intersection(1, –1, 4) is

Equation of the line PQ : r̅ = – 2î – 4ĵ+ 7k̂ + (3î + 3ĵ – 3k̂ ) or, r̅ = – 2î – 4ĵ+ 7k̂ + s(î + ĵ – k̂ ) .