See below CBSE Class 12 Mathematics Sample Paper Set E with solutions. We have provided CBSE Sample Papers for Class 12 Mathematics as per the latest paper pattern issued by CBSE for the current academic year. All sample papers provided by our Class 12 Mathematics teachers are with answers. You can see the sample paper given below and use them for more practice for Class 12 Mathematics examination.

**SECTION – A**

**Followings are multiple choice questions. Select the correct options **

**Q.1 If A is any square matrix of order 3×3 such that | A | = 3 , then the value of |adj. A| is**

(a) 3

(b) 1/3

(c) 9

(d) 27

## Answer

C

**Q.2 Suppose P and Q are two different matrices of order 3xn and nxp respectively, then the order of the matrix PxQ is**

(a) 3xp

(b) px3

(c) nx n

(d) 3×3

## Answer

A

**Q.3 If (2î + 6ĵ+ 27k̂)x(î + pĵ+ qk̂ ) = 0 ^{→} , then the value of p and q are**

(a) p = 6, q = 27

(b) p 3, q 27/2 = =

(c) p 6, q 27/2 = =

(d) p = 3, q = 27

## Answer

B

**Q.4 If A and B are two events such that P(A) = 0.2, P(B) = 0.4 and P(A∪B) = 0.5 , then value of P(A | B) is**

(a) 0.1

(b) 0.25

(c) 0.5

(d) 0.08

## Answer

B

**Q.5 The point which doesn’t lie in the half plane 2x + 3y −12 ≤ 0 is**

(a) (1, 2)

(b) (2, 1)

(c) (2, 3)

(d) (–3, 2)

## Answer

C

**Q.6 If sin ^{−1} x+sin 1 y = 2π/3 , then the value of cos^{−1} x+cos^{−1} y is**

(a) 2π/3

(b) π/3

(c) π/2

(d) π

## Answer

B

**Q.7 An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random.****Probability that they are of the different colours is**

(a) 2/5

(b) 1/15

(c) 8/15

(d) 4/15

## Answer

C

**Q.8 **

## Answer

B

**Q.9 What is the distance (in units) between the two planes 3x + 5y + 7z = 3 and 9x +15y + 21z = 9 ?**

(a) 0

(b) 3

(c) 6/√83

(d) 6

## Answer

A

**Q.10 The equation of the line in vector form passing through the point (–1, 3, 5) and parallel to line x−3/2 y−4/3 , z = 2, is**

(a) r̅ = (−î + 3ĵ+ 5k̂) + λ(2î + 3ĵ+ k̂ )

(b) r̅ = (−î + 3ĵ+ 5k̂) + λ(2î + 3ĵ)

(c) r̅ = (2î + 3ĵ− 2k̂ ) + λ(−î + 3ĵ+ 5k̂ )

(d) r̅ = (2î + 3ĵ) + λ(−î + 3ĵ+ 5k̂ )

## Answer

B

**Fill in the blanks in the following **

**Q.11 If f be the greatest integer function defined as f (x) = [x] and g be the modulus function defined as g(x) = x , then the value of gof {−5/4} is ______.**

## Answer

gof {–5/4} =g {[–5/4]} =g(–2) = |–2| 2 .

**Q.12 If the function**

**is given to be continuous at x =1, then the value of k is _________.****Answer :**

**Q.13 If**

**, then the value of y is _________.****Answer :**

**Q.14 If tangent to the curve y ^{2} + 3x − 7 = 0 at the point (h, k) is parallel to the line x − y = 4 , then the value of k is____________.**

**Answer :**

**OR**

**For the curve y = 5x − 2x ^{3} , if x increases at the rate of 2 units/sec, then at x = 3 , the slope of**

**the curve is changing at _________.**

**Answer :**

**Q.15 The magnitude of projection of (2î − ĵ+ k̂ ) on (î − 2ĵ+ 2k̂) is __________.****Answer :**

**OR**

**Vector of magnitude 5 units and in the direction opposite to 2î + 3ĵ− 6k̂ is__________.****Answer :**

**Q.16 Check whether (l + m+ n) is a factor of the determinant**

**reason.****Answer :**

**Q.17 Evaluate ∫− _{2}^{2} (x^{3}+ 1)dx .**

**Answer :**

**Q.18 Find ∫ 3+3 cos x / x+sin x dx .****Answer :**

**OR**

**Find ∫(cos ^{2} 2x −sin^{2} 2x)dx .**

**Answer :**We have ∫ (cos

^{2}2x sin–

^{2}2x)dx = ∫ cos 4xdx = 1/4 sin 4x+C .

**Q.19 Find xe ^{(1+x2)} dx ∫ .**

**Answer :**

**Q.20 Write the general solution of differential equation dy/dx = e ^{x+y} .**

**Answer :**

**SECTION – B**

**Q.21 Express sin ^{−1} **

**Answer :**

**OR**

**Let R be the relation in the set Z of integers given by R = {(a, b) : 2 divides a − b}. Show that the relation R is transitive. Write the equivalence class [0].****Answer :**

**Q.22 If y = ae ^{2x} + be^{−x} , then show that d^{2}y/dx^{2} dy/dx −2y = 0 .**

**Answer :**

**Q.23 A particle moves along the curve x ^{2} = 2y . At what point, ordinate increases at the same rate as abscissa increases?**

**Answer :**

**Q.24 For three non-zero vectors [a̅ , b̅ and c̅ , prove that [a̅ − b̅ b̅ − c̅ c̅− a̅] = 0 ^{–} .**

**Answer :**

**OR**

**If a̅ + b̅ + c̅ = 0 ^{–} and |a̅|= 3, |b̅| = 5, |c̅| = 7 then, find the value of a̅ .b̅ + b̅.c̅ + c̅.a .**

**Answer :**

**Q.25 Find the acute angle between the lines x−4/3 y+3/4 z+1/5 and x−1/4 = y+1/−3 + z+10/5 .****Answer :**

**Q.26 A speaks truth in 80% cases and B speaks truth in 90% cases. In what percentage of cases are they likely to agree with each other in stating the same fact?****Answer :** Let A and B denote the event that A speaks truth and B speaks the truth, respectively.

We have P(A) = 80%, P(B) = 90% ∴P(A̅) =100%−80% = 20%, P(B̅) =10%.

Therefore, P(A and B agree) = P(both speak truth or both lie)

⇒ = P(AB or A̅B̅) = P(A)P(B) + P(A̅)P(B̅)

∴ P(A and B agree) 80/100 x 90/100 + 20/100 x 10/100 = 74/100 i.e.,74% .

Therefore, in 74% of the cases A and B are likely to agree with each other in stating the same fact.

**SECTION – C**

**Q.27 Let f :A→B be a function defined as f (x) = 2×3/x−3 where A = R −{3} and B = R −{2} . Is the function f one-one and onto? Is f invertible? If yes, then find its inverse.****Answer :**

**Q.28 If √1− x ^{2} + √1− y^{2} = a(x − y), then prove that dy/dx √1−y^{2}/√1−x^{2 }.**

**Answer :**Put x = sin α, y = sin β ⇒ α sin

^{−1}x,β = sinn

^{−1}y…(i)

**OR**

**If x = a(cos 2θ+ 2θsin 2θ) and y = a(sin 2θ− 2θcos 2θ) , find d ^{2}y/dx^{2} at θ = π/8 .**

**Answer :**

**Q.29 Solve the differential equation xdy − ydx = √x ^{2} + y^{2} dx .**

**Answer :**

**Q.30 Evaluate ∫ _{1}^{3} |x^{2} − 2x| dx .**

**Answer :**

**Q.31 Two numbers are selected at random (without replacement) from first 7 natural numbers. If X denotes the smallest of the two numbers obtained, find the probability distribution of X.****Also find mean of the distribution.****Answer : **Here X denotes the smallest of the two numbers obtained from first 7 natural numbers.

So, X can take values 1, 2, 3, 4, 5, 6.

Total no. of possible ways in which 2 nos. can be selected from 7 natural nos. is ^{7}C_{2} = 21.

Table for probability distribution is given as below :

**OR**

**There are three coins, one is a two headed coin (having head on both the faces), another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows head, what is probability that it was the two headed coin?****Answer :** Let E : the coin shows head.

Also let E_{1}, E_{2}, E_{3} : the coin is a two headed, a biased coin and unbiased coin, respectively.

**Q.32 Two tailors A and B earn ₹150 and ₹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 an L.P.P. to minimize the labour cost to produce (stitch) at least 60 shirts and 32 pants and solve it graphically.****Answer :** Let the number of days for which the tailors A and B work be x and y, respectively.

To minimize : Z = ₹ (150x + 200y)

Since the feasible region is unbounded so, Z =1350 may or may not be the minimum value.

To check, let 150x + 200y < 1350 i.e., 3x + 4y < 27 .

As there’s no common point between 3x + 4y < 27 and the feasible region so, Z =1350 is the minimum value.

**SECTION – D**

**Q.33 Using the properties of determinants, prove that**

**Answer :**

**OR**

**Q.33 **

**Answer : **

**Q.34 Using integration, find the area of the region {(x, y) : x ^{2} + y^{2} ≤1, x + y ≥1, x ≥ 0, y ≥ 0} .**

**Answer :**We have {(x, y) : x

^{2}+ y

^{2}≤1, x + y ≥1, x ≥ 0, y ≥ 0}

Consider x

^{2}+ y

^{2}=1…(i), x + y = 1…(ii) and x = 0, y = 0 .

Curve (i) is a circle whose centre is at origin and radius is of 1 unit.

Also the line (ii) cuts off the intercepts of 1 unit on both the axes in the 1st quadrant.

**Q.35 A given quantity of metal is to be cast into a solid half circular cylinder with a rectangular base and semi-circular ends. Show that in order that total surface area is minimum, the ratio of length of the cylinder to the diameter of its semi-circular ends is π: (π + 2) .****Answer :** Given volume V 1/2 πr^{2}h

⇒ h = 2V/πr^{2} …(i)

**OR**

**Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.****Answer :**

**Q.36 Find the equation of a plane passing through the points A(2, 1, 2) and B(4, –2, 1) and perpendicular to plane r̅ .(î − 2k̂ ) = 5 . Also find the coordinates of the point, where the line passing through the points (3, 4, 1) and (5, 1, 6) crosses the plane thus obtained.****Answer : **