The document is a mathematics sample test paper for Class XII, consisting of 29 questions divided into three sections: A (10 questions of 1 mark each), B (12 questions of 4 marks each), and C (7 questions of 6 marks each), with internal choices in some questions. The contents cover various mathematical topics including calculus, matrices, probability, and geometry, requiring both theoretical knowledge and problem-solving skills. Use of calculators is not permitted, and all questions are compulsory.

1. MATHEMATICS
SAMPLE TEST PAPER
CLASS XII
Class:12
Time 3hrs
Max Mks:100
No of pages: 3
General Instructions:
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Ò All questions are compulsory.
Ò The question paper consists of 29 questions divided into three sections - A, B and C.
Ò Section - A comprises of 10 questions of one mark each, Section B is of 12 questions of four
marks each, Section C comprises of 7 questions of six marks each.
Ò Internal choice has been provided in four marks question and six marks question. You have to
attempt any one of the alternatives in all such questions
Ò use of calculator not permitted.
1. Find the value of x , if
w
w
sin2x
2. Evaluate: ∫ ( 3+ secx )dx
.e
du
SECTION A
Question number 1 to 10 carry 1 mark each
w
3. If A is irreversible matrix or order 5 and ∣ A∣ = 5, then find [adj,A]
a
a b
b
b
i
k
4. Find the projection of ⃗ on ⃗ , if ⃗ . ⃗ = 6 and ⃗ =4 ⃗ + ⃗j +8 ⃗
2
2x
5. Evaluate: ∫ ( 1+ 3x ) dx
3π
6. what is the value of tan-1 4
7. write the order of the matrix
8. write the position vector of the mid point of the vector join g the points p(2,3,4 and q (4,-1,2))
9. write the distance from following plane from the origin2x-y+2z+1 = 0
0
i
k
i
k
10. Find λ if (2 ⃗ +6 ⃗j +14 ⃗ )*( ⃗ -λ ⃗j +7 ⃗ ) = ⃗

2. SECTION B
Question numbers 11 to 22 carry 4 marks each.
11. State whether the function is one -one, onto or objective. Justify your answer
a) f:R→R defined by f (x) = 4+5x
b) f:R→R defined by f (x) = 4x+5x2
dy
12. use mathematical induction prove that dx (xn)=nx(n-1)for all n belongs N for all positive
integers n.
13.
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1+
1−
√ x− √ x π
14. prove that tan-1( √ x+ √ x )= 2
1+
1−
15. Find the derivative of the function given by f(x) = (1+x) (1+x2) (1+x4)(1+x8) and hence find
f(3)
16. using properties of determinants, prove that
w
prove that
w
w
.e
du
or
17. solve the differential equation
2
2
x dy - y dx= √x + y
or
dy
2
dx = log(x+2)
18. A speaks truth 10 times out of 15 times. A dice is tossed. He reports that it was 6. What is the
probability it was actually 6.'
sin− 1( x)
d2y
dy
19. If y =
, show that (1-x2) dx2 -3x dx -y = 0
2
√1− x
20. Express the following matrix as the sum of symmetric and skew symmetric matrix, and verify
your answer.

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21. On multiplying choice examination with three possible answers for each of the five questions,
what is the probability that a candidate would get four or more correct answer just by guessing?
22. Finds the shortest between the following lines:
r
⃗ = (1+λ) ⃗ +(2-λ) ⃗j + (λ+1)7 ⃗ ; ⃗ = (2 ⃗ - ⃗j + - ⃗ )+μ(2 ⃗ + ⃗j +2 ⃗ )
i
i
i
k r
k
k
SECTION C
Question numbers 23 to 29 carry 6 marks each.
23. solve using matrices:
2x-y+3z = 5
3x+2y-z = 7
4x+5y-5z = 9
or
using matrix method solve the following system of equations:
2x+y+z = 3
3x-y+z = 0
x-2y+3z = -6
du
24. Find the area of the region bounded by y2 = 4x, x = 1, x = 4 and x -a xis in the first quadrant
.e
or
Evaluate lim 0− 2 ∫
as limit of a sum.
25. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of
1
height h is 3 h.
w
w
w
x 2 + x+ 1 dx
26. Find the equation of the plane passing through the (-1,-1,2) are perpendicular to each of the
following planes 2x+3y-3z = 2 and 5x-4y+z = 6
or
Find the point on the curve y2 = 4x which is nearest to the point (-,-8)
27. Sketch the graph of y = Ix + 5I and evaluate the area under the curve y = x + 5 above x-axis
and between x = -7 to x = O.
28. Using the method of integration, find the area of region bounded by the lines 2x+y = 4, 3x-2y =
6, x-3y+5 = 0
29. A can hit a target 4 times in 5 shots,B 3 times in 4 shots and C, 2 times in 3 shots. Calculate the
probability that:
a)A,B,C all may hit b) none of them ii hit the target