MATHS FOR CLASS 12TH. Your shortcut to success. SCHOLAR'S ACADEMY
BOOK CO. ☎9868220237. 1. Sample Paper-2. Class XII. Maths. Time : 3 Hours.
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MATHS FOR CLASS 12TH
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Sample Paper -2 Class XII Maths www.yogenius.com
by Tarun Gaur
Sample Paper-2 Class XII Maths Time : 3 Hours Maximum marks 100 General Instruction: 1. All Question are compulsory 2. There are three - sections of 29 questions i.e, section A, section B, section C. 3. Section A contain 10 question of 1 mark each, Section B of 12 question of 4 marks each and section C contains 7 question of 6 marks each. 4. There is no overall choice. However internal choice has been provided in 04 question of 4 marks and 02 question of 6 marks. You have to attempt only are of the alternatives in all such question.
Q1
Section - A If f(x) = e and g(x) = log x. Show that fog = gof for x>0. x
( )
( )
Q4
FInd the principal value of cos-1 cos 7π + cos-1 cos 5π 3 6 1 If A= 2 , B = [2 -3 4] , find AB. 3 Define skew symmetric matrix with an example.
Q5
For what value of ‘k’, the matrix
Q6
Differentiate, log10 (sinx) w.r.t. x.
Q7
Evaluate
Q8
→ ∧ ∧ ∧ If vector AB = 3 i - j + 5 k and co-ordinates of terminal point are (0, 1, 3). Find co-ordinates of initial point.
Q2 Q3
K 3 2 4
in singular matrix.
∫ sin xdx. cos x 2
2
∧ ∧ ∧ ∧ ∧ ∧ Find the value of ‘p’ such that the vectors 2 i - j + k and i + p j -5 k are perpendicular to each other. ∧ ∧ ∧ ∧ ∧ ∧ → ∧ ∧ ∧ Q 10 Find the angle between line→ r = ( 2 i - j +3 k) + λ ( 3 i - j +2 k) and the plane r . ( i + j + k) = 3.
Q9
Q 11
Section - B Consider f : R →[-5, ∞ ) given by f(x) = 9x2 + 6x -5, show that f is invatible also find f-1. OR +
Consider the function f : 0, π 2
→R
given by f(x) = sin x and g : 0, π 2
→R
given by
g(x) = cos x. Show that ‘f’ and ‘g’ are one-one but not f + g. Q 12 If cos-1 x + cos-1 y = α , prove that a b
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x2 - 2xy cosα + y2 = sin2α . ab b2 a2
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MATHS FOR CLASS 12TH
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Q 13 If x, y and z are different and x x2 1 + x3 y y2 1 + y3 z z2 1 + z3
= 0 , then show that xyz = -1.
OR a b c a2 b2 c2 bc ca ab
Prove that :
= (a - b) (b - c) (c - a) (ab + bc + ca).
Q 14 For what value of ‘k’, is the function f(x) continous at x = 0?
{ sinx +xx cos x
, when x ≠ 0 and
f(x) =
f(x) = k, when x = 0. Q 15 Verify Lagrange’s mean value theovean for the function f(x) = (x - 3) (x - 6) (x - 9) in [3, 5]. Q 16 Find the approximate value of f (5.001), where f(x) = x3 - 7x2 + 15. OR Find the internal in which the function f(x) = sin x + cos x ; 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing. 3 2 x sin πx . dx. Q 17 Evaluate I = -1 dy Q 18 Find the particular solution of the differential equation + y cot x = 2x + x2 cot x ; (x ≠ 0) dx given that y = 0, when x = π . 2 y2 + y + 1 dy = 0 is given by (x + y Q 19 Show that the general solution of different equation + 2 dx x + x + 1 + 1) = A (1 - x - y - 2xy) where ‘A’ in perametre. → → → → → → → → → → → → → Q 20 If → a × b = c × d and a × c = b × d, prove that a − d is parallel to b − c , provided a ≠ d → and → b≠c .
∫
OR → Show that → a×b =
√a
2
→ b2 - ( → a.b )
2
Q 21 Prove that the agnle between two diognals of a cube is cos-1
( 13 ).
Q 22 A and B throw a die alternatively till one of them gets a ‘6’ and wins the game. Find their respective probability of winning if ‘A’ starts first. Section - C Q 23 An electric assembly consists of two subsystems, say A and B. From previous testing procedures, the SCHOLAR’S ACADEMY BOOK CO.
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MATHS FOR CLASS 12TH
following probabilities are assumed to be known. P(A fails) = 0.2 , P(B fails alone) = 0.15 Find the following probability (i) P( A fails / B has failed)
Your shortcut to success
P(A and B fail) = 0.5
(ii) P( A fails alone).
Q 24 Find the magnitude and equation of the line of shortest distance between the lines. ∧ ∧ ∧ → (λ - 1) ∧i + (λ + 1) ∧j - (λ + 1) ∧k. and → r = (1 - µ) i + (2 µ - 1) j + ( µ + 2 ) k . r = OR Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38. Q 25 Find the area of region bounded by curve y = √1 - x2 , line y = x and positive x-axis. OR Using integration, find the area of region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9. π log (1 + cos x) . dx. Q 26 Evaluate : I = 0 Q 27 Show that the volume of greatest cylinder that can be inscribed in a cone of height ‘h’ and semivertical angle ‘α’ is 4 πh3 tan2 α. 27
∫
Q 28 The sum of three numbers in 6. If we multiply the third number by 3 and add second number to it we get 11. By adding first and third numbers, we get double of second number. Represent the information algebrically and find the numbers using matrix method. Q 29 A toy company manufactures two types of dolls, A and B. Market test and available resources have indicated that the combined production level should not exceed 1200 dolls per week and demand for dolls of type B is atmost half of that for dolls of type A. Further the production level of dolls of type A can exceed three times the production of dolls of other type by atmost 600 units. If the company makes profit of Rs.12 and Rs.16 per doll respectively on dolls A and B, how much of each should be produced weekly in order to maximise the profit?
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