jiyg;gpw;f fpno cs;s ypq;if fpspf; nra;j fotpy; ,izatk;! ) · 12/10/2019 · if adj(adj a) = a 9,...
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( jiyg;gpw;F fPNo cs;s ypq;if fpspf; nra;J FOtpy; ,izaTk;! )
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SHRI KRISHNA ACADEMY NEET,JEE AND BOARD EXAM COACHING CENTRE
SBM SCHOOL CAMPUS,TRICHY MAIN ROAD,NAMAKKAL
CELL:9965531727-9443231727
SUBJECT: MATHEMATICS TIME: 2.30 hrs
MARKS: 100
PART- A Note : (i) All questions are compulsory. 20x1=20 (ii) Choose the most suitable answer from the given four alternatives and write the option code and the corresponding answer.
1. If adj(adj A) = A 9, then the order of the square matrix A is
(1) 3 (2) 4 (3) 2 (4) 5
2. Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj λA = λn adj(A)
(iv) 𝐴 adj A = adj A 𝐴 = 𝐴 𝐼
(1) Only (i) (2) (ii) and (iii) (3) (iii) and (iv) (4) (i),(ii) and (iv)
3. The area of the triangle formed by the complex numbers z,iz and z+iz in the
Argand’s diagram is
(1) 1
2 z 2 (2) z 2 (3)
3
2 z 2 (4) 2 z 2
4. The principal argument of sin40° + icos40° 5 is
(1) – 110° (2) – 70° (3) 70° (4) 110°
5. A zero of x3+64 is
(1) 0 (2) 4 (3) 4i (4) -4
6. sin−1 3
5− cos−1 12
13+ sec−1 5
3− cosec−1 13
12 is equal to
(1) 2π (2) π (3) 0 (4) tan−1 12
65
7. The equation of the circle passing through (1,5) and (4,1) and touching y -axis is
x2+y
2 −5x −6y+9+λ(4x+3y −19) =0 where λ is equal to
(1) 0,−40
9 (2) 0 (3)
40
9 (4)
−40
9
8. The eccentricity of the ellipse 𝑥 − 3 2 + 𝑦 − 4 2 =y2
9 is
(1) 3
2 (2)
1
3 (3)
1
3 2 (4)
1
3
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9. If a . b = b . c = c . a = 0 then the value of a , b , c is
(1) a b c (2) 1
3 a b c (3) 1 (4) −1
10. The angle between the line 𝑟 = 𝑖 + 2j − 3k + 𝑡 2𝑖 + j − 2k and the plane line
𝑟 . 𝑖 + j + 4 = 0 is
(1) 0° (2) 30° (3) 45° (4) 90°
11. The volume of a sphere is increasing in volume at the rate of 3π cm3/sec.
The rate of change of its radius when radius is 1
2 cm
(1) 3 cm/s (2) 2 cm/s (3) 1 cm/s (4) 1
2 cm/s
12. If u( x, y) =𝑒𝑥2+𝑦2,then u
x
is equal to
(1) 𝑒𝑥2+𝑦2 (2) 2xu (3) x
2u (4) y
2u
13. The value of 𝑥 2
−1 𝑑𝑥 is
(1) 1
2 (2)
3
2 (3)
5
2 (4)
7
2
14. The order and degree of the differential equation
𝑠𝑖𝑛 𝑥 (dx + dy) = 𝑐𝑜𝑠 𝑥 (dx − dy) is
(1) 1, 2 (2) 2, 2 (3) 1, 1 (4) 2, 1
15. The solution of the differential equation 2x 𝑑𝑦
𝑑𝑥 − y = 3 represents
(1) straight lines (2) circles (3) parabola (4) ellipse
16. If in 6 trials, X is a binomial variable which follows the relation 9P(X=4) = P(X=2),
then the probability of success is
(1)0.125 (2) 0.25 (3) 0.375 (4) 0.75
17. Which one of the following is not true?
(1) Negation of a negation of a statement is the statement itself.
(2) If the last column of the truth table contains only T then it is a tautology.
(3) If the last column of its truth table contains only F then it is a contradiction
(4) If p and q are any two statements then p↔q is a tautology.
18. If i2 = - 1, then i
1 + i
2 + i
3 + … + up to 1000 terms is equal to
(1) 1 (2) - 1 (3) i (4) 0
19. The area enclosed by the curve y = 𝑥2
2, the x- axis and the lines x = 1, x= 3 is
(1) 4 (2) 82
3 (3) 13 (4) 4
1
3
20. In eight throws of a die, 1 or 3 is considered a success. Then the mean number of
success is
(1) 8
3 (2)
4
3 (3)
2
3 (4)
5
3
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PART – B
Note: (i) Answer any 7 questions. 7X2 = 14
(ii) Questions No. 30 is compulsory and choose any six questions
from the remaining.
21. State and prove Right Cancellation Law of square matrices.
22. Simplify :
12
1
n
n
i
23. If α, β, and γ are the roots of the equation x3 +px
2+qx+r= 0, find the value of
1
in terms of the coefficients.
24. Find the general equation of a circle with centre (-3,-4) and radius 3 units.
25. Prove by vector method that an angle in a semi-circle is a right angle.
26. Use the linear approximation to find approximate values of 154
27. Evaluate : (3𝑥2 − 4𝑥 + 5) 𝑑𝑥3
0.
28. Determine the order and degree (if exists) of the following differential equations:
𝑑2𝑦
𝑑𝑥 2+ 3
𝑑𝑦
𝑑𝑥
2= 𝑥2𝑙𝑜𝑔
𝑑2𝑦
𝑑𝑥 2
29. Suppose X is the number of tails occurred when three fair coins are tossed once
simultaneously. Find the values of the random variable X and number of points in its
inverse images.
30. Find the maximum and minimum values of f(x) = 𝑥 + 3 ∀ x ∈ R.
PART – C
Note : (i) Answer any 7 questions. 7 X 3 = 21
(ii) Questions No. 40 is compulsory and choose any six questions
from the remaining..
31. If A is a non-singular square matrix of order n , then prove that adj A T = adj AT
32. Find the values of the real numbers x and y, if the complex numbers
(3-i) x- (2-i) y+2i+5 and 2x+ (-1+2i) y+3+2i are equal.
33. Solve: cot−1 x − cot−1(x + 2) =π
12, x > 0
34. Find the equation of the circle through the points (1,0) , (−1,0) and (0,1).
35. If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by,
𝑚 𝑥 = 3x then what is the rate of change of mass with respect to the length when
it is x = 3 and x = 27 metres.
36. Find the linear approximation for f (x) = 1 + x, x ≥ −1, at x0 = 3. Use the linear
approximation to estimate f (3.2).
37. Find an approximate value of 𝑥2𝑑𝑥1.5
1 by applying the right-end rule with the
partition {1.1, 1.2, 1.3, 1.4, 1.5}.
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38. Assume that a spherical rain drop evaporates at a rate proportional to its surface area.
Form a differential equation involving the rate of change of the radius of the rain
drop.
39. Verify the (i) closure property, (ii) commutative property, (iii) associative property
(iv) existence of identity and (v) existence of inverse for the arithmetic operation
+ on Z .
40. Solve: 𝑥 − 1 4 + 𝑥 − 5 4 = 82.
PART – D
Note: Answer all questions. 7 X 5 = 35
41. (a) If A = 4 32 5
, find x and y such that A2 +xA +yI2 = O2. Hence, find A
−1. (OR)
(b) If cos α + cos β + cos γ = sin α + sin β + sin γ = 0 then show that
(i) cos3 α + cos 3β + cos 3γ = 3cos α +β + γ and
(ii) sin3 α + sin 3β + sin3γ = 3sin α +β + γ
42. (a) Form the equation whose roots are the squares of the roots of the cubic equation
x3 +ax
2+bx+c= 0. (OR)
(b) If a1, a2, a3, ... an is an arithmetic progression with common difference d, prove
that tan tan−1 d
1+a1a2 + tan−1
d
1+a2a3 + … . . + tan−1
d
1+an an−1 =
an −a1
1+a1an
43. (a) The orbit of Halley’s Comet (Fig) is an
ellipse 36.18 astronomical units long
and by 9.12 astronomical units wide.
Find its eccentricity.
(OR)
(b) Show that the straight lines x+1= 2y = −12z and x = y+2 = 6z−6 are skew and
hence find the shortest distance between them.
44. (a) If we blow air into a balloon of spherical shape at a rate of 1000 cm3 per second. At
what rate the radius of the baloon changes when the radius is 7cm? Also compute
the rate at which the surface area changes. (OR)
(b) Let 𝑓(x, y) = sin(xy2) + ex3+5y for all (x, y)ϵℝ2. Calculate
f
x
,
f
y
,
2 f
y x
and
2 f
x y
.
45. (a) Evaluate the following integrals as the limits of sums: (4𝑥2 − 1) 𝑑𝑥2
1 (OR)
(b) Solve 1 + 𝑥 + 𝑥𝑦2 𝑑𝑦
𝑑𝑥+ 𝑦 + 𝑦3 = 0
46. (a) A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on
remaining three faces. The die is thrown twice. If X denotes the total score in two
throws, find
(i) the probability mass function (ii) the cumulative distribution function
(iii) P(4≤X<10) (iv) P(X≥6) (OR)
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(b) (i) Define an operation∗ on ℚ as follows: a ∗ b = a+b
2 ; a, b ∊ ℚ. Examine the
closure, commutative, and associative properties satisfied by ∗ on ℚ.
(ii) Define an operation∗ on ℚ as follows: a ∗ b = a+b
2 ; a, b ∊ ℚ. Examine the
existence of identity and the existence of inverse for the operation ∗ on ℚ.
47. (a) Find all the roots 2 − 2𝑖 1
3 and also find the product of its roots. (OR)
(b) Find the area bounded by x = at2, y = 2at between the ordinates corresponding to
t = 1 and t= 2,
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