TN HSE Class 12 Model Paper Maths Free PDF Download! DGE TN 2023 Sample Paper
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Mathematics
PART – I
1.) A square matrix A of order n has inverse if and only if :
(a) ρ(A) > n (b) ρ(A)=n (c) ρ(A) ≠ n (d) ρ(A) < n
2.) Distance from the origin to the plane 3x−6y+2z+7=0 is :
(a) 2 (b) 0 (c) 3 (d) 1
3.) If 3 cos−1x = cos−1 (4x 3−3x)
(a) x ∈ (1/2, 1)
(b) x ∈ [1/2, 1]
(c) x ∈ (− ∞, 1]
(d) x ∈ [1/2, ∞)
4.) The general solution of the differential equation dy/d= y /x = is :
(a) y=kx (b) xy=k (c) logy=kx (d) y=k logx
5.) The number of normals that can be drawn from a point to the parabola y2=4ax is :
(a) 3 (b) 2 (c) 0 (d) 1
6.) If a and b are parallel vectors then [a, c, b] is equal to:
(a) 1
(b) 2
(c) 0
(d) -1
7.) The number of real numbers in [0, 2π] satisfying sin4 x−2 sin2 x+1 is :
(a) 1 (b) 2 (c) ∞ (d) 4
8.) Suppose that X takes on one of the values 0, 1, 2. If for some constant k,
P (X=i) = kP (X=i−1) for i=1, 2 and p (x = 0) = 1/7, then the value of K is:
(a) 3 (b) 1 (c) 4 (d) 2
9.) The maximum value of the function x2 e−2x, x > 0 is:
(a) 1 /e2 (b) 1/ e (c) 4 /e4 (d) 1/ 2e
10.) The operation * defined by a*b= ab/ 7 = is not a binary operation on :
(a) R (b) Q+ (c) C (d) Z
11.) The area between y 2=4x and its latus rectum is :
(a) 8/ 3 (b) 2/ 3 (c) 5/ 3 (d) 4/ 3
12.) Angle between the curves y2 = x and x 2 = y at the origin is :
(a) π/2
(b) tan-1 (3/4)
(c) π/4
(d) tan-1 (4/3)
13.) |adj (adjA)| = |A|16, then the order of the square matrix A is:
(a) 2
(b) 3
(c) 5
(d) 4
14.) The value of (1+i/√2)8+(1-i/√2)8 is :
(a) 8 (b) 4 (c) 2 (d) 6
15.) If |z|=1, then the value of 1+z /1 +z is :
(a) 1 /z (b) z (c) 1 (d) z
16.) The abscissa of the point on the curve f (x )=√8- 2x at which the slope of the tangent is −0.25?
(a) −2 (b) −8 (c) 0 (d) −4
17.) The value of π/3∫0 tanx d xis :
(a) −log 2 (b) log 2 (c) −log 3 (d) log 3
(18)
(a) < n (b) 0 (c) r (d) n
19.) The Principal value of sin-1(-1/2) is
(a) –π/6 (b) 0 (c) –π/2 (d)π/ 2
20.) Area of the greatest rectangle inscribed in the ellipse x2/a2+y2/b2= 1 is
(a) √ab (b) 2ab (c) a/b (d) ab
PART – II
Answer any seven questions. Question No. 30 is Compulsory.
21.) If |z|=2, Show that 3 ≤|z+3+4i|≤7
22.) If p and q are the roots of the equation 1x 2+nx+n=0, show that √p/q +√ q/ p+√n/1 =0
23.) If y=4x+c is a tangent to the circle x2+y 2=9, find c.
24.) If the radius of a sphere with radius 10 cm, has to decrease by 0.1 cm, approximately how much will its volume decrease ?
25.) Evaluate :∞∫b 1/a2+x2 dx, a>0,b ∈R
26.) Find the vector equation of a plane which is at a distance of 7 units from the origin having 3, −4, 5 as direction ratios of a normal to it.
(27)
(28)
29.) Find the equation of tangent to the curve y=x 2+3x−2 at the point (1, 2).
30.) Express ecosθ+isinθ in a +ib form.
PART – III
Answer any seven questions. Question No. 40 is Compulsory.
31.) Find the equation of the parabola with vertex (−1, −2), axis parallel to y-axis and passing through (3, 6).
32.) The maximum and minimum distances of the Earth from the Sun respectively are 152×106 km and 94.5×106 km. The Sun is at one focus of the elliptical orbit. Find the distance from the Sun to the other focus.
33.) For what value of x, the inequality π/2<cos-1(3x-1)<π holds?
34.) Find the angle made by the straight line 2 x+3/2= y-1/2=-z with coordinate axes.
35.) Use the linear approximation to find an approximate value of (123)2 /3
36.) Solve : x cosy dy=ex(x logx+1)dx
(37)
38.) Show that p → q and q → p are not equivalent.
39.) If z=(2+3i) (1−i), then find z−1.
40.) If a+b+c=0 and a, b, c are rational numbers then, prove that the roots of the equation (b+c−a)x2+(c+a−b)x+(a+b−c)=0 are rational numbers.
Also See: Samacheer Kalvi 10th Social Science Solutions
PART – IV
Answer all the questions.
41.) (a) Solve the equation z3+8i=0, where z e C.
OR
(b) Solve: (1+x+xy2)dy/dx+(y+y3)=0
42.) (a) Using vector method, prove that cos(α−β)=cosα cosβ+sinα sinβ
OR
(b) Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density
function of random variable X is f (x) {K 200 ≤ x ≤ 600; 0 otherwise}
Find (i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and 500 litres.
43.) (a) Identify the type of conic and find centre, foci and vertices of 18x2+12y2−144x+48y+120=0
OR
(b) If cos−1x+cos−1y+cos−1z=π and 0 < x, y, z < 1, show that x2+y2+z 2+2xyz=1
44.) (a) A boy is walking along the path y=ax2+bx+c through the points (−6, 8), (−2, −12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend ? (Use Gaussian Elimination method)
OR
(b) Prove that the ellipse x 2+4y 2=8 and the hyperbola x 2−2y 2=4 intersect orthogonally.
45.) (a) Find the parametric form of Vector equation and Cartesian equations of the plane containing the line r = (î – ĵ + 3k̂) + t(2î – ĵ + 4k̂) and perpendicular to the plane r (î + 2ĵ + k̂) = 8
OR
(b) Solve the equation 6x 4−5x 3−38x 2−5x+6=0 if it is known that 1/3 is a solution.
46.) Prove that p→(¬q∨r) ≡¬p∨(¬q∨r) using truth table.
OR
(b) Suppose a person deposits ₹ 10,000 in a bank account at the rate of 5% per annum compounded continuously. How much money will be in his bank account 18 months later?
47.) (a) Find the maximum value of logx/x
OR
(b) Find the area of the region common to the ellipse x2/a2 +y2/b2=1 and the straight line x/a+y/b=1
Also See: Samacheer Kalvi 6th Social Science Solutions Pdf
The complete question paper is as follows.
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