Junior Inter Model Paper, Mathematics Paper – I (B).
Section – A
I. (i) Very short answer type questions.
(ii) Answer ALL questions.
(iii) Each question carries TWO marks. 10 × 2 = 20
1) Find ‘k’, if the lines 6x + 5y – 1 = 0, 2x + ky + 2 = 0, 4x – 3y – 7 = 0 are concurrent.
2) Find the equation of the line passing through (4, -3) and perpendicular to the line 5x – 3y + 1 = 0
3) (4, 2, 2) is the centroid of a tetrahedon whose three vertices are (6,2,5), (4, 1, 1), (3, 2, -1). Find the fourth vertex.
4) Find the equation of the plane passing through the point (α, β, γ) and parallel to the plane ax + by + cz = 0.
5) Evaluate: Lt x→0 x2.sin(1/x).
6) Evaluate: Lt ⎯(3x ⎯-1) x x→0 1-cos x
7) Discuss the continuity of x – [x]
8) Differentiate sin x.sin 2x.sin3x w.r.t.x.
9) Find Δ y and dy when y = ⎯1 x ; x = 2; Δ x = 0.002.
10) Find the equation of normal to the curve x2 y2 ⎯ + ⎯ = 1 a2 b2
at the point (a cos θ , b sin θ).
Section – B
II. (i) Short answer type questions.
(ii) Answer any FIVE questions.
(iii) Each question carries FOUR marks. 5 × 4 = 20
11) Find the equation to the locus of a point p which forms a triangle PAB of area 9 square units with the points A(5, 3) and B(3, -2).
12) When the origin is shifted to the point (2, 3), the transformed equation of a curve is observed to be x2 + 3xy – 2y2 + 17x- 7y – 11
= 0. Find the original equation of the curve.
13) Prove that the altitudes of a triangle are concurrent.
14) Find ⎯dy dx , if y = Tan-1 ( ⎯c⎯o⎯s⎯x cosx+1)
15) Find ⎯dy dx , if xy = ex-y
16) Water is dripping out from a conical funnel, at a uniform rate of 2 cc./ sec. through a tiny hole at the vertex at the bottom. When the slant height of water is 4 cm., find the rate of decrease of the slant height of the water given that vertical angle of the funnel is 120o.
17) If F = x. g(y/x) + y. h (y/x), then show that x2 Fxx + 2xy. Fxy + y2.Fyy = 0
Section – C
III. (i) Long answer type questions.
(ii) Answer any FIVE questions.
(iii) Each question carries SEVEN marks. 5 × 7 = 35
18) If p and q are the length of the perpendiculars from the origin to the straight lines x sec α + y cosec α = a and x cos α – y sin
α = a cos 2α , then prove that 4p2+q2 = a2.
19) Find the equation to the pair of lines passing through the point (2, -1) and (i) Parallel to the pair of lines 6×2 – 13xy – 5y2 = 0 and (ii) perpendicular to the pair of lines 6×2 – 13xy – 5y2 = 0.
20) If the equation 2×2 + kxy – 6y2 + 3x + y + 1 = 0 represents a pair of lines, find (i) k (ii) the point of intersection and (iii) the angle of intersection of the lines.
21) Find the angle between the lines whose direction cosines satisfy the relations 3l + m + 5n = 0 and 5 lm – 2 ln + 6 mn = 0.
22) If f(x) = sin-1 √ x-β ; g(x) = tan-1√ x-β α-β α-x , then prove that f'(x) = g'(x) where β < x < α
23) Show that the curves y2 = 4(x+1) and y2 = 36(9-x) intersect each other orthogonally.
24) Show that the maximum rectangle inscribed in a circle is a square.