A) x+2y= 0 B) x+2y +√(288/15)= 0
C) x+2y+√(1/15)= 0 D) none
2) If the area of the triangle included between the axes and any tangent to the curve xⁿy= aⁿ is constant, then n is equal to
A) 1 B) 2 C) 3/2 D) 1/2
3) if OT and ON are perpendiculars dropped from the origin to the tangent and normal to the curve x= a sin³t, y= a cos³t at an arbitrary point, then
A) 4OT²+ ON²= a²
B) the length of the tangent |y/cost|
C) the length of the normal |y/sint| D) none
4) The normal to the curve represented parametrically by x= a(cost + t sint) and y= a(sint - t cost) at any point t, is such that it
A) makes a constant angle with x-axis
B) is at a constant distance from the origin.
C) passes through the origin
D) satisfies all the three conditions.
5) The equation of the tangent to the curve x= t cost, y= t sint at the origin is.
A) x= 0 B) y=0 C) x+y= 0 D) x-y= 0
6) The length of the normal to the curve y= a{(e⁻ˣ⁾ᵃ + eˣ⁾ᵃ)/2} at any point varies as the
A) abscissa of the point
B) ordinates of the point
C) square of the abscissa of the point
D) square of the ordinate of the point
7) The slope of the tangent to the curve represent by x= t²+3t-8 and y= 2t² - 2t -5 at the point M(2,-1) is..
A) 7/6 B) 2/3 C) 3/2 D) 6/7
8) The equations of the tangent to the curve y = x⁴ from the point (2,0) not on the curve, are given by
A) y= 0 B) y-1=5(x-1)
C) y- 4096/81=2048/27 (x -8/3)
D) y- 32/243=80/81 (x - 2/3)
9) The value of a for which the equation x³ - 3x +a= 0 has two distinct roots in [0,1] is given by
A) -1 B) 1 C) 3. D) none
10) If the tangent to the curve 2y³ = ax² + x³ at the point (a,a) cuts off intercepts m, n on the coordinate axes such that m²+n²= 61, then a=
A)± 30 B) ±5 C) ±6 D) ±61
11) The number of real roots of the equation eˣ⁻¹+ x -2= 0 is
A) 1. B) 2. C) 3 D) 4
12) The equation of the tangent to the curve y= 1 - eˣ⁾² at the point of intersection with the y-axis is
A) x+2y= 0. B) 2x+y= 0.
C) x - y= 2. D) none
13) The normal to the curve x= a(1+ cost), y= a sint at t always passes through the fixed point.
A) (a,a) B) (a,0) C)(0,a) D) none
14) If the two curves y= aˣ & y=bˣ intersect at an angle m, then tan m equal to
A) (loga - log b)/(1+loga log b)
B) (loga + log b)/(1-loga log b)
C) (loga - log b)/(1-loga log b)
D) none
15) For the curve x= t² -1, y= t²- t, the tangent line is perpendicular to x-axis where.
A) t= 0 B) t=∞
C) t= 1/√3 D) t= -1/√3
16) The normal to the parabola y²= 4ax at (at₁²,2at₁) meets it again at (at₂², 2at₂), then
A) t₁t₂=1 B)t₂ =t₁-2/t₁ C)2t₁=t₂ D) n
17) if the line ax+ by+ c= 0 is normal to xy= 1, then
A) a>0, b>0 B) a>0, b< 0
C) b> 0, a< 0. D) a< 0, b<0
18) the tangent to a given curve is perpendicular to x-axis if
A) dy/dx= 0. B) dy/dx= 1
C) dx/dy= 0. D) dx/dy= 1
19) the normal to a given curve is parallel to x-axis if
A) dy/dx= 0. B) dy/dx= 1
C) dx/dy= 0. D) dx/dy= 1
20) the line x/a + y/b =1 touches the curve y= b ⁻ˣ⁾ᵃ at the point
A) (a,b/a). B) (-a,b/a) C) (a,a/b) D) n
21) The normal to the curve x= a(cost +t sin t), y=a(sint -t sin t) at any t is such that
A) it makes a constant angle with x-axis
B) it passes through the origin
C) it is at a distant from the origin
D) none
22) the tangent to the curve y=e²ˣ at the point (0,1) meets x-axis at
A) (0,2) B)(2,0) C) (-1/2,0) D) n
23) the point P of the curve y²= 2x³ such that the tangent at P is perpendicular to the line 4x -3y+2= 0 is given by
A) (2,4) B) (1,√2). C) (1/2, -1/2) D) (1/8, -1/16)
24) the equation of the one of the tangent to the curve y= cos(x+y), -2π≤x ≤2π that is parallel to the line x+ 2y= 0, is
A) x+ 2y= 1 B) x+ 2y= π/2
C) x+ 2y= π/4. D) none
25) the equation of the tangent at the origin to the curve y²= x²(1+x) are
A) y=± x B)c=± y C) y=± 2x D) n
26) the coordinates of the point on the curve x= a(t + sint), y= a(1- cost) where tangent is inclined at an angle π/4 to the x-axis are.
A) (a,a) B) {a(π/2-1), a}
C) {a(π/2 +1),a} D) {a, a(π/2+1)}
27) The chord joining the points where x= p and x= q on the curve y= ax²+ bx +c is parallel to the tangent at the point on the curve whose abscissa is
A) 1/2 (p+q) B) 1/2 (p-q) C) PQ/2 D) n
28) if the tangent to the curve xy + ax + by= 0 at (1,1) is inclined at an angle tan⁻¹2 with x-axis, then
A) a=1, b=2 B) a=1, b= -2
C) a=-1, b=2 D) a=-1, b=-2
29) All points on the curve y²= 4a(x+a sin(x/a)) at which the tangent are parallel to the x-axis of x, lie on a
A) circle B) parabola C) line D) n
30) If the tangent to the curve x= a(t + sint), y= a(1+cost) at t= π/3 makes an angle k with x-axis, then k=
A) π/3 B) 2π/3 C) π/6 D) 5π/6
31) the fixed point P on the curve y= x² - 4x+5 such that the tangent at P is perpendicular to the line x+2y-7= 0 is given by
A) (3,2) B) (1,2) C) (2,1) D) n
32) the point of contact of the tangents drawn from the origin to the curve y= sinx lie on the curve
A) x²- y²= xy. B) x²+ y²= x²y²
C) x²- y²= x²y² D) none
33) If the area of the triangle, included between axes and any tangent to the curve xyⁿ = aⁿ⁺¹ is constant, then the value of n is
A) -1 B) -2 C) 1 D) 2
34) The abscissa of the point on the curve ay² = x³, the normal at which cuts off equal intercepts from the coordinates axes is
A) 2a/9 B) 4a/9 C) -4a/9 D) -2a/9
35) If the curve x²/l²+ y²/m²= 1 and x²/a²- y²/b²= 1cuts each other orthogonally, then
A) a²+ b²= l² + m² B) a²- b²= l²- m²
C) a²- b²= l² + m² D) a²+ b²= l²- m²
36) The curves ax²+ by²= 1 and Ax²+ By²= 1intersect orthogonally then
A) 1/a + 1/A= 1/b + 1/B
B) 1/a - 1/A= 1/b - 1/B
C) 1/a + 1/b= 1/B - 1/A. D) none
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