M. A- R-1
2) If α , β be the roots of the equation ax²+ bx+ c= 0 and γ, δ those of the equation px²+ qx + r= 0, show that ac/pr = b²/q², if αδ = βγ .
3) If n be the positive integer greater than unity, then show that 49ⁿ - 16n -1 is divisible by 64.
4) If the sum of the first 2n terms of a GP is twice the sum of the reciprocals of the terms, then show that the continued product of the terms is equal to 2ⁿ.
5) How many numbers of four digits can be formed from the numbers 1,2,3, 4? Find the sum of all such numbers (digits being used once only). 24, 66660
6) If 9α=π, find the value of sinα sin2α sin3α sin4α. 3/16
7) If tan θ = (tanα - tan β)/(1- tanα tan β), then show that sin2θ = (sin2α - sin2 β)/(1- sin2α sin2β).
8) If 8R²= a²+ b²+ c² (or, cos²A+ cos²B + cos²C= 1), then show that the triangle ABC is right angled.
9) Solve: cos³ θ cos3 θ+ sin³ θsin3 θ = 1/8. nπ± π/6
10) If {m tan (x- y)}/cos²y= n tan y/cos²(x - y), show that y= (1/2)[x - tan⁻¹{n - m)/(n+ m)} tan x].
11) if lx + m y=0 be the perpendicular bisector of the segment joining the point (a,b) and (c,d). then prove that (c - a)/l = (d - b)/m = 2(la+ mb)/(l²+ m²).
12) Show that the two circles x²+ y²+ 2gx + 2fy=0 and x²+ y²+ 2g'x + 2f'y=0 will touch each other if f'g= g'f.
13) Find the equation and the latus rectum on the parabola whose focus is (5,3) and vertex is (3,1). x²+ y²- 2xy - 20x -12y+ 68=0; 8√2
14) if α and β be the eccentric angles of the extremities of a focal chord of the ellipse x²/a²+ y²/b²=1.
Show that tan(α/2) tan(β/2) = (e -1)/(e+1) or (e +1)/(e-1).
15) Find dy/dx when y= ₓx²+ ₐx². ₓx²+1+ (1+ 2 logx)+ ₐx² log a
16) Evaluate lim ₓ→₀ (tan2x - x)/(3x - sinx). 1/2
17) if y= xⁿ{a cos(logx)+ b sin(logx)}, show that x² d²y/dx²+ (1- 2n) x dy/dx + (1+ n²)y= 0.
18) Show that eˢᶦⁿˣ - e⁻ˢᶦⁿˣ = 4 has no real solution.
19) find the derivative of x² cosx. 2x cosx - x² sinx
20) prove ᵗᵃⁿˣ₁/ₑ ∫ t dt/(1+ t²)+ ᶜᵒᵗˣ₁/ₑ ∫ dt/t(1+ t²)= 1.
21) solve: (x + y)² dy/dx = 2x + 2y+5. y= log{(x + y+1)²+4} + (3/2) tan⁻¹{(x+ y+1)/2}+ c
22) Evaluate lim ₙ→∞ (1/n)[sec²(π/4n) + sec²(2π/4n + .....+ sec²(nπ/4n)]. 4/π
23) integrate: ∫ (cosx - sinx)(2+ 2 sin2x)/(cosx + sinx) dx. Sin2x + c
24) solve: d²y/dx² - 2a dy/dx + a²y= 0, given y= a and dy/dx = 0 when x= 0. y= a(1- ax)eᵃˣ
25) Shade the region above the x-axis , included between the parabola y²= 4x and the circle (x -4)= 4 cos θ , y= 4 sin θ. Find the area of the region by integration. (4π - 32/3) Square.unit
26) Show that the maximum value of the function x + 1/x is less than its minimum value.
27) Show that the line lx + my = n is a normal to the ellipse x²/a²+ y²/b² = 1, if a²/l²+ b²/m² = (a²- b²)²/n².
28) Show that log(1 + x)> (tan⁻¹x)/(1+ x) for all x > 0.
29) prove that ³√(2+ √5)+ ³√(2 - √5)= 1.
30) determine the sign of the expression
(x - 1)(x -2)(x - 3)(x - 4) + 5 for real value of x. Positive
31) If cot x = 2 and cot y=3, then find (x - y). π/4
32) Find when the solution of the equation acos x + b sin x = c is possible. c≤ √(a²+ b²)
33) Find the square root of 4ab - 2i(a²- b²). ±(a+ b) i(a- b)
34) if θ (x)= (x -1)eˣ + 1, Show that θ (x) is positive for all the values of x> 0.
35) If y= f(x)= (x+1)/(x+2) show that, f(y)= (2x+3)/(3x +5).
36) Evaluate: ¹₋₁∫ sin³x cos²x dx. 0
37) Is it possible to draw a tangent from the point (-2,-1) to the x²+ y²- 4x + 6y - 12= 0 ? give reasons . No
38) If f(x)= tan(x - π/4), find f(x) x f(-x). 1
M. A- R-2
1) If x= {q - √(p²- 4q)}/{q+ √(p²- 4q)} , show that (q²- p²+ 4q)(x²+1) - 2(p²+ q²- 4q)x = 0.
2) show that log₂₀3 lies between 1/2 and 1/3.
3) In the expansion of (√x - √k/x²)¹⁰, the term independent of the x be 405; find the value of k. 9
4) if the equation x²+ bx + ca = 0 and x²+ cx + ab = 0 have a common root, prove that their other roots will satisfy the equation x²+ ax + bc = 0.
5) if x - 1/x = 2i sin θ, Show that x⁴- 1/x⁴ = 2i sin4θ.
6) prove that tan 6° tan 42° tan 66° tan 78°=1.
7) Show that time 20 tan 40 tan18 = tan60.
8) if a cosθ + b sinθ = a cos α+ b sinα, show that sin(α +θ)= 2ab/(a²+ b²).
9) Solve : sin⁸x + cos⁸x = 17/32. (2n +1)π/8
10) If a²(1- sinα) + b²(1+ sinα)= 2ab cosα , show that a/b - b/a = 2 tanα.
11) In a triangle ABC, prove that (bc - r₂r₃)/r₁ = (ca - r₃r₁)/r₂ = (ab - r₁r₂)/r₃.
12) y= mx is the equation of a chord of the circle x²+ y²- 2ax =0. Prove that the equation of the circle on this chord as diameter is (1+ m²)(x²+ y²) - 2a(x + my)= 0.
13) The point (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices lie on the line y= 2x + c. Find c and the remaining vertices. -4, (2,0),(4,4)
14) P is a variable point on the hyperbola x²- y²= a² and A is the fixed point (2 a, 0). Show that the locus of the midpoint of the line segment AP is another hyperbola.
15) Given the ellipse 4x²+ 9y²= 36, find the equation of the chord which is bisected at (2,1). 8x - 9y=25
16) A function f(x) is defined as follows:
f(x)= 2 - |x|/x, when -2≤ x ≤ 2.
a) Draw the graph of f(x) and discuss the continuity at x=0. discontinuous at x= 0.
b) Does the limit of f(x) exist when x= 0? Explain.
17) Find derivative of tan(√x). (Sec²√x)/2√x.
18) If cosy = x cos(a+ y), show that dy/dx = (cos²(a+ y)/sina.
19) Evaluate: lim ₓ→₀ (x log(1+ x))/(1- cosx). 2
20) lim ₓ→₁ (1+ cosπx)/(1- x)². π²/2
21) If x= eᶻ, then show that x² d²y/dx² = d²y/dz² - dy/dz.
22) Evaluate: lim ₙ→∞ [1/(n +1) + 1/(n +2)+ 1/(n +3)+ .....+ 1/6n]. Log6
23) ∫ dx/[(x - a)√{(x - a)(b - x)}]. 2/(a - b) √{(b - x)/(x - a)}+ c
24) ∫ (x cosx)/sin³x dx. (-1/2) (x cosec²x + cotx)
25) Show that: π⁾⁴₀∫ sin²x cos²x/(sun³x + cos³x)² dx = 1/6
26) Solve: xy dy/dx = (1+ y²)(1+ x + x²)/(1+ x²). (1/2) (1+ y²)= Logx + tan⁻¹x + c
27) find the value of ᵇₐ∫ e⁻ˣ dx. e⁻ᵃ - e⁻ᵇ
28) Find the area bounded by the upper half of the ellipse x²/25 + y²/9= 1, the x-axis and the straight lines x=3 and x= 4. (15/2) (Sin⁻¹ - sin⁻¹(3/5)) Sq. Unit
29) Prove that the normal chord of the parabola y²= ax which is normal at the point (a/2, a/√2) subtends a right angle at the vertex.
30) The eccentric angles of two points on the ellipse x²/a²+ y²/b²= 1 are α and β. If the tangents at those two points and intersect at (h,k), show that (h²/a²+ k²/b²)² sin²(α - β)= 4(h²/a²+ k²/b² -1).
31) Prove that the extremum of u(x)/v(x) is given by u've v - uv'= 0 and the extremum is a maximum or a minimum according as u" v - uv" is < or > 0.
32) which term of the following two series is equal?
2+ 7+12+...... And 101+97 +93 +....12th
33) The roots of (a²+ b²)x²+ 2(a+ b)x +2= 0 are always:
a) real b) imaginary c) positive d) equal
34) If tan200 = a find (sin110 - cos 250)/(cosec 160+ sec 340) in terms of a. a/(1+ a²)
35) evaluate: tan⁻¹(a/b) - tan⁻¹{(a-b)/(a+ b)}. π/4
36) Find the minimum value of ligₐx + logₓ a for 0< a < x. 2
37) If y= c₁x⁻¹ + c₂x², the value of x² d²y/dx² is:
a) x b) y c) 2x d) 2y
38) The centre of a circle is (3,4) and the length of a tangent drawn from (-2,-2) to the circle is 6. Find the radius of the circle. 5
39) π₀∫ f(x) dx = 0, then f(π - x) is equals to:
a) f(x) b) f(-x) c) - f(x) d) none of these
40) Find the differential equation of the straight lines which pass through the origin. y= x dy/dx
41) If f(x)= x⁹- 6x⁸ - 2x⁷+ 12x⁶+ x⁴ - 7x³+ 6x²+ x -3, then find the value of f(6). 3
M. A - R- 3
α
β λ θ γ δ ⁻¹
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