1) Show that (5+ √5)/√(5+ 3√5)= ⁴√20.
2) If α, β be the roots of the equation ax²+ bx + c = 0 and γ, δ those of the equation px¹+ qx +r=0, show that, ac/pr = b²/q², if αδ = βγ..
3) If n positive integer greater than unity, then prove 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 reciprocal of the terms , then show that 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).
6) If 9α=π, find the value of sinα sin2α sin3α sin4α.
7) If tanx = (tanα - tanβ)(1- tanα tanβ), then show that
Sin2x= (sin2α - sin2β)/(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: cos³x cos3x + sin³x sin3x = 1/8.
10) If m(tan(α- β)/cos²β = ntanβ/cos²(α- β), show that
β = (1/2) [α - tan⁻¹{(n - m)/(n + m)} tanα].
11) If lx + my = 0 be the perpendicular bisector the statement joining the points (a,b) and (c,d), then show that
(c - a)/I = (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 of the parabola whose focus is (5,3) and vertex is (3,1).
14) If α and β be the eccentric angles of the extremities of a focal chord of the ellipse x²/a² + y²/b² = 1.
prove that, tan(α/2) tan(β/2) = (e -1)/(e +1) or (e +1)/(e -1(.
15) The base of the right prism is a regular hexagon . If the height and the area of the whole surface of the prism be respectively 8√3 ft. and 576√3 square.ft., find the volume of the prism .
16) Find dy/dx when y= ₓx² + ₐx².
17) Evaluate: lim ₓ→₀ (tan2x - x)/(3x - sin x).
18) If y= xⁿ {a cos(logx) + b sin(logx)}, show that
x² d²y/dx² + (1+ 2n) x dy/dx + (1+ n²)y=0.
19) Show that the equation eˢᶦⁿˣ - w⁻ˢⁱⁿˣ = 4 has no real solution.
20) Find derivative of x² cosx.
21) ᵗᵃⁿˣ₁/ₑ∫ t dt/(1+ t²) + ᶜᵒᵗˣ₁/ₑ∫ dt/{t(1+ t²)= 1.
22) Solve: (x + y)¹ dy/dx = 2x + 2y +5.
23) Evaluate:
lim ₓ→∞ (1/n) [sec²(π/4n) + sec²(2π/4n +.....+ Sec²(nπ/4n)]
24) ∫ (cosx- sinx)(2+ 2 sin2x)/(cosx + sinx) dx.
25) Solve: d²y/dx² - 2a dy/dx + a²y = 0, given y= a and dy/dx = 0 when x = 0.
26) Shade the region above the x-axis, included between parabola y²= 4x and the circle (x -4)= 4 cosθ, y= 4 sinθ. Find the area of the region by integration.
27) Show that the maximum value of the function x + 1/x is less than its minimum value.
28) A ball projected vertically upwards is at a height h ft. from the point of projection after t seconds, where h= 32t - 16t².
a) What is the maximum height reached ?
b) What is the velocity of the ball at a height 12 ft above the ground ?
c) What is the velocity of projection ?
29) 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.
30) Show that log(1+x)> (tan⁻¹x)/(1+ x) for all x > 0.
31) Show that: ³√(2+ √5) + ³√(2- √5)= 1.
32) Determine the sign of the expression
(x -1)(x -2)(x -3)(x - 4)+ 5 for real values of x.
33) If cotx = 2 and cot y = 3, then find (x + y).
34) Find when the solution of the equation a cos x + b sin x = c is possible.
35) Find square root of 4ab - 2i(a²- b²).
36) If θ{x}= (x -1) eˣ +1, show that θ{x} is positive for all the values of x > 0.
37) If y= f{x}= (x +1)/(x +2), Show that, f(y)= (2x +3)/(3x +5).
38) Evaluate ¹₋₁∫ sin³x cos²x dx.
39) is it possible to draw a tangent from the point (-2,-1) to the circle x½+ y²- 4x + 6y - 12=0? give reasons.
40) If f(x)= tan(x - π/4), find f(x) . f(-x).
TEST PAPER - 2
So that so that be to win in the expansion the term end dependent the find the value of k the equation have a common root proof that other roots will be satisfy the equation prove that prove that solve so that in a triangle prove that the equation of the god of the circle prove that the equation all the circle on this 51@gyan Dadar to bhatija live on the find the remaining vertices is a variable point in the hyperbola and the fixed point so that the locus of the midpoint of the line segment is another hyperbola given the lips find the equation of the chord which is bisector at 21 the volume on the lateral surface would like to is in equilateral triangle respectively 60 180 find the height of the prisoner function is depend as follows draw the graph and a discuss the continuity does the limit exist when explain evaluate so that valuable to valuate so that's all find the definition the value of find the area bounded by the per up of succession the straight line Prove the normal chord of the parabola which is at the point subtend the right angles of the vertex A particle is moving in a straight line subject to a resistance they are being know other force on it is the resistance produces retardation where is the velocity of the particle is a constant so that the velocity of the particle is reduced to have estimation after transverse in the distance the centric angle of 2 points on the ellipse of the tangent of this points intersect prove that prove that HTML given and the extremism maximum are minimum according as which turn of the following two series equal to 7120197 93 the roots of are always real image positive and 6340 value find the minimum value the centre of the circle 34 in the length of the tangent drawn from 22 to the circle find the radius of the circle
If the roots equation so that has proved that an engine without wagon can go 24 miles and hour and speed is diminished by a point quantities varies of square root of the number of wagon attached with four guys and speed is 20 miles and hour find the greatest number to that find the number that different combination permutation that can be made out of the letters taken 3 at the time the rain such that a real solution show that the locus of the food of the perpendicular drop from the original on the line passing through a fixed in the circle the length of the chord of the circle interceptor on the straight line so that is a double ordinate of the parabola find the locus of its point of price and given the point find the locus of the point then sub the custom of pyramid of triangle find the first principle with prove so that the function integrate by the method of any valuate
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