2) If n be positive integer greater than unity, then show 49ⁿ -16n -1 is divisible by 64.
3) 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.
4) 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).
5) If 9x =π, find the value of sinx sin2x sin3x sin4x.
6) If tanx= (tana - tanb)/(1- tana tanb), then show sin2x= (sin2a - sin2b)/(1- sin2a sin2b).
7) Solve: cos³x cos3x+ sin³x sin3x =1/8.
8) If {m tan(x-y)}/cos²y = (n tany)/cos²(x - y), then show that y= 1/2 {x - tan⁻¹{(n - m)/(n+ m). Tanx}
9) If lx+ my=0 be the perpendicular bisector of the segment joining the points (a,b) and (c,d), then show that (c- a)/l = (d - b)/m = 2(la+ mb)/(l²+ m²).
10) 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.
11) Find the equation of latus rectum of the parabola whose focus is (5,3) and vertex is (3,1).
12) If m, n be the eccentric angles of the extremities of a focal chord of the ellipse x²/a² + y²/b² =1.
Prove that, tan(m/2) tan(n/2)= (e-1)/(e+1) or (e+1)/(e -1).
13) Find dy/dx: y= ₓx² + ₐx².
14) Evaluate: lim ₓ→₀ (tan2x - x)/(3x - sinx).
15) If y= xⁿ{a cos(logx)+ bsin(logx)}, show that x² d²y/dx² + (1- 2n) x dy/dx + (1+ n²)y=0.
16) Show that the equation ₑsinx - ₑ-sinx =4 has no real solution.
17) Find dy/dx of x² cosx.
18) ∫ tdt/(1+t²) at (tanx, 1/e) + ∫dt/t(1+ t²) at (cotx, 1/e) = 1.
19) Solve: (x+ y)² dy/dx = 2x +2y+5.
20) Evaluate:
lim ₓ→∞ 1/n[sec²(π/4n)+ sec²(2π/4n)+ ......sec²(nπ/4n].
21) ∫ (cosx - sinx)(2+ 2sin2x)/(cosx + sinx) dx.
22) Solve: d²y/dx² - 2a dy/dx + a²y=0, given y= a and dy/dx=0 when x=0.
23) Shade the region above the x-axis, included between the parabola y²= 4x and the circle (x -4)=4 cosk, y= 4sink. Find the area of the region by integration.
24) Show that the maximum value of the function x+ 1/x is less than its minimum value.
25) 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.
26) Determine the sign of the expression (x-1)(x-2)(x-3)(x-4)+5 for real values of x.
27) If cotx = 2 and coty=3, then find (x+ y).
28) If y= f(x)= (x+1)/(x+2), show that, f(y)= (2x+C)/(3x+5).
29) ¹₋₁∫sin³x cos²x dx.
30) Is it possible to draw a tangent from the point (-2,-1) to the circle x²+ y²- 4x (6y -12=0 ? Give reasons.
31) If f(x)=tan(x - π/4), find f(x) . f(- x).