The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tabk is at the same level as the to of the cylinder as shown in the figure. The density of the water is ρ. In equilibrium, the height H of the water column in the cylinder satisfies.
Three randomly chosen nonnegative integers x,yandzare found to satisfy the equation x+y+z=10.Then the probability that zis even, is:125 (b) 21 (c) 116 (d) 5536
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover cards numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isa.264 b. 265 c. 53 d. 67
Let `f(x)``=x+log_ex-xlog_ex ,x(0,oo)dot`Column 1 contains information about zeros of `f^(prime)(x)f^(prime)(x)a n df^(x)dot`Column 2 contains information about the limiting behaviour of `f^(prime)(x)f^(prime)(x)a n df^(x)`at infinity.Column 2 contains information about the increasing/decreasing nature of `f(x)a n df^(prime)(x)dot`Column I, Column 2, Column 3I, `f(x)=0forsom ex(l , e^2)`, (i), `("lim")_("x"vecoo"")f^(prime)(x)=0`, (P), `f`is increasing in (0,1)II, `f'(x)=0forsom ex(l , e)`, (ii), `("lim")_("x"vecoo"")f^(x)=-oo`, (Q), `f`is decreasing in `(e ,e^2)`III, `f'(x)=0forsom ex(0,1)`, , `("lim")_("x"vecoo"")f^(prime)(x)=-oo`, (R), `f`is increasing in (0,1)IV, `f^(' '(x))=0forsom ex(1, e)`, , `("lim")_("x"vecoo"")f^prime^'(x)=0`, (S), `f`is decreasing in (`e , e^2`)Which of the following options is the only CORRECT combination?(I) (ii) (P) (b) (IV) (iv) (S) (III) (iii) (R) (d) (II) (ii) (Q)Which of the following option is the only incorrect combination?(III) (i) (R) (b) (I) (iii) (P)(II) (iii) (P) (d) (II) (iv) (Q)Which of the following options is the only CORRECT combination?(I) (ii) (R) (b) (II) (iii) (S)(III) (iv) (P) (d) (IV) (i) (S)
Which of the following is (are) NOT the square of a 3×3 matrix with real entries? (a)⎣⎡10001000−1⎦⎤ (b) ⎣⎡−1000−1000−1⎦⎤ (c)⎣⎡100010001⎦⎤ (d) ⎣⎡1000−1000−1⎦⎤
Let f:(0,∞)R be given by f(x)=∫x1xte−(t+t1)dt, then (a)f(x) is monotonically increasing on [1,∞)(b)f(x) is monotonically decreasing on (0,1)(c)f(2x) is an odd function of x on R
A line L : y = mx + 3 meets y-axis at E (0, 3) and the arc of the parabola y2=16x 0≤y≤6 at the point art F(x0,y0). The tangent to the parabola at F(X0,Y0) intersects the y-axis at G(0,y). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum P) m= Q) = Maximum area of △EFG is (R) y0= (S) y1=
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8:15is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60