Let Xbe a set with exactly 5 elements and Ybe a set with exactly 7 elements. If αis the number of one-one function from Xto Yand βis the number of onto function from Yto X, then the value of 5!1(β−α)is _____.
Let Obe the origin and let PQR be an arbitrary triangle. The point S is such thatOPO˙Q+ORO˙S=ORO˙P+OQO˙S=OQ.OR+OPO˙SThen the triangle PQ has S as its:circumcentre (b) orthocentre (c) incentre (d) centroid
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
The circle C1:x2+y2=3, with centre at O, intersects the parabola x2=2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2andC3atR2andR3, respectively. Suppose C2andC3 have equal radii 23 and centres at Q2 and Q3 respectively. If Q2 and Q3 lie on the y-axis, then (a)Q2Q3=12(b)R2R3=46(c)area of triangle OR2R3 is 62(d)area of triangle PQ2Q3is=42
The number of real solutions of the equation sin−1(i=1∑∞xi+1−xi=1∑∞(2x)i)=2π−cos−1(i=1∑∞(−2x)i−i=1∑∞(−x)i)lying in the interval (−21,21)is ____. (Here, the inverse trigonometric function =sin−1xand cos−1xassume values in [2π,2π]and [0, π], respectively.)
Let P=⎣⎡1416014001⎦⎤and I be the identity matrix of order 3. If Q=[q()ij] is a matrix, such that P50−Q=I, then q21q31+q32 equals
For any positive integer n, define fn:(0,∞)→Ras fn(x)=j=1∑ntan−1(1+(x+j)(x+j−1)1)for all x∈(0,∞).Here, the inverse trigonometric function tan−1xassumes values in (−2π,2π)˙Then, which of the following statement(s) is (are) TRUE?j=1∑5tan2(fj(0))=55(b) j=1∑10(1+fj′(0))sec2(fj(0))=10(c) For any fixed positive integer n, (lim)x→∞tan(fn(x))=n1(d) For any fixed positive integer n, (lim)x→∞sec2(fn(x))=1