Matrix - Match TypeColumn IColumn IIA.The minimum value of y=4sec2x+cos2x for permissible real values of x, is equal to1.2B.If m,n are positive integers and m+n2=41+242, then (m+n) is equal to2.3C.The number of solutions of the equation logt(sin3x−sinx)=logtcos2x, where t=(79x−x2−14), is3.4D.Consider an arithmetic sequence of positive integers. If the sum of the first ten terms is equal to the 58th term, then the least possible value of the first term is equal to4.55.7

Question

Matrix - Match TypeColumn IColumn IIA.The minimum value of y=4sec2x+cos2x for permissible real values of x, is equal to1.2B.If m,n are positive integers and m+n2​=41+242​​, then (m+n) is equal to2.3C.The number of solutions of the equation logt​(sin3x−sinx)=logt​cos2x, where t=(79x−x2−14​), is3.4D.Consider an arithmetic sequence of positive integers. If the sum of the first ten terms is equal to the 58th term, then the least possible value of the first term is equal to4.55.7

Filo · Accepted Answer

A) cosx=aby using A.M≥G.M2a24​+a2​≥a24​∗a2​⇒a24​+a2≥4B)41+242​ ​=(42​)2+2.(42​).3+32​=42​+3=>m=3,n=4=>m+n=7C)Base>0⇒x2−9x+14cos2x=0orsinx=21​cos2x=0=>2x=2π ​,23π ​,25π ​,27π ​,29π ​...orx=4π ​,43π ​,45π ​,47π ​,49π ​...sinx=0⇒x=6π ​,65π ​,613π ​,617π ​...The values satisfying  the  range  from  2  to  7  are : 43π ​,45π ​,47π ​,65π ​,613π ​ Hence, there are 5 values.D)Let the first term =a and the common diference =d.210​×(2a+9d)=a+57d⇒10a+45d=a+57d⇒9a=12d⇒a=34d​ Since all are positive integers and we want the least valued=3,a=4Hence, the answer is A−4,B−5,C−4,D−3

Text solutionVerified

Practice questions from similar books

	Column I		Column II
A.	The minimum value of $y = 4 se c^{2} x + co s^{2} x$ for permissible real values of $x$ , is equal to	1.	2
B.	If $m, n$ are positive integers and $m + n 2 =$ $41 + 242$ , then $(m + n)$ is equal to	2.	3
C.	The number of solutions of the equation $l o g_{t} (s in 3 x - s in x)$ $= l o g_{t} cos 2 x$ , where $t = (\frac{9 x - x ^{2} - 14}{7})$ , is	3.	4
D.	Consider an arithmetic sequence of positive integers. If the sum of the first ten terms is equal to the $5 8^{t h}$ term, then the least possible value of the first term is equal to	4.	5
		5.	7