MATHEMATICS-19-11- 11th (PQRS) space

REVIEW TEST-6 Class : XI (PQRS) Time : 100 min Max. Marks : 100 General Remarks: INSTRUCTIONS 1. The question paper contain 20 questions and 24 pages. All questions are compulsory. Please ensure that the Question Paper you have received contains all the QUESTIONS and Pages. If you found some mistake like missing questions or pages then contact immediately to the Invigilator. 2. Each question should be done only in the space provided for it, otherwise the solution will not be checked. 3. Use of Calculator, Log table and Mobile is not permitted. 4. Legibility and clarity in answering the question will be appreciated. 5. Put a cross ( × ) on the rough work done by you. 6. Last three pages are Extra pages. You may use them for any unfinished question(s) mentioning the page number with remark "continued on page " Name Father's Name Class : Batch : B.C. Roll No. Invigilator's Full Name For Office Use ……………………………. Total Marks Obtained………………… Q.No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Marks Q.1 Consider the quadratic polynomial f (x) = x2 – 4ax + 5a2 – 6a. (a) Find the smallest positive integral value of 'a' for which f (x) is positive for every real x. (b) Find the largest distance between the roots of the equation f (x) = 0. [2.5 + 2.5] Q.2(a) Find the greatest value of c such that system of equations x2 + y2 = 25 x + y = c has a real solution. (b) The equations to a pair of opposite sides of a parallelogram are x2 – 7x + 6 = 0 and y2 – 14y + 40 = 0 find the equations to its diagonals. [2.5+2.5] Q.3 Find the equation of the straight line with gradient 2 if it intercepts a chord of length 4 on the circle x2 + y2 – 6x – 10y + 9 = 0. [5] Q.4 The value of the expression, cos3 2x + 3cos 2x cos6 x  sin6 x wherever defined is independent of x. Without allotting a particular value of x, find the value of this constant. [5] Q.5 Find the general solution of the equation sin3x(1 + cot x) + cos3x(1 + tan x) = cos 2x. [5] Q.6 If the third and fourth terms of an arithmetic sequence are increased by 3 and 8 respectively, then the first four terms form a geometric sequence. Find (i) the sum of the first four terms of A.P. (ii) second term of the G.P. [2.5+2.5] Q.7(a) Let x = 1 or x = – 15 satisfies the equation, log (kx2 + wx + f ) = 2. If k, w and f are relatively prime 3 8 positive integers then find the value of k + w + f. (b) The quadratic equation x2 + mx + n = 0 has roots which are twice those of x2 + px + m = 0 and n m, n and p  0. Find the value of p . [2.5+2.5] Q.8 Line x + y = 1 intersects the x and y axes at M and N respectively. If the coordinates of the point P 6 8 lying inside the triangle OMN (where 'O' is origin) are (a, b) such that the areas of the triangle POM, PON and PMN are equal. Find (a) the coordinates of the point P and (b) the radius of the circle escribed opposite to the angle N. [2.5+2.5] Q.9 Starting at the origin, a beam of light hits a mirror (in the form of a line) at the point A(4, 8) and is reflected at the point B(8, 12). Compute the slope of the mirror. [5] Q.10 Find the solution set of inequality, logx+3 (x2  x) < 1. [5] Q.11 If the first 3 consecutive terms of a geometrical progression are the roots of the equation 2x3 – 19x2 + 57x – 54 = 0 find the sum to infinite number of terms of G.P. [5] Q.12 Find the equation to the straight lines joining the origin to the points of intersection of the straight line x + y = 1 and the circle 5(x2 + y2 + bx + ay) = 9ab. Also find the linear relation between a and b so that a b these straight lines may be at right angle. [3+2] Q.13 Let f (x) = | x – 2 | + | x – 4 | – | 2x – 6 |. Find the sum of the largest and smallest values of f (x) if x  [2, 8]. [5] Q.14 If x +1 x + 2 x + 3 x + 2 x + 3 x + 4 x + a x + b x + c = 0 then all lines represented by ax + by + c = 0 pass through a fixed point. Find the coordinates of that fixed point. [5] Q.15 If S1, S2, S3, ... Sn, .... are the sums of infinite geometric series whose first terms are 1, 2, 3, ... n, ... and 1 1 whose common ratios are 1 1 , ...., 2n1 , ... respectively, then find the value of 2 . [5] 2 , 3 , 4 n +1 Sr r=1 A Q.16 In any triangle if tan 2 5 B = 6 and tan 2 20 = 37 then find the value of tan C. [5] Q.17 The radii r1, r2, r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides. [5] Q.18 Find the equation of a circle passing through the origin if the line pair, xy – 3x + 2y – 6 = 0 is orthogonal to it. If this circle is orthogonal to the circle x2 + y2 – kx + 2ky – 8 = 0 then find the value of k. [5] Q.19 Find the locus of the centres of the circles which bisects the circumference of the circles x2 + y2 = 4 and x2 + y2 – 2x + 6y + 1 = 0. [5] Q.20 Find the equation of the circle whose radius is 3 and which touches the circle x2 + y2 – 4x – 6y – 12=0 internally at the point (–1, – 1). [5] EXTRA PAGE EXTRA PAGE EXTRA PAGE