MATHEMATICS-25-02- 11th (PQRS & J) Space

FINAL TEST Class : XI Time : 90 min. Max. Marks : 90 General Remarks: INSTRUCTIONS 1. The question paper contain 15 questions and 22 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. Write your BC roll no. on top right corner of page #3 of answer sheet also. 3. Each question should be done only in the space provided for it, otherwise the solution will not be checked. 4. Use of Calculator, Log table and Mobile is not permitted. 5. Legibility and clarity in answering the question will be appreciated. 6. Put a cross ( × ) on the rough work done by you. 7. You can tear off the LAST PAGE carefully along the scissor marked line and take it as Final Test. 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 PART-A Q.1 Three straight lines l1, l2 and l3 have slopes 1/2, 1/3 and 1/4 respectively. All three lines have the same y-intercept. If the sum of the x-intercept of three lines is 36 then find the y-intercept. [5] BC Roll. No. Q.2 Find the general solution of the equation 1 sin x  sin 2 x  sin 3 x  ...... sin n x   1 sin x  sin 2 x  sin 3 x ..(1)n sin n x   4 = 1 tan2 x  where x  k + 2 , k  I. [5] B C 1  b  c sin A Q.3 In a triangle ABC if 2 cos 2 cos 2 = +   2  a  2 then find the measure of angle A. [5] Q.4 Let p & q be the two roots of the equation, mx2 + x(2  m) + 3 = 0. Let m1, m2 be the two values of m satisfying p  q = 2 . Determine the numerical value of m1  m2 . [5] q p 3 2 2 Q.5 If Cr denotes the combinatorial co-efficient in the expansion of (1 + x)n, n  N then using algebraic approach prove that C + C1 0 2 + C2 3 + ........ + Cn n 1 = 2n1 1 n 1 . [5] Q.6 Let 'A' denotes the real part of the complex number z = 19  7i 9  i + 20  5i 7  6i and 'B' denotes the sum of the imaginary parts of the roots of the equation z2 – 8(1 – i)z + 63 – 16i = 0 and 'C' denotes the sum of the series, 1 + i + i2 + i3 + + i2008 where i = 1 . and 'D' denotes the value of the product (1 + )(1 + 2)(1 + 4)(1 + 8) where  is the imaginary cube root of unity. Find the value of A  B . [5] C  D PART-B Q.7 (a) If  and  are the roots of the equation x2 + 5x – 49 = 0 then find the value of cot(cot–1 + cot–1). (b) Prove that the sum to 'n' terms of the series, 1 1 1 1 1 1 1 1  n  2  –1 tan 3 + tan 7 + tan 13 + tan 21 +...... = cot  n  [3+3] Q.8 In acute angled triangle ABC, a semicircle with radius ra is constructed with its base on BC and tangent to the other two sides. rb and rc are defined similarly. If r is the radius of the incircle of triangle ABC then prove that 2 = 1  1  1 [6] r ra rb rc Q.9 Find the equation of the circle passing through the point (–6 , 0) if the power of the point (1, 1) w.r.t. the circle is 5 and it cuts the circle x2 + y2 – 4x – 6y – 3 = 0 orthogonally. [6] Q.10 Ten dogs encounter 8 biscuits. Dogs do not share biscuits. In how many different ways can the biscuits be consumed (a) if we assume that the dogs are distinguishable, but the biscuits are not. (b) if we assume that both dogs and biscuits are different and any dog can receive any number of biscuits. (c) if dogs and biscuits are different and every dog can get atmost one biscuit. [2+2+2] Q.11 Given 35 sin 5k = tan m  , where angles are measured in degrees, and m and n are relatively prime n k1   positive integers that satisfy m < 90, find the value of (m + n). [6] n PART-C Q.12 A point moving around circle (x + 4)2 + (y + 2)2 = 25 with centre C broke away fromit either at the point A or point B on the circle and moved along a tangent to the circle passing through the point D (3, – 3). Find the following. (i) Equation of the tangents at Aand B. (ii) Coordinates of the points A and B. (iii) Equation of the circle circumscribing the DAB and also the intercepts made by this circle on the coordinate axes. [2.5 + 2.5 + 2.5]  k  Q.13 Let f (n) =  r  . Find the total number of divisors of f (9). [7.5] r0 kr Q.14 Given below is a partial graph of an even periodic function f whose period is 8. If [*] denotes greatest integer function then find the value of the expression.  f  7  f (–3) + 2 | f (–1) | +   8  + f (0) + arc cos  f (2) + f (–7) + f (20) [7.5]    Q.15 Asquare ABCD lying in I-quadrant has area 36 sq. units and is such that its side AB is parallel to x-axis. Vertices A, B and C are on the graph of y = logax, y = 2 logax and y = 3 logax respectively then find the value of 'a'. [7.5] MATHEMATICS FINAL TEST PART-A Q.1 Three straight lines l1, l2 and l3 have slopes 1/2, 1/3 and 1/4 respectively. All three lines have the same y-intercept. If the sum of the x-intercept of three lines is 36 then find the y-intercept. [5] Q.2 Find the general solution of the equation 1 sin x  sin 2 x  sin 3 x  ...... sin n x   1 sin x  sin 2 x  sin 3 x ..(1)n sin n x   4 = 1 tan2 x  where x  k + 2 , k  I. [5] B C 1  b  c sin A Q.3 In a triangle ABC if 2 cos 2 cos 2 = +   2  a  2 then find the measure of angle A. [5] Q.4 Let p & q be the two roots of the equation, mx2 + x(2  m) + 3 = 0. Let m1, m2 be the two values of m satisfying p  q 2 m1  m2 . [5] q p = 3 . Determine the numerical value of m2 2 Q.5 If Cr denotes the combinatorial co-efficient in the expansion of (1 + x)n, n  N then using algebraic approach prove that C + C1 0 2 + C2 3 + ........ + Cn n 1 = 2n1 1 n 1 . [5] Q.6 Let 'A' denotes the real part of the complex number z = 19  7i 9  i + 20  5i 7  6i and 'B' denotes the sum of the imaginary parts of the roots of the equation z2 – 8(1 – i)z + 63 – 16i = 0 and 'C' denotes the sum of the series, 1 + i + i2 + i3 + + i2008 where i = 1 . and 'D' denotes the value of the product (1 + )(1 + 2)(1 + 4)(1 + 8) where  is the imaginary cube root of unity. Find the value of A  B . [5] C  D PART-B Q.7 (a) If  and  are the roots of the equation x2 + 5x – 49 = 0 then find the value of cot(cot–1 + cot–1). (b) Prove that the sum to 'n' terms of the series, 1 1 1 1 1 1 1 1  n  2  –1 tan 3 + tan 7 + tan 13 + tan 21 +...... = cot  n  [3+3] Q.8 In acute angled triangle ABC, a semicircle with radius ra is constructed with its base on BC and tangent to the other two sides. rb and rc are defined similarly. If r is the radius of the incircle of triangle ABC then prove that 2 = 1  1  1 [6] r ra rb rc Q.9 Find the equation of the circle passing through the point (–6 , 0) if the power of the point (1, 1) w.r.t. the circle is 5 and it cuts the circle x2 + y2 – 4x – 6y – 3 = 0 orthogonally. [6] Q.10 Ten dogs encounter 8 biscuits. Dogs do not share biscuits. In how many different ways can the biscuits be consumed (a) if we assume that the dogs are distinguishable, but the biscuits are not. (b) if we assume that both dogs and biscuits are different and any dog can receive any number of biscuits. (c) if dogs and biscuits are different and every dog can get atmost one biscuit. [2+2+2] Q.11 Given 35 sin 5k = tan m  , where angles are measured in degrees, and m and n are relatively prime n k1   positive integers that satisfy m < 90, find the value of (m + n). [6] n PART-C Q.12 A point moving around circle (x + 4)2 + (y + 2)2 = 25 with centre C broke away fromit either at the point A or point B on the circle and moved along a tangent to the circle passing through the point D (3, – 3). Find the following. (i) Equation of the tangents at Aand B. (ii) Coordinates of the points A and B. (iii) Equation of the circle circumscribing the DAB and also the intercepts made by this circle on the coordinate axes. [2.5 + 2.5 + 2.5]  k  Q.13 Let f (n) =  r  . Find the total number of divisors of f (9). [7.5] r0 kr Q.14 Given below is a partial graph of an even periodic function f whose period is 8. If [*] denotes greatest integer function then find the value of the expression.  f  7  f (–3) + 2 | f (–1) | +   8  + f (0) + arc cos  f (2) + f (–7) + f (20) [7.5]    Q.15 Asquare ABCD lying in I-quadrant has area 36 sq. units and is such that its side AB is parallel to x-axis. Vertices A, B and C are on the graph of y = logax, y = 2 logax and y = 3 logax respectively then find the value of 'a'. [7.5]