Report

STUDY INNOVATIONSFollow

Mar. 28, 2023•0 likes## 0 likes

•14 views## views

Be the first to like this

Show More

Total views

0

On Slideshare

0

From embeds

0

Number of embeds

0

Mar. 28, 2023•0 likes## 0 likes

•14 views## views

Be the first to like this

Show More

Total views

0

On Slideshare

0

From embeds

0

Number of embeds

0

Download to read offline

Report

Education

2log21/ 4 x 3log27 (x2 +1)3 2x dy Q.1 Let y = 74 log49 x – x 1 and dx = ax + b, find the value of a and b. [4] Q.2 Show that cos2A + cos2(A + B) + 2 cosA cos(180° + B) · cos(360° + A + B) is independent of A. Hence find its value when B = 810°. [4] Q.3 Find the product of the roots of the equation, | x2 | + | x | – 6 = 0. [4] Q.4 One root of mx2 – 10x + 3 = 0 is two third of the other root. Find the sum of the roots. [4] Q.5 Suppose x and y are real numbers such that tan x + tan y = 42 and cot x + cot y = 49. Find the value of tan(x + y). [4] Q.6 Find the solution set of k so that y = kx is secant to the curve y = x2 + k. [4] Q.7 A quadratic polynomial p(x) has 1 + and 1 – as roots and it satisfies p(1) = 2. Find the quadratic polynomial. [4] Q.8 Solve the equation 0.5log x (x2 x) log 4 3 . [4] 3 5 Q.9 Find the sum of the series, cos 2n +1 + cos 2n +1 + cos 2n +1 + upto n terms. Do not use any direct formula of summation. [5] Q.10 Find the minimum and maximum value of f (x, y) = 7x2 + 4xy + 3y2 subjected to x2 + y2 = 1. [5] Q.11 Find the minimum & maximum value of (sin x – cos x – 1) (sin x + cos x – 1) x R. [5] Q.12 Given that log2a = s, log4b = s2 and logc2 (8) = 2 s3 +1 a 2b5 . Write log2 c4 as a function of 's' (a, b, c > 0, c 1). [5] x2 2x 8 Q.13 Find the range of the expression y = x2 4x 5 , for all permissible value of x. [5] Q.14 Find whether a triangle ABC can exists with the tangents of its interior angle satisfying, tan A= x, tan B = x + 1 and tan C = 1 – x for some real value of x. Justify your assertion with adequate reasoning. [6] Q.15 Solve the equation, 5 sin x + 5 – 5 = 2 sin2x + 2 sin x 1 2 sin2 x if x (0, ). [6] Q.16 Find the value of x, y, z satisfying the equations log2x + log4y + log4z = 2 log9x + log3y + log9z = 2 and log16x + log16y + log4z = 2. [6] x + 9 Q.1 Integrate: x3 + 9x dx DPP-30 TIME : 45 Min. Q.2 Find the domain of definition of the function, f (x) = log4 log3 (log2 (x2 2x + 3) 1+x2 log 2 (2x 1)). Q.3 Integrate : 2+x2 dx . 2 Q.4 Evaluate : 0 dx 2 + cos 2x . Q.5 Examine the function f (x) = Limit n x 1+ (4 sin2 x) n for continuity in [0, ]. Plot its graph and state the nature of discontinuity and jump of discontinuity if applicable. 2 1 2 Q.6 Evaluate : 0 dx . Q.1 Integrate: (p3 + 6p) sin p dp DPP-31 TIME : 45 Min. 1 Q.2 Find the range of the function f (x) = sin–1 x2 + [{ln x [x]}] + cot–1 1 + 2 x2 Where {*} & [*] are fractional part function & greatest integer function respectively. 2 x x Q.3 Evaluate : e 2 sin + dx . Q.4 Integrate: dx 0 2 4 x1 4 + 5 Q.5 Integrate : x 16 dx , Q.6 Let f : (0, ) 2 be defined as, f (x) = arc tan(ln x) (a) Prove that f is invertible, (b) If g is the inverse of f, find g' ( 4) e (c) Sketch the graph of f (x), (

STUDY INNOVATIONSFollow

- Q.1 Let y = 1 x 7 x 2 3 2 x log 4 ) 1 x ( log x log 49 3 2 27 4 / 1 2 and dx dy = ax + b, find the value of a and b. [4] Q.2 Show that cos2A + cos2(A + B) + 2 cosA cos(180° + B) · cos(360° + A + B) is independent of A. Hence findits value when B= 810°. [4] Q.3 Find the product of the roots of the equation, | x2 | + | x | – 6 = 0. [4] Q.4 One root of mx2 – 10x + 3 = 0 is two third of the other root. Find the sum of the roots. [4] Q.5 Suppose x and yare real numbers such that tanx + tan y= 42 andcot x + cot y= 49. Find the value of tan(x + y). [4] Q.6 Find the solution set of k so that y= kx is secant to the curve y = x2 + k. [4] Q.7 A quadratic polynomial p(x) has 1 + 5 and 1 – 5 as roots and it satisfies p(1) = 2. Find the quadraticpolynomial. [4] Q.8 Solve theequation 4 log ) x x ( log 5 . 0 9 2 x 3 x . [4] Q.9 Find the sum of the series, cos 1 n 2 + cos 1 n 2 3 + cos 1 n 2 5 + ........ upto n terms. Donotuseanydirect formulaofsummation. [5] Q.10 Find the minimum and maximum value of f (x, y) = 7x2 + 4xy + 3y2 subjected to x2 + y2 = 1. [5] Q.11 Find the minimum & maximum value of (sin x – cos x – 1) (sin x + cos x –1) x R. [5] Q.12 Given that log2a = s, log4b = s2 and ) 8 ( log 2 c = 1 s 2 3 . Write log2 4 5 2 c b a as a function of 's' (a, b, c > 0, c 1). [5] Q.13 Find the range of the expression y= 5 x 4 x 8 x 2 x 2 2 , for all permissible valueof x. [5] Q.14 FindwhetheratriangleABC canexists with thetangentsofits interior anglesatisfying, tanA=x, tan B = x + 1 and tan C = 1 – x for some real value of x. Justifyyour assertion with adequate reasoning. [6] Q.15 Solve the equation, 5 sin x + x sin 2 5 – 5 = 2 sin2x + x sin 2 1 2 if x (0, ). [6] Q.16 Findthevalueofx,y,zsatisfyingtheequations log2x + log4y+ log4z = 2 log9x + log3y+ log9z = 2 and log16x + log16y+ log4z = 2. [6] CLASS : XII (ALL) TIME : 60 Min. DPP. NO.-29
- DPP-30 TIME : 45 Min. Q.1 Integrate: dx x 9 x 9 x 3 Q.2 Findthedomainofdefinitionofthefunction, f(x) = log4 log3 log log ( ) ( ) 2 2 2 2 3 2 1 x x x . Q.3 Integrate : dx x 2 x 1 2 2 . Q.4 Evaluate: 2 0 x 2 cos 2 dx . Q.5 Examinethefunction f(x)= n 2 n x sin 4 1 x Limit forcontinuityin[0,].Plotits graphandstatethe natureofdiscontinuityandjumpofdiscontinuityifapplicable. Q.6 Evaluate : 2 1 2 0 2 x x ) 1 x 2 ( dx . DPP-31 TIME : 45 Min. Q.1 Integrate: dp p sin ) p 6 p ( 3 Q.2 Find the range of the function f (x) = sin–1 x2 + ] [ ] x [ x n l + cot–1 1 1 2 2 x Where {*} &[*] arefractionalpart function& greatest integerfunction respectively. Q.3 Evaluate: 2 0 2 x dx 4 2 x sin e . Q.4 Integrate: dx x 4 2 Q.5 Integrate : dx 16 x 5 x 4 1 Q.6 Let f : (0, ) 2 , 2 be defined as, f (x) = arc tan(ln x) (a) Prove that f is invertible, (b) If g is the inverse of f, find g' 4 (c) Sketch the graph of f (x), (d) evaluate e 1 dx x ) x ( f . CLASS : XII (ALL) DPP. NO.-30, 31
- Q.1 If (sin x + cos x)2 + k sin x cos x = 1 holds x R then find the value of k. [3] Q.2 Iftheexpression cos 2 3 x + sin x 2 3 + sin (32 + x) – 18 cos(19 – x) + cos(56 + x) – 9 sin(x + 17) is expressed in the form of a sin x + b cos x find the value of a + b. [3] Q.3 3 statements are given below each of which is either True or False. State whether True or False with appropriatereasoning.Marks will beallottedonlyifappropriate reasoningis given. I (log3169)(log13243) = 10 II cos(cos ) = cos (cos 0°) III cos x + x cos 1 = 2 3 [3] Q.4 Prove the identity cos4t = 8 3 + 2 1 cos 2t + 8 1 cos 4t. [3] Q.5 Suppose that for some angles x andythe equations sin2x + cos2y = 2 a 3 and cos2x + sin2y = 2 a2 holdsimultaneously.Determinethepossiblevaluesofa. [3] Q.6 Find the sum ofall the solutions of the equation (log27x3)2 = log27x6. [3] Q.7 If – 2 < x < 2 and y= log10(tan x + sec x). Then the expression E = 2 10 10 y y simplifies to one of thesixtrigonometricfunctions.findthetrigonometricfunction. [3] Q.8 If ) x (log log log 2 2 2 = 2 then find the number of digits in x.You mayuse log102 = 0.3010. [3] Q.9 Assumingthatx andyareboth+ve satisfyingtheequation log(x +y)=logx +log yfind yin terms of x. Base of the logarithm is 10 everywhere. [3] Q.10 If x = 7.5° then find the value of x sin x 3 sin x 3 cos x cos . [3] Q.11 Find the solutions of the equation, ) x cos 1 ( log x sin 2 = 2 in the interval x [0, 2]. [4] CLASS : XII (ALL) TIME : 45 Min. DPP. NO.-32
- Q.12 Given that ) 1 a ( log 2 a2 = 16 find the value of ) a 1 a ( log 32 a . [4] Q.13 If cos = 5 4 find the values of (i) cos 3 (ii)tan 2 [4] Q.14 If log1227 =a find the value of log616 in term of a. [5] Q.15 Provetheidentity, 1 x cos x sin 1 x cos x sin = x cos x sin 1 =tan 2 x 4 ,whereveritisdefined.Startingwith left handsideonly. [5] Q.16 Find the exact value of cos 24° – cos 12° + cos 48° – cos 84°. [5] Q.17 Solve the system of equations 5(logxy+ logyx) = 26 and xy = 64. [6] Q.18 Prove that 4 r 1 r 4 8 ) 1 r 2 ( sin = 4 r 1 r 4 8 ) 1 r 2 ( cos . Alsofindtheirexactnumericalvalue. [6] Q.19 Solve for x: log2 (4x) + log (4x) . log 2 1 x 2 log2 2 1 x = 0. [6]
- CLASS : XII (ABCD) DPP. NO.-33 DPP OF THE WEEK . NOTE: Leave Star () marked problems. PART -A Select the correct alternative. (Only one is correct) [26 × 3 = 78] Q.1 Number of zeros of the cubic f (x) = x3 + 2x + k k R, is (A) 0 (B) 1 (C) 2 (D) 3 Q.2 The value of x 3 3 x dr ) 1 r )( 1 r ( r dx d Lim , is (A) 0 (B) 1 (C) 1/2 (D) non existent Q.3 There are two numbers x making the value of the determinant x 2 4 0 1 x 2 5 2 1 equal to 86. The sum of these two numbers, is (A) – 4 (B) 5 (C) – 3 (D) 9 Q.4 Afunction f (x) takes a domain D onto a range R if for each y R, there is some x D for which f (x) = y. Number of function that can be defined from the domain D = {1, 2, 3} onto the range R = {4, 5} is (A) 5 (B) 6 (C) 7 (D) 8 Q.5 Suppose f , f ' and f '' are continuous on [0, e] and that f ' (e) = f (e) = f (1) = 1 and e 1 2 dx x ) x ( f = 2 1 , then the value of dx x n ) x ( ' ' e 1 l f equals (A) e 1 2 5 (B) e 1 2 3 (C) e 1 2 1 (D) e 1 1 Q.6 A circle with centre C (1, 1) passes through the origin and intersect the x-axis at A and y-axis at B. The area of the part of the circle that lies in the first quadrant is (A) + 2 (B) 2 – 1 (C) 2 – 2 (D) + 1 *Q.7 The planes 2x – 3y + z = 4 and x + 2y – 5z = 11 intersect in a line L. Then a vector parallel to L, is (A) k̂ 7 j ˆ 11 î 13 (B) k̂ 7 j ˆ 11 î 13 (C) k̂ 7 j ˆ 11 î 13 (D) k̂ 5 j ˆ 2 î *Q.8 A fair dice is thrown 3 times. The probability that the product of the three outcomes is a prime number, is (A) 1/24 (B) 1/36 (C) 1/32 (D) 1/8 Q.9 Period of the function, f (x) = [x] + [2x] + [3x] + ....... + [nx] – 2 ) 1 n ( n x where n N and [ ] denotes the greatest integer function, is (A) 1 (B) n (C) 1/n (D) non periodic Q.10 Let Z be a complex number given by, Z = i 1 10 1 i 3 1 i i 2 the statement which does not hold good, is (A) Z is purely real (B) Z is purely imaginary (C) Z is not imaginary (D) Z is complex with sum of its real and imaginary part equals to 10 Q.11 Let f (x, y) = xy2 if x and y satisfy x2 + y2 = 9 then the minimum value of f (x, y) is (A) 0 (B) – 3 3 (C) – 3 6 (D) – 6 3
- Q.12 x 101 1 ) x 1 ( x 1 x 3 1 Lim 101 3 0 x has the value equal to (A) – 5050 3 (B) – 5050 1 (C) 5051 1 (D) 4950 1 Q.13 Number of positive solution which satisfy the equation log2x · log4x · log6x = log2x · log4x + log2x · log6x + log4x · log6x? (A) 0 (B) 1 (C) 2 (D) infinite Q.14 Number of real solution of equation 16 sin–1x tan–1x cosec–1x = 3 is/are (A) 0 (B) 1 (C) 2 (D) infinite Q.15 Length of the perpendicular from the centre of the ellipse 27x2 + 9y2 = 243 on a tangent drawn to it which makes equal intercepts on the coordinates axes is (A) 3/2 (B) 2 3 (C) 2 3 (D) 6 Q.16 Let f (x) = cos–1 2 2 x 1 x 1 + tan–1 2 x 1 x 2 where x (–1, 0) then f simplifies to (A) 0 (B) /4 (C) /2 (D) *Q.17Aperson throws four standard six sided distinguishable dice. Number of ways in which he can throw if the product of the four number shown on the upper faces is 144, is (A) 24 (B) 36 (C) 42 (D) 48 Q.18 Let A = z y x r q p c b a and suppose that det.(A) = 2 then the det.(B) equals, where B = r c 2 z 4 q b 2 y 4 p a 2 x 4 (A) det(B) = – 2 (B) det(B) = – 8 (C) det(B) = – 16 (D) det(B) = 8 Q.19 The digit at the unit place of the number (2003)2003 is (A) 1 (B) 3 (C) 7 (D) 9 Q.20 Let ABCDEFGHIJKL be a regular dodecagon, then the value of AF AB + AB AF is (A) 4 (B) 3 2 (C) 2 2 (D) 2 *Q.21 Urn A contains 9 red balls and 11 white balls. Urn B contains 12 red balls and 3 white balls. One is to roll a single fair die. If the result is a one or a two, then one is to randomly select a ball from urnA. Otherwise one is to randomly select a ball form urn B. The probability of obtaining a red ball, is (A) 41/60 (B) 19/60 (C) 21/35 (D) 35/60 Q.22 Let f be a real valued function of real and positive argument such that f (x) + 3x f x 1 = 2(x + 1) for all real x > 0. The value of f (10099) is (A) 550 (B) 505 (C) 5050 (D) 10010 Q.23 If and be the roots of the equation x2 + 3x + 1 = 0 then the value of 2 2 1 1 is equal to (A) 15 (B) 18 (C) 21 (D) none Q.24 The equation (x – 1)(x – 2)(x – 3) = 24 has the real root equal to 'a' and the complex roots b and c. Then the value of a bc , is (A) 1/5 (B) – 1/5 (C) 6/5 (D) – 6/5 Q.25 If m and n are positive integers satisfying 1 + cos 2 + cos 4 + cos 6 + cos 8 + cos 10 = sin n sin · m cos then m + n is equal to (A) 9 (B) 10 (C) 11 (D) 12
- Q.26 Acircleofradius320unitsistangenttotheinsideofacircleofradius1000.Thesmallercircleistangenttoadiameter of the larger circle at the point P. Least distance of the point Pfrom the circumference of the larger circle is (A) 300 (B) 360 (C) 400 (D) 420 Select the correct alternative. (More than one are correct) [8 × 4 = 32] Q.27 In which of the following cases limit exists at the indicated points. (A) f (x) = x ] | x | x [ at x = 0 (B) f (x) = x 1 x 1 e 1 e x at x = 0 where [x] denotes the greatest integer functions. (C) f (x) = (x – 3)1/5 Sgn(x – 3) at x = 3, (D) f (x) = x | x | tan 1 at x = 0. where Sgn stands for Signum function. *Q.28 Let A and B are two independent events. If P(A) = 0.3 and P(B) = 0.6, then (A) P(A and B) = 0.18 (B) P(A) is equal to P(A/B) (C) P(A or B) = 0 (D) P(A or B) = 0.72 Q.29 Let T be the triangle with vertices (0, 0), (0, c2) and (c, c2) and let R be the region between y = cx and y = x2 where c > 0 then (A) Area (R)= 6 c3 (B) Area of R= 3 c3 (C) ) R ( Area ) T ( Area Lim 0 c =3 (D) ) R ( Area ) T ( Area Lim 0 c = 2 3 Q.30 Consider the graph of the function f (x) = 1 x 3 x n e l . Then which of the following is correct. (A) range of the function is (1, ) (B) f (x) has no zeroes. (C) graph lies completely above the x-axis. (D) domain of f is (– , – 3) (–1, ) Q.31 Let f1(x) = x, f2(x) = 1 – x; f3(x) = x 1 , f4(x) = x 1 1 ; f5(x) = 1 x x ; f6(x) = x 1 x Suppose that ) x ( m 6 f f = f4(x) and ) x ( 4 n f f = f3(x) then (A) m = 5 (B) n = 5 (C) m = 6 (D) n = 6 Q.32 The graph of the parabolas y = – (x – 2)2 – 1 and y = (x – 2)2 – 1 are shown. Use these graphs to decide which of the statements below are true. (A) Both function have the same domain. (B) Both functions have the same range. (C) Both graphs have the same vertex. (D) Both graphs have the same y-intercepts. Q.33 Consider the function f (x) = x 2 bx 1 ax where a2 + b2 0 then ) x ( Lim x f (A) exists for all values of a and b (B) is zero for a < b (C) is non existent for a > b (D) is a 1 e or b 1 e if a = b Q.34 Which of the following function(s) would represent a non singular mapping. (A) f : R R f (x) = | x | Sgn x (B) g : R R g (x) = x3/5 where Sgn denotes Signum function (C) h : R R h (x) = x4 + 3x2 + 1 (D) k : R R k (x) = 2 x x 6 x 7 x 3 2 2 MATCH THE COLUMN PART -B [4 × 4 = 16] INSTRUCTIONS: Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-II. One or more than one entries of column-I may have the matching with the same entries of column-II and one entry of column-I may have one or more than one matching with entries of column-II. Q.1 Column I Column II (A) Constant function f (x) = c, c R (P) Bound (B) The function g (x) = x 1 t dt (x > 0), is (Q) periodic (C) The function h (x) = arc tan x is (R) Monotonic (D) The function k (x) = arc cot x is (S) neither odd nor even
- Q.2 Column I Column II (A) cot–1 ) 37 tan( (P) 143° (B) cos–1 ) 233 cos( (Q) 127° (C) sin 9 1 cos 2 1 1 (R) 4 3 (D) cos 8 1 cos arc 2 1 (S) 3 2 Q.3 Column I Column II (A) Number of integral values of x satisfying the inequality 3 x 1 x 4 x 2 x (P) 1 (B) The quadratic equations 2006 x2 + 2007 x + 1 = 0 and x2 + 2007x + 2006 = 0 have a root in common. Then the product of the uncommon roots is (Q) – 2 (C) Suppose sin – cos = 1 then the value of sin3 – cos3 is ( R) (R) – 1 (D) The value of the limit, ) x 1 ( n x tan 2 x 2 sin Lim 3 0 x l is (S) 0 Q.4 A quadratic polynomial f (x) = x2 + ax + b is formed with one of its zeros being 3 2 3 3 4 where a and b are integers. Also g (x) = x4 + 2x3 – 10x2 + 4x – 10 is a biquadratic polynomial such that 3 2 3 3 4 g = d 3 c where c and d are also integers. Column I Column II (A) a is equal to (P) 4 (B) b is equal to (Q) 2 (C) c is equal to (R) – 1 (D) d is equal to (S) – 11 SUBJECTIVE: PART -C [3 × 8 = 24] Q.1 Let y = sin–1(sin 8) – tan–1(tan 10) + cos–1(cos 12) – sec–1(sec 9) + cot–1(cot 6) – cosec–1(cosec 7). If y simplifies to a + b then find (a – b). Q.2 Suppose a cubic polynomial f (x) = x3 + px2 + qx + 72 is divisible by both x2 + ax + b and x2 + bx + a (where a, b, p, q are constants and a b). Find the sum of the squares of the roots of the cubic polynomial. Q.3 The set of real values of 'x' satisfying the equality x 3 + x 4 = 5 (where [ ] denotes the greatest integer function) belongs to the interval c b , a where a, b, c N and c b is in its lowest form. Find the value of a + b + c + abc.
- Q.1 Let F (x) = x 1 2 dt t 4 and G (x) = 1 x 2 dt t 4 then compute the value of (FG)' (0) where dash denotes thederivative. Q.2 10 identical balls are to be distributed in 5 different boxes kept in a row and labelledA, B, C, D and E. Find the number of ways in which the balls can be distributed in the boxes if no two adjacent boxes remainempty. Q.3 If f (x) = 4x2 + ax + (a – 3) is negative for atleast one negative x, find all possible values of a. Q.4 Let f (x) = sin6x + cos6x + k(sin4x + cos4x) for some real number k. Determine (a) all real numbers k for which f (x)is constant for all values of x. (b) all real numbers k for which there exists a real number 'c' such that f (c) = 0. (c) If k = – 0.7, determine all solutions to the equation f (x) = 0. Q.5 Let x0 = 2 cos 6 and xn = 1 n x 2 , n = 1, 2, 3, .......... find n ) 1 n ( n x 2 · 2 Lim . Q.6 Let f (x)= 3 9 9 x x then find the value of the sum f 2006 1 + f 2006 2 + f 2006 3 +....+ f 2006 2005 Q.7 2 0 2 2 dx x sin 8 x sin x . Q.8 For a> 0, findthe minimum valueofthe integral a 1 0 ax 2 5 3 dx e ) x a x 4 a ( . CLASS : XII (ABCD) TIME: 60 Min. DPP. NO.-34
- CLASS : XII (ABCD) DPP. NO.-35, 36 DPP-35 DATE : 16-17/08/2006 TIME : 45 Min. Q.1 If y = x 1 2 t n x dt, find 3 3 dx y d at x = e . Q.2 Find the equation of the normal to the curve y = (1+ x)y + sin1 (sin² x) at x = 0. Q.3 Find the real number 'a' such that 6 + x a 2 t dt ) t ( f = x 2 . Q.4 The tangent to y = ax2 + bx + 7 2 at (1, 2) is parallel to the normal at the point (–2, 2) on the curve y = x2 + 6x + 10. Find the value of a and b. Q.5 Let f bearealvalued functionsatisfying f(x)+f(x +4)=f(x+2)+f(x +6) then provethatthefunction g(x) = x x 8 f(t) dt is a constant function . Q.6 A tangent drawnto the curve C1 y= x2 + 4x + 8 at its point P touches the curve C2 y = x2 + 8x + 4 at its point Q. Find the coordinates of the point P and Q, on the curves C1 and C2. DPP-36 DATE : 16-17/08/2006 TIME : 45 Min. Q.1 Given real numbers a and r, considerthe following 20 numbers : ar, ar2, ar3, ar4,......, ar20. Ifthe sum of the 20 numbers is 2006 and thesum of the reciprocal of the 20 numberis 1003, find the product of the 20numbers. Q.2 Letf(x)andg(x)aredifferentiablefunctionssatisfyingtheconditions; (i) f(0) = 2 ; g(0) = 1 (ii) f (x) = g(x) & (iii) g (x) = f(x). Findthefunctions f(x)and g(x). Q.3 Let f (x) = 3 x 1 , 3 x 2 1 x 0 , 2 b 3 b 1 b b b x 2 2 3 3 Find all possible real values of b such that f(x) has the smallest value at x = 1. Q.4 Thereisafunctionfdefinedandcontinuous forall realx,whichsatisfies an equationoftheform f t dt x ( ) 0 = t f t dt x 2 1 ( ) + x x C 16 18 8 9 , where C is a constant. Find an explicit formula for f(x) and also the value of the constant. Q.5 Given f tx dt ( ) 0 1 = nf(x)thenfind f(x)wherex > 0. Q.6 Tangent at a point P1 [other than (0,0)] on the curve y= x3 meets thecurve again at P2.The tangent at P2 meets the curve at P3 & so on. Show that the abscissae of P1, P2, P3, ......... Pn, form a GP.Also find the ratio ) P P P ( area ) P P P ( area 4 3 2 3 2 1 .
- Q.1 The sum ofthe first fiveterms of a geometricseries is 189, thesum of the first six terms is 381,and the sum ofthe first seven termsis 765. What is thecommon ratio inthis series. [4] Q.2 Form aquadraticequation withrational coefficients if one of itsroot is cot218°. [4] Q.3 Let and be the roots of thequadratic equation (x – 2)(x – 3)+(x – 3)(x +1)+(x + 1)(x – 2)=0. Find the value of ) 1 )( 1 ( 1 + ) 2 )( 2 ( 1 + ) 3 )( 3 ( 1 . [4] Q.4 If a sin2x + b lies in the interval [–2, 8] for everyx R then find the value of (a – b). [4] Q.5 For x 0, what is the smallest possible value of the expression log(x3 – 4x2 + x + 26) – log(x + 2)? [4] Q.6 The coefficients of the equation ax2 + bx + c = 0 where a 0, satisfythe inequality (a + b + c)(4a – 2b + c) < 0. Prove that this equation has 2 distinct real solutions. [4] Q.7 In an arithmetic progression, the third term is 15 and the eleventh term is 55.An infinite geometric progression can be formed beginning with the eighth term of thisA.P. and followed bythe fourth and second term. Findthe sum of this geometricprogression upto n terms.Also compute S if it exists. [5] Q.8 Find the solution set of this equation log|sin x|(x2 – 8x + 23) > log|sin x|(8) in x [0, 2]. [5] Q.9 Find the positive integers p, q, r, s satisfying tan 24 = q p s r . [5] Q.10 Find thesum to n terms of the series. ........ 32 5 16 4 8 3 4 2 2 1 Alsofindthesum ifit exist ifn . [5] Q.11 If sin x, sin22x and cos x · sin 4x form an increasing geometric sequence, find the numerial value of cos 2x.Alsofindthecommonratioofgeometric sequence. [5] Q.12 Find all possible parameters 'a' for which, f(x) = (a2 + a – 2)x2 – (a + 5)x – 2 is non positive for every x [0, 1]. [5] Q.13 The 1st, 2nd and 3rd terms of an arithmeticseries are a, b and a2 where 'a'is negative. The 1st, 2nd and 3rd terms of a geometric series are a, a2 and b find the (a) value of a and b (b) sum ofinfinite geometricseries ifit exists. Ifnothenfind thesumto n terms oftheG.P. (c) sum ofthe40 term ofthe arithmeticseries. [5] Q.14 The nth term, an of a sequence of numbers is given by the formula an = an – 1 + 2n for n 2 and a1 =1.Findanequation expressingan asapolynomialinn.Also findthesumtonterms ofthesequence. [8] Q.15 Let f (x) denote the sum of the infinite trigonometric series, f (x) = 1 n n n 3 x sin 3 x 2 sin .Find f (x) (independent of n) also evaluate the sum of the solutions of the equation f (x) = 0 lying in the interval (0, 629). [8] CLASS : XII (ABCD) TIME : 100 Min. Max. Marks: 75 DPP. NO.-37
- CLASS : XII (ABCD) TIME: 60 Min. DPP. NO.-38 Q.1 Find the value of a and b where a < b, for which the integral b a 2 1 2 dx ) x x 2 24 ( has the largest value. Q.2 Solve the differential eqaution: y' + x cos e x cos x sin x y= x cos e 1 x . Q.3 Integrate : dx ) x cos x sin x )( x sin x cos x ( x2 Q.4 In a ABC, given sinA: sin B : sin C = 4 : 5 : 6 and cosA: cos B : cos C = x : y: z. Find the ordered pair that(x, y) that satisfiesthis extended proportion. Q.5 1 0 2 1 dx 1 x x x sin Q.6 Find the general solution of the equation , 2 + tan x · cot 2 x + cot x · tan 2 x = 0 Q.7 Let , be the distinct positive roots of the equation tan x = 2x then evaluate 1 0 dx ) x sin · x (sin , independent of and .
- CLASS : XII (ABCD) TIME: 60 Min. DPP. NO.-39 Q.1 Findtheset of values of'a' for which thequadraticpolynomial (a + 4)x2 – 2ax + 2a – 6 < 0 x R. [3] Q.2 Solvetheinequalitybyusingmethodof interval, 1 x 5 x 1 x 1 x . [3] Q.3 Find the minimum vertical distance between the graphs of y= 2 + sin x and y=cos x. [3] Q.4 Solve: x cos x cos 4 3 dx d 3 when x = 18°. [3] Q.5 If p, q are the roots of the quadratic equation x2 + 2bx + c = 0, prove that 2 log q y p y = log 2 + log c by 2 y b y 2 . [4] Q.6 Find the maximum and minimum value of y= 3 x 2 x 9 x 14 x 2 2 x R. [4] Q.7 Suppose that a and b are positive real numbers such that log27a + log9b = 7/2 and log27b + log9a = 2/3. Find the value of the ab. [4] Q.8 Given sin2y=sinx ·sin zwhere x, y, z areinanA.P.Find all possible values ofthe common difference of theA.P.andevaluatethesumof all the common differences which lie in theinterval (0, 315). [4] Q.9 Prove that tan 8 tan = (1 + sec2) (1 + sec4) (1 + sec8). [4] Q.10 Find the exact value of tan2 16 + tan2 16 3 + tan2 16 5 + tan2 16 7 . [4] Q.11 Evaluate 89 1 n 2 ) n (tan 1 1 . [5] Q.12 Find the value of k for which one root of the equation of x2 – (k + 1)x + k2 + k–8=0 exceed 2 and other is smallerthan2. [5] Q.13 Let an bethenth termof anarithmetic progression.Let Sn bethesumofthefirst n termsofthearithmetic progression with a1 = 1 and a3 = 3a8. Find the largest possible value of Sn. [5] Q.14(a)IfA+B+C = & sin 2 C A = k sin 2 C , then find the value of tan 2 A ·tan 2 B in terms of k. (b)Solve the inequality, log0.5 4 x x x log 2 6 < 0. [2 + 4] Q.15 Given the product p of sines of the angles of a triangle & product q of their cosines, find the cubic equation, whose coefficients are functions of p & q & whose roots are the tangents of the angles of the triangle. [6] Q.16 If each pairof the equations 0 q x p x 0 q x p x 0 q x p x 3 3 2 2 2 2 1 1 2 has exactlyone root in commonthen show that (p1 + p2 + p3)2 = 4(p1p2 + p2p3 + p3p1 – q1 – q2 – q3). [6]
- CLASS : XII (ABCD) TIME : 100 Min. DPP. NO.-40-41 DPP-40 TIME : 50 Min. Q.1 Let f (x) = 1 – x – x3. Find all real values of x satisfying the inequality, 1 – f (x) – f 3(x) > f (1 – 5x) Q.2 Integrate: dx ) x sin x cos e )( x cos x sin e ( 1 e e x x x x 2 Q.3 The circle C : x2 + y2 + kx + (1 + k)y – (k + 1) = 0 passes through the same two points for every real number k.Find (i) the coordinates ofthese two points. (ii) theminimum valueoftheradiusofacircleC. Q.4 Comment uponthenatureofroots ofthequadraticequation x2 +2x = k+ 1 0 dt | k t | dependingonthe value of k R. Q.5 Given n 1 n n 2 n n 3 n C C Lim = b a where a and b are relativelyprime, find the value of (a + b). DPP-41 TIME : 50 Min. Q.1 Let a, b, c be three sides of a triangle. Suppose a and b are the roots of the equation x2 – (c + 4)x + 4(c + 2) = 0 and the largest angle of the triangle is degrees. Find . Q.2 Findthevalue ofthedefiniteintegral 0 dx x cos 2 x sin 2 . Q.3 Let tan · tan = 2005 1 . Find the value of (1003 – 1002 cos 2)(1003 – 1002 cos 2) Q.4 dx x 1 x 1 n 1 x x 1 x 2 5 1 1 2 4 2 l Q.5 Two vectors 1 e and 2 e with | e | 1 = 2 and | e | 2 = 1 and angle between 1 e and 2 e is 60°. The angle between 2t 1 e + 7 2 e and 1 e + t 2 e belongs to the interval (90°, 180°). Find the range of t. Q.6 A function f (x) continuous on R and periodicwith period 2satisfies f (x) + sin x · f (x + ) = sin2x. Find f (x) and evaluate dx ) x ( f .
- PART -A Select the correct alternative. (Only one is correct) [16 × 3 = 48] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q.1 ) 3 x 4 x cos( 1 ) 3 x x x ( sin Lim 2 2 3 2 1 x has the value equal to (A) 18 (B) 9/2 (C) 9 (D) none Q.2 Let f (x) = x 3 2 4 13 t 3 t dt . If g (x) is the inverse of f (x) then g'(0) has the value equal to (A) 1/11 (B) 11 (C) 13 (D) 13 1 Q.3 The function f (x) has the property that for each real number x in its domain, 1/x is also in its domain and f (x) + x 1 f = x. The largest set of real numbers that can be in the domain of f (x), is (A) { x | x 0) (B) { x | x > 0) (C) { x | x –1 and x 0 and x 1) (D) {–1, 1} Q.4 Let w = 1 z 6 z 3 z2 and z = 1 + i, then | w | and amp w respectively are (A) 2, – 4 (B) 2 , – 4 (C) 2, 4 3 (D) 2 , 4 3 Q.5 If 2 a sin 2 a tan a cos 1 2 2 = a cos p w a cos k where k, w and p have no common factor other than 1, then the value of k2 + w2 + p2 is equal to (A) 3 (B) 4 (C) 5 (D) 6 Q.6 In a birthday party, each man shook hands with everyone except his spouse, and no handshakes took place between women. If 13 married couples attended, how many handshakes were there among these 26 people? (A) 185 (B) 234 (C) 312 (D) 325 Q.7 If x and y are real numbers such that x2 + y2 = 8, the maximum possible value of x – y, is (A) 2 (B) 2 (C) 2 2 (D) 4 Q.8 Let u(x) and v(x) are differentiable functions such that ) x ( v ) x ( u = 7. If ) x ( ' v ) x ( ' u = p and ' ) x ( v ) x ( u = q, then q p q p has the value equal to (A) 1 (B) 0 (C) 7 (D) – 7 Q.9 The coefficient of x9 when 30 x 2 x is expanded and simplified is (A) 30C14 · 29 (B) 30C16 · 214 (C) 30C9 · 221 (D) 10C9 Q.10 Let C be the circle described by (x – a)2 + y2 = r2 where 0 < r < a. Let m be the slope of the line through the origin that is tangent to C at a point in the first quadrant. Then (A) m = 2 2 r a r (B) m = r r a 2 2 (C) m = a r (D) m = r a Q.11 What can one say about the local extrema of the function f (x) = x + (1/x)? (A) The local maximum of f (x) is greater than the local minimum of f (x). (B) The local minimum of f (x) greater than the local maximum of f (x). (C) The function f (x) does not have any local extrema. (D) f (x) has one asymptote. Q.12 tan ) 5 tan( arc 3 2 tan arc equals (A) – 3 (B) – 1 (C) 1 (D) 3 Q.13 A line passes through (2, 2) and cuts a triangle of area 9 square units from the first quadrant. The sum of CLASS : XII (ABCD) DPP. NO.-42, 43
- all possible values for the slope of such a line, is (A) – 2.5 (B) – 2 (C) – 1.5 (D) – 1 Q.14 Which of the following statement is/are true concerning the general cubic f (x) = ax3 + bx2 + cx + d (a 0 & a, b, c, d R) I The cubic always has at least one real root II The cubic always has exactly one point of inflection (A) Only I (B) Only II (C) Both I and II are true (D) Neither I nor II is true Q.15 If S = 12 + 32 + 52 + ....... + (99)2 then the value of the sum 22 + 42 + 62 + ....... + (100)2 is (A) S + 2550 (B) 2S (C) 4S (D) S + 5050 Q.16 Through the focus of the parabola y2 = 2px (p > 0) a line is drawn which intersects the curve at A(x1, y1) and B(x2, y2). The ratio 2 1 2 1 x x y y equals (A) 2 (B) – 1 (C) – 4 (D) some function of p Select the correct alternative. (Only one is correct) [9 × 4 = 36] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q.17 If n 1 n n n n 3 3 · n ) 2 x ( n 3 · n Lim = 3 1 then the range of x is (n N) (A) [2, 5) (B) (1, 5) (C) (–1, 5) (D) (– , ) Q.18 The area of the region(s) enclosed by the curves y = x2 and y = | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1 Q.19 Suppose that the domain of the function f (x) is set D and the range is the set R, where D and R are the subsets of real numbers. Consider the functions: f (2x), f (x + 2), 2 f (x), 2 x f , 2 (x) f – 2. If m is the number of functions listed above that must have the same domain as f and n is the number of functions that must have the same range as f (x), then the ordered pair (m, n) is (A) (1, 5) (B) (2, 3) (C) (3, 2) (D) (3, 3) Q.20 f : R R is defined as f (x) = 0 x for 1 mx 0 x for 1 mx 2 x2 . If f (x) is one-one then m must lies in the interval (A) (– , 0) (B) (– , 0] (C) (0, ) (D) [0, ) Q.21 Let A = { x | x2 + (m – 1)x – 2(m + 1) = 0, x R}; B = { x | (m – 1)x2 + mx + 1 = 0, x R} Number of values of m such that A B has exactly 3 distinct elements, is (A) 4 (B) 5 (C) 6 (D) 7 Q.22 If the function f (x) = 4x2 – 4x – tan2 has the minimum value equal to – 4 then the most general values of '' are given by (A) 2n + /3 (B) 2n – /3 (C) n ± /3 (D) 3 n 2 where n I Direction for Q.23 to Q.25. Consider the function defined on [0, 1] R, f (x) = 2 x x cos x x sin if x 0 and f (0) = 0 Q.23 The function f (x) (A) has a removable discontinuity at x = 0 (B) has a non removable finite discontinuity at x=0 (C) has a non removable infinite discontinuity at x = 0 (D) is continuous at x = 0 Q.24 1 0 dx ) x ( f equals (A) 1 – sin (1) (B) sin (1) – 1 (C) sin (1) (D) – sin (1) Q.25 t 0 2 0 t dx ) x ( t 1 Lim f equals (A) 1/3 (B) 1/6 (C) 1/12 (D) 1/24 DPP-43 TIME : 60 Min.
- Select the correct alternative. (More than one are correct) [7 × 4 = 28] There is NO NEGATIVE marking. Marks will be awarded only if all the correct alternatives are selected. Q.26 Let f (x) = 0 x x x x 0 x xe 3 2 x then the correct statement is (A) f is continuous and differentiable for all x. (B) f is continuous but not differentiable at x = 0. (C) f ' is continuous and differentiable for all x. (D) f ' is continuous but not differentiable at x = 0. Q.27 Suppose f is defined from R [–1, 1] as f (x) = 1 x 1 x 2 2 where R is the set of real number. Then the statement which does not hold is (A) f is many one onto (B) f increases for x > 0 and decrease for x < 0 (C) minimum value is not attained even though f is bounded (D) the area included by the curve y = f (x) and the line y = 1 is sq. units. Q.28 The value of the definite integral 2 0 dx x cos 3 x cos 3 n xl , is (A) 2 0 dx x cos 3 x cos 3 n l (B) 0 dx x cos 3 x cos 3 n 2 l (C) zero (D) 0 dx x cos 3 x cos 3 n 2 l Q.29 f : [0, 1] R is defined as f (x) = 0 x if 0 1 x 0 if x 1 sin ) x 1 ( x 2 3 , then (A) f is continuous but not derivable in [0, 1] (B) f is differentiable in [0, 1] (C) f is bounded in [0, 1] (D) f ' is bounded in [0, 1] Q.30 Let 2 sin x + 3 cos y = 3 and 3 sin y + 2 cos x = 4 then (A) x + y = (4n + 1)/2, n I (B) x + y = (2n + 1)/2, n I (C) x and y can be the two non right angles of a 3-4-5 triangle with x > y. (D) x and y can be the two non right angles of a 3-4-5 triangle with y > x. Q.31 The equation cosec x + sec x = 2 2 has (A) no solution in 4 , 0 (B) a solution in 2 , 4 (C) no solution in 4 3 , 2 (D) a solution in , 4 3 Q.32 For the quadratic polynomial f (x) = 4x2 – 8kx + k, the statements which hold good are (A) there is only one integral k for which f (x) is non negative x R (B) for k < 0 the number zero lies between the zeros of the polynomial. (C) f (x) = 0 has two distinct solutions in (0, 1) for k (1/4, 4/7) (D) Minimum value of y k R is k(1 + 12k) PART -B MATCH THE COLUMN [3 × 8 = 24] Q.1 Column-I contain four functions and column-II contain their properties. Match every entry of column-I with one or more entries of column-II. Column-I Column-II (A) f (x) = sin–1(sin x) + cos–1(cos x) (P) range is [0, ] (B) g (x) = sin–1| x | + 2 tan–1| x | (Q) is increasing x (0, 1) (C) h (x) = 2sin–1 2 x 1 x 2 , x [0, 1] (R) period is 2 (D) k (x) = cot(cot–1x) (S) is decreasing x (0, 1) Q.2 Column-I Column-II
- (A) is origin, the is O ly where respective and , vectors position have and , edges s coterminou 3 whose iped parallelop the of Centre c b a OC OB OA (P) c b a (B) are s p.v.' whose P point a at concurrent are face opposite the of centroid the with vertex each joining Segments ly. respective c and b , a are C and B A, points angular its of vectors Positions origin. the is O where n tetrahedro a is OABC (Q) 3 c b a (C) is triangle the of e orthocentr the of p.v. the then | a c | | c b | | b a | If ly. respective c and b , a are points angular its of vectors position the triangle a be ABC Let (R) 4 c b a (D) by given is x Then c ) a x ( c b ) c x ( b a ) b x ( a equation the satisfies x vector unknown an If magnitude. same the of vectors lar perpendicu mutually 3 be c , b , a Let . 0 (S) 2 c b a Q.3 Column-I Column-II (A) If 3 3 3 c b a c b a 1 1 1 = (a – b)(b – c)(c – a)(a + b + c) then the solution (P) 3 c b a of the equation ) b x )( a x ( ) a x )( c x ( ) c x )( b x ( ) c x ( ) b x ( ) a x ( 1 1 1 2 2 2 =0, is (Q) 1 (B) The value of the limit, x Lim x ) c x ( ) b x ( ) a x ( 3 , is (R) 2 c b a (C) x 1 x x x 0 x 3 c b a Lim equals (S) 3 abc (D) Let a, b, c are distinct reals satisfying a3 + b3 + c3 = 3abc. If the quadratic equation (a + b – c)x2 + (b + c – a)x + (c + a – b) = 0 has equal roots then a root of the quadratic equation is PART -C SUBJECTIVE: [4 × 6 = 24] Q.1 Let f (x) = (x + 1)(x + 2)(x + 3)(x + 4) + 5 where x [–6, 6]. If the range of the function is [a, b] where a, b N then find the value of (a + b). Q.2 Let I = 4 / 0 (x – 4x2) ln(1 + tan x)dx. If the value of I = k 2 n 3 l where k N, find k. Q.3 Suppose f and g are two functions such that f, g : R R, f (x) = ln 2 x 1 1 and g (x) = ln 2 x 1 x then find the value of x eg(x) ) x ( ' g x 1 f ' at x = 1. Q.4 If the value of limit n 2 k 1 n ) 1 k ( k ) 2 k )( 1 k ( k ) 1 k ( 1 cos Lim is equal to k 120 , find the value of k.
- CLASS : XII (ABCD) DPP. NO.-44 This is the test paper of Class-XI (PQRS) held on 17-09-2006.Take exactly 75 minutes. Q.1 Evaluate n 1 r n 1 s s r rs 5 · 2 · where rs = s r if 1 s r if 0 . Will thesum holdifn ? [4] Q.2 Find the general solution of the equation, 2 + tan x · cot 2 x + cot x · tan 2 x = 0. [4] Q.3 Given that 3 sin x + 4 cos x = 5 where x 2 , 0 . Find the value of 2 sin x + cos x + 4 tan x. [4] Q.4 Findthe integral solution of theinequality 8 x x 2 ) 1 x ( log 2 3 . 0 0. [4] Q.5 In ABC, suppose AB = 5 cm, AC = 7 cm, ABC = 3 (a) Find the lengthof the side BC. (b) Find the area of ABC. [4] Q.6 The sides of a triangle are n – 1, n and n + 1 and the area is n n . Determine n. [4] Q.7 Withusualnotions,provethat inatriangleABC, r + r1 + r2 – r3 = 4R cos C. [5] Q.8 Find the general solution of the equation, sin x + cos x = 0. Also find the sum of all solutions in [0, 100]. [5] Q.9 Findall negativevaluesof'a'whichmakesthequadraticinequality sin2x + a cos x + a2 1 + cos x true for every x R. [5] Q.10 Solve for x, 2 x log 2 x log 2 4 2 2 3 5 = 1 x log x 2 log 2 2 2 2 5 3 . [5] Q.11 In a triangleABC if a2 + b2 = 101c2 then find the value of B cot A cot C cot . [5] Q.12 Solve the equation for x, ) x (sin log 2 1 2 1 5 5 5 = ) x (cos log 2 1 15 15 [5] Q.13 Evaluate thesum 1 n n 2 6 n . [5] Q.14 Suppose that P(x) is a quadratic polynomial such that P(0) = cos340°, P(1) = (cos 40°)(sin240°) and P(2) = 0. Find the value of P(3). [8] Q.15 If l, m, n are 3 numbers in G.P. prove that the first term of anA.P. whose lth, mth, nth terms are in H.P. is to the common difference as (m + 1) to 1. [8]
- Q.1 Let a, b, c, d, e, f R such that ad + be + cf = ) f e d )( c b a ( 2 2 2 2 2 2 use vectors or otherwise to prove that, 2 2 2 c b a c b a = 2 2 2 f e d f e d . Q.2 Let the equation x3 – 4x2 + 5x – 1.9 = 0 has real roots r, s, t. Find the area of the triangle with sides r, s, and t. Q.3 Suppose x3 + ax2 + bx + c satisfies f (–2) = – 10 and takes the extreme value 27 50 where x = 3 2 . Find the value of a, b and c. Q.4 Let I = dx xy x n y 1 1 x l and J = dy y 1 xy x n 1 x l where y x = xy. Show that I · J = (x + d)(y + c) where c, d R. Hence show that ) J I ( dx d = I + dx dy J . Q.5 Let ai, i = 1, 2, 3, 4, be real numbers such that a1 + a2 + a3 + a4 = 0. Show that for arbitrary real numbers bi, i = 1, 2, 3 the equation a1 + b1x + 3a2x2 + b2x3 + 5a3x4 + b3x5 + 7a4x6 = 0 has at least one real root which lies on the interval – 1 x 1. Q.6 Evaluate: dx 1 x x 3 x x 1 x 3 1 2 3 4 2 Q.7 Let x, y R in the interval (0, 1) and x + y= 1. Find the minimum value of the expression xx + yy. Q.8 dx ) x sin 2 )( x sin 1 ( ) x sin 2 )( x sin 1 ( CLASS : XII (ABCD) TIME: 55 to 60 Min. DPP. NO.-45
- CLASS : XII (ABCD) DPP. NO.-46 This is the test paper of Class-XI (J-Batch) held on 24-09-2006.Take exactly 75 minutes. Q.1 If tan , tan are the roots of x2 – px + q = 0 and cot , cot are the roots of x2 – rx + s = 0 then find the value of rs in terms of p and q. [4] Q.2 Let P(x)=ax2 +bx + 8 isaquadratic polynomial. Iftheminimum valueofP(x)is 6when x =2,find the values of a and b. [4] Q.3 Let P = 1 n 2 1 1 n 10 then find log0.01(P). [4] Q.4 Prove the identity 1 A 4 sec 1 A 8 sec = A 2 tan A 8 tan . [4] Q.5 Find the general solution set of the equation logtan x(2 + 4 cos2x) = 2. [4] Q.6 Find the value of 17 cos ........ 5 cos 3 cos cos 17 sin ......... 5 sin 3 sin sin when = 24 . [4] Q.7(a) Sumthefollowingseriestoinfinity 7 · 4 · 1 1 + 10 · 7 · 4 1 + 13 · 10 · 7 1 + ........... (b) Sum thefollowingseries upton-terms. 1 · 2 · 3 · 4 + 2 · 3 · 4 · 5 + 3 · 4 · 5 · 6 + ............. [3 + 3] Q.8 The equation cos2x – sin x + a = 0 has roots when x (0, /2) find 'a'. [6] Q.9 A, B and C are distinct positiveintegers, less than or equal to 10.The arithmeticmean ofAand B is 9. The geometric mean ofAand C is 2 6 . Findthe harmonic mean of B and C. [6] Q.10 Express cos 5x in terms ofcos x andhence find general solution ofthe equation cos 5x = 16 cos5x. [6] Q.11 If x is real and 4y2 + 4xy+ x + 6 = 0, then find the complete set of values of x for which yis real. [6] Q.12 Findthesumofalltheintegralsolutionsoftheinequality [6] 2 log3x – 4 logx27 5. Q.13 If + + = 2 , show that 2 tan 1 2 tan 1 2 tan 1 2 tan 1 2 tan 1 2 tan 1 = cos cos cos 1 sin sin sin . [7] Q.14(a) In any ABC prove that c2 = (a – b)2cos2 2 C + (a + b)2sin2 2 C . (b) In any ABC prove that a3cos(B – C) + b3cos(C – A) + c3cos(A – B) = 3 abc. [4 + 4]
- PART -A Select the correct alternative. (Only one is correct) [24 × 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q.1 The area of the region of the plane bounded above by the graph of x2 + y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 |, is (A) /4 (B) 2/4 (C) /2 (D) Q.2 Consider straight line ax + by = c where a, b, c R+ and a, b, c are distinct. This line meets the coordinate axes at P and Q respectively. If area of OPQ, 'O' being origin does not depend upon a, b and c, then (A) a, b, c are in G.P. (B) a, c, b are in G.P. (C) a, b, c are in A.P. (D) a, c, b are in A.P. Q.3 If x and y are real numbers and x2 + y2 = 1, then the maximum value of (x + y)2 is (A) 3 (B) 2 (C) 3/2 (D) 5 Q.4 The value of the definite integral 0 2 a ) x 1 )( x 1 ( dx (a > 0) is (A) /4 (B) /2 (C) (D) some function of a. Q.5 Let a, b, c are non zero constant number then r c sin r b sin r c cos r b cos r a cos Lim r equals (A) bc 2 c b a 2 2 2 (B) bc 2 b a c 2 2 2 (C) bc 2 a c b 2 2 2 (D) independent of a, b and c Q.6 A curve y = f (x) such that f ''(x) = 4x at each point (x, y) on it and crosses the x-axis at (–2, 0) at an angle of 45°. The value of f (1), is (A) – 5 (B) – 15 (C) – 3 55 (D) – 3 35 Q.7 The minimum value of the function f (x) = x cos 1 x sin 2 + x sin 1 x cos 2 + 1 x sec x tan 2 + 1 x cosec x cot 2 as x varies over all numbers in the largest possible domain of f (x) is (A) 4 (B) – 2 (C) 0 (D) 2 Q.8 A non zero polynomial with real coefficients has the property that f (x) = f ' (x) · f ''(x). The leading coefficient of f (x) is (A) 1/6 (B) 1/9 (C) 1/12 (D) 1/18 Q.9 Let Cn = n 1 1 n 1 1 1 dx ) nx ( sin ) nx ( tan then n 2 n C · n Lim equals (A) 1 (B) 0 (C) – 1 (D) 1/2 Q.10 Let z1, z2, z3 be complex numbers such that z1 + z2 + z3 = 0 and | z1 | = | z2 | = | z3 | = 1 then 2 3 2 2 2 1 z z z , is (A) greater than zero (B) equal to 3 (C) equal to zero (D) equal to 1 CLASS : XII (ABCD) DPP. NO.-47
- Q.11 Number of rectangles with sides parallel to the coordinate axes whose vertices are all of the form (a, b) with a and b integers such that 0 a, b n, is (n N) (A) 4 ) 1 n ( n 2 2 (B) 4 n ) 1 n ( 2 2 (C) 4 ) 1 n ( 2 (D) n2 Q.12 Number of roots of the function f (x) = 3 ) 1 x ( 1 – 3x + sin x is (A) 0 (B) 1 (C) 2 (D) more than 2 Q.13 If p (x) = ax2 + bx + c leaves a remainder of 4 when divided by x, a remainder of 3 when divided by x + 1, and a remainder of 1 when divided by x – 1 then p(2) is (A) 3 (B) 6 (C) – 3 (D) – 6 Q.14 Let f (x) be a function that has a continuous derivative on [a, b], f (a) and f (b) have opposite signs, and f ' (x) 0 for all numbers x between a and b, (a < x < b). Number of solutions does the equation f (x) = 0 have (a < x < b). (A) 1 (B) 0 (C) 2 (D) cannot be determined Q.15 Which of the following definite integral has a positive value? (A) 3 2 0 dx ) x 3 sin( (B) 0 3 2 dx ) x 3 sin( (C) 0 2 3 dx ) x 3 sin( (D) 2 3 0 dx ) x 3 sin( Q.16 Let set A consists of 5 elements and set B consists of 3 elements. Number of functions that can be defined fromA to B which are neither injective nor surjective, is (A) 99 (B) 93 (C) 123 (D) none Q.17 A circle with center A and radius 7 is tangent to the sides of an angle of 60°. A larger circle with center B is tangent to the sides of the angle and to the first circle. The radius of the larger circle is (A) 3 30 (B) 21 (C) 3 20 (D) 30 Q.18 The value of the scalar s r · q p can be expressed in the determinant form as (A) s · p r · p s · q r · q (B) r · q s · q s · p r · p (C) s · p r · q s · q r · p (D) s · q r · q s · p r · p Q.19 If x 1 0 1 x 1 0 1 x n · x Lim x l = – 5, where , , are finite real numbers then (A) = 2, =1, R (B) =2, =2, = 5 (C) R, =1, R (D) R, = 1, = 5 Q.20 If f (x, y) = sin–1( | x | + | y | ), then the area of the domain of f, is (A) 2 (B) 2 2 (C) 4 (D) 1 Q.21 A, B and C are distinct positive integers, less than or equal to 10. The arithmetic mean ofAand B is 9. The geometric mean of A and C is 2 6 . The harmonic mean of B and C is (A) 19 9 9 (B) 9 8 8 (C) 19 7 2 (D) 17 8 2 Q.22 If x is real and 4y2 + 4xy + x + 6 = 0, then the complete set of values of x for which y is real, is (A) x 2 or x 3 (B) x – 2 or x 3 (C) – 3 x 2 (D) x – 3 or x 2 Q.23 I alternatively toss a fair coin and throw a fair die until I, either toss a head or throw a 2. If I toss the coin first, the probability that I throw a 2 before I toss a head, is (A) 1/7 (B) 7/12 (C) 5/12 (D) 5/7 Q.24 Let A, B, C, D be (not necessarily square) real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix S = ABCD, consider the two statements. I S3 = S II S2 = S4 (A) II is true but not I (B) I is true but not II (C) both I and II are true (D) both I and II are false.
- PART-A Select the correct alternative. (More than one is/are correct) [3 × 6 = 18] There is NEGATIVE marking. 1 mark will be deducted for eachwrong answer. Q.1 The function f (x) is defined for x 0 and has its inverse g (x) which is differentiable. If f (x) satisfies ) x ( g 0 dt ) t ( f = x2 and g (0) = 0 then (A) f (x)isan odd linear polynomial (B) f (x)issomequadraticpolynomial (C) f (2) = 1 (D) g (2) = 4 Q.2 Consider a triangleABC in xyplanewith D,E and F as the middle points of the sides BC, CAandAB respectively. If thecoordinates of the points D, E and Fare (3/2, 3/2); (7/2, 0) and (0, –1/2) then which ofthefollowingarecorrect? (A)circumcentreofthetriangleABCdoes not lie insidethetriangle. (B) orthocentre, centroid, circumcentreand incentre of triangleDEF are collinear but of triangleABC arenon collinear. (C) Equation ofa line passes through the orthocentre of triangleABC and perpendicular to its plane is k̂ ) j ˆ î ( 2 r (D) distancebetween centroid and orthocentre of the triangleABC is 3 2 5 . Q.3 Ifacontinuousfunctionf (x)satisfies therelation, x 0 dt ) t x ( t f = x 0 dt ) t ( f +sinx +cosx –x –1, for all real numbersx, thenwhichof thefollowingdoesnot hold good? (A) f (0) = 1 (B) f ' (0) = 0 (C) f '' (0) = 2 (D) 0 dx ) x ( f = e PART-B MATCH THE COLUMN [3 × 8 = 24] There is NEGATIVE marking.0.5 mark will be deductedfor each wrong match within a question. Q.1 ColumnI ColumnII (A) x e x 3 t n dt x x n Lim l l is (P) 0 (B) 1 x 1 x x 2 4 e e Lim is (Q) 2 1 (C) n 4 ) 1 n ( sin 1 n 5 . 0 n sin ) 1 ( Lim 2 n n is where n N (R) 1 (D) The valueof the integral 1 0 2 1 1 dx 2 x 2 x 2 1 tan 1 x x tan is (S) nonexistent CLASS : XII (ABCD) DPP. NO.-48
- Q.2 Consider the matricesA= 1 1 4 3 andB= 1 0 b a andlet Pbeanyorthogonal matrix and Q= PAP APT and R = PTQKP also S = PBPT and T = PTSKP ColumnI ColumnII (A) If we varyK from 1 to n then the first row (P) G.P. with common ratio a firstcolumnelementsatRwillform (B) If we varyK from 1 to n then the 2nd row 2nd (Q) A.P. with common difference 2 columnelementsatRwillform (C) If we varyK from 1 to n then the first row first (R) G.P. with common ratio b columnelementsofTwillform (D) If we varyK from 3 to n then the first row 2nd column (S) A.P. with common difference– 2. elements ofT willrepresent thesum of Q.3 ColumnI ColumnII (A) Given two vectors b and a such that | b | | a | = | b a | = 1 (P) 30° The angle between the vectors b a 2 and a is (B) In a scalene triangleABC, if a cosA= b cos B (Q) 45° then C equals (C) In a triangleABC, BC = 1 andAC = 2.The maximum possible (R) 60° value which the Acanhave is (D) In a ABC B = 75° and BC = 2AD whereAD is the (S) 90° altitude fromA, then C equals PART-C SUBJECTIVE: [5 × 10 = 50] Q.1 Suppose V = 2 0 2 dx 2 1 x sin x , find the value of V 96 . Q.2 One of the roots of the equation 2000x6 + 100x5 + 10x3 + x – 2 = 0 is of the form r n m , where m is non zero integer and n and r are relativelyprime natural numbers. Find the value of m + n + r. Q.3 Acircle C is tangent to the x and yaxis in the first quadrant at the points Pand Q respectively. BC and AD are parallel tangents to thecircle with slope – 1. If the pointsAand B areon the y-axis while C and Dareonthex-axisandtheareaofthefigureABCDis 900 2 sq.units thenfindtheradius ofthecircle. Q.4 Let f (x) = ax2 –4ax + b(a>0) be definedin 1 x 5. Supposethe average ofthemaximum value and theminimumvalueofthefunctionis14,andthedifferencebetweenthemaximumvalueand minimum value is 18. Find the value of a2 + b2. Q.5 If the bx 1 ax 1 x 1 1 x 1 Lim 3 0 x existsand hasthevalueequal to l, thenfind thevalueof b 3 2 a 1 l .
- CLASS : XII (ABCD) DATE : 04-07/10/2006 TIME: 40 Min. for each DPP. NO.-49, 50 DPP-49 Q.1 8 claytargets have been arranged in vertical column, 3 being in the first column, 2 in the second, and 3 in thethird. In how manyways cantheybeshot (one at a time) if no target below it has been shot. [4] Q.2 Evaluate: 0 2 2 dx ) x (cos cos ) x (sin sin x [4] Q.3 Evaluate: 0 2 2 dx ) x cos(sin ) x sin(cos x [4] Q.4 2 0 2 dx x cos x sin x x [6] Q.5 Prove that 0 n 2 n 3 1 1 n 3 1 = 3 3 [9] DPP-50 Q.1 If cosA, cos B and cos C are the roots of the cubic x3 + ax2 + bx + c = 0 whereA, B, C arethe angles of a triangle then find the value of a2 – 2b – 2c. [4] Q.2 Find all functions, f : R R satisfying 2 x ) x ( F ) x ( F 2 ) x ( x f = 0 x R where f (x) = F'(x). [4] Q.3 2 2 3 2 1 dx x 3 1 x [4] Q.4 For a > 0, b > 0 verify that 0 2 dx a bx ax x n l reduces to zero by a substitution x = 1/t. Using this or otherwiseevaluate: 0 2 dx 4 x 2 x x n l [7] Q.5 0 3 1 dx x x tan [8]
- Select the correct alternative. (Only one is correct) There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 Findthesum oftheinfiniteseries ..... 63 1 45 1 30 1 18 1 9 1 (A) 3 1 (B) 4 1 (C) 5 1 (D) 3 2 Q.2 Numberofdegreesin thesmallest positiveanglex such that 8 sin x cos5x – 8 sin5x cos x = 1, is (A) 5° (B) 7.5° (C) 10° (D) 15° Q.3 There exist positive integersA, Band C with no common factors greater than 1, such that Alog2005 + B log2002 = C. The sumA+ B + C equals (A) 5 (B) 6 (C) 7 (D) 8 Q.4 A trianglewith sides 5, 12 and 13 has both inscribed and circumscribed circles. The distance between the centres of these circles is (A) 2 (B) 2 5 (C) 65 (D) 2 65 Q.5 Thegraphofacertaincubic polynomialis as shown. Ifthepolynomial can be written in the form f (x) = x3 + ax2 + bx + c, then (A) c = 0 (B) c < 0 (C) c > 0 (D) c = – 1 Q.6 Thesidesofa triangleare6 and8andtheangle betweenthesesidesvariessuch that0° < < 90°.The length of 3rd side x is (A) 2 < x < 14 (B) 0 < x < 10 (C) 2 < x < 10 (D) 0 < x < 14 Q.7 The sequence a1, a2, a3, .... satisfies a1 = 19, a9 = 99, and for all n 3, an is the arithmetic mean of the first n – 1 terms. Then a2 is equal to (A) 179 (B) 99 (C) 79 (D) 59 Q.8 If b is the arithmetic mean between a and x; b is the geometric mean between 'a' and y; 'b' is the harmonic mean between a and z, (a, b, x, y, z > 0) then the value of xyz is (A) a3 (B) b3 (C) a b 2 ) b a 2 ( b3 (D) b a 2 ) a b 2 ( b3 Q.9 GivenA(0, 0),ABCD is a rhombus of side 5 units where the slope ofAB is 2 and the slope ofAD is 1/2. Thesum of abscissa and ordinate of the point C is (A) 5 4 (B) 5 5 (C) 5 6 (D) 5 8 CLASS : XII (ALL) DATE : 10-11/11/2006 TIME: 60 Min. DPP. NO.-51
- Q.10 A circle of finite radius with points (–2, –2), (1, 4) and (k, 2006)can exist for (A) no value of k (B) exactlyone value of k (C) exactlytwovalues of k (D)infinitevaluesofk Q.11 If a ABC is formed by3 staright lines u = 2x + y – 3 = 0; v = x – y = 0 and w = x – 2 = 0 then for k = – 1 the line u + kv = 0 passes throughits (A)incentre (B)centroid (C) orthocentre (D)circumcentre Q.12 If a,bandc arenumbers for whichtheequation 2 2 ) 3 x ( x 36 x 10 x = x a + 3 x b + 2 ) 3 x ( c is an identity, , then a + b + c equals (A) 2 (B) 3 (C) 10 (D) 8 Q.13 If a, b, c are in G.P. then a b 1 , b 2 1 , c b 1 are in (A)A.P. (B) G.P. (C) H.P. (D) none Q.14 How manyterms are there in the G.P. 5, 20, 80, .........20480. (A) 6 (B) 5 (C) 7 (D) 8 Q.15 The sum of the first 14 terms of the sequence x 1 1 + x 1 1 + x 1 1 + ....... is (A) x 1 ) x 11 2 ( 7 (B) x 1 ) x 7 1 ( 7 (C) ) x 1 )( x 1 )( x 1 ( 14 (D) none Q.16 If x, y> 0, logyx + logxy= 3 10 and xy = 144, then arithmetic mean of x and y is (A) 24 (B) 36 (C) 2 12 (D) 3 13 Q.17 Acircleofradius R is circumscribedabout aright triangleABC. If ris theradius ofincircleinscribed in triangle thenthearea ofthetriangle is (A) r(2r + R) (B) r(r + 2R) (C) R(r + 2R) (D) R(2r + R) Q.18 The simplest form of a 1 1 1 a 1 is (A) a for a 1 (B) a for a 0 and a 1 (C) – a for a 0 and a 1 (D) 1 for a 1 Select the correct alternatives. (More than one are correct) Q.19 If the quadratic equation ax2 + bx + c = 0 (a > 0) has sec2 and cosec2 as its roots then which of the followingmustholdgood? (A) b + c = 0 (B) b2 – 4ac 0 (C) c 4a (D) 4a + b 0
- Q.20 Which of the followingequations can have sec2 and cosec2 as its roots ( R)? (A) x2 – 3x + 3 = 0 (B) x2 – 6x + 6 = 0 (C) x2 – 9x + 9 = 0 (D) x2 – 2x + 2 = 0 Q.21 The equation 1 x 10 2 | 2 x | = x 3 | 2 x | has (A)3integralsolutions (B)4 real solutions (C)1primesolution (D)noirrational solution Q.22 Whichofthefollowingstatementsholdgood? (A)If M is the maximum and m is the minimum valueof y= 3 sin2x + 3 sinx · cosx + 7 cos2x then the mean of M and m is 5. (B)The value of cosec 18 – 3 sec 18 is a rational which is not integral. (C) If x lies in the third quadrant, then the expression x 2 sin x sin 4 2 4 + 4 cos2 2 x 4 is independent ofx. (D) There are exactly 2 values of in [0, 2] which satisfy4 cos2 2 2 cos 1 = 0. PART-B MATCH THE COLUMN INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Column-I Column-II Q.1 (A) Area ofthetriangle formedbythe straight lines (P) 1 x + 2y – 5 = 0, 2x + y – 7 = 0 and x – y + 1 = 0 insquareunits is equal to (Q) 3/4 (B) Abscissaofthe orthocentre of thetriangle whose vertices are the points (–2, –1); (6, – 1) and (2, 5) (R) 2 (C) Variable line 3x( + 1) + 4y( – 1) – 3( – 1) = 0 for different values of areconcurrent at the (S) 3/2 point (a, b). The sum (a + b) is (D) The equation ax2 + 3xy – 2y2 – 5x + 5y + c = 0 represents twostraight lines perpendiculartoeach other, then | a + c | equals Column-I Column-II Q.2 (A) In a triangleABC,AB = 3 2 , BC = 6 2 ,AC > 6, (P) 60° and area of the triangleABC is 6 3 . B equals (Q) 90° (B) In a triangleABC is b = 3 , c = 1 andA= 30° (R) 120° thenangleBequals (C) In a ABC if (a + b + c)(b + c – a) = 3bc (S) 75° then Aequals (D) Area of a triangleABC is 6sq. units. If the radii of its excircles are2, 3 and 6then largest angle of the triangle is
- Column-I Column-II Q.3 (A) The sequence a, b, 10, c, d is an arithmetic progression. (P) 10 The value of a + b + c + d (B) Thesidesofright triangleform athreeterm geometric (Q) 20 sequence. Theshortest side has length2. Thelength of the hypotenuse is of the form b a where a N (R) 26 and b is a surd, then a2 + b2 equals (C) Thesumoffirst threeconsecutivenumbers ofan (S) 40 infinite G..P.is70, ifthetwoextremes bemultipled each by4, and the mean by5, the products are inA.P. The first term of the G.P. is (D) The diagonals ofaparallelogram haveameasure of 4 and6 metres. Theycut off formingan angleof 60°. If theperimeterofthe parallelogram is b a 2 where a, b N then (a + b) equals
- This is the test paper of Class-XI (PQRS & J) held on 19-11-2006. Take exactly 75 minutes. Q.1 Consider the quadratic polynomial f (x) = x2 – 4ax + 5a2 – 6a. (a) Findthesmallest positiveintegral value of'a'for which f(x) is positive foreveryreal 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 areal solution. (b) The equations to a pair of opposite sides of a parallelogram are x2 – 7x + 6 = 0 and y2 – 14y + 40 = 0 findtheequationstoits diagonals. [2.5+2.5] Q.3 Find the equationof the straight linewith gradient 2 ifit intercepts a chord of length 5 4 on the circle x2 + y2 – 6x – 10y + 9 = 0. [5] Q.4 Thevalueoftheexpression, x sin x cos x 2 cos 3 x 2 cos 6 6 3 whereverdefinedisindependentof x.Withoutallotting aparticularvalueofx, find thevalueofthis constant. [5] Q.5 Findthegeneralsolutionoftheequation sin3x(1 + cot x) + cos3x(1 + tan x) = cos 2x. [5] Q.6 Ifthethirdandfourthtermsofanarithmeticsequenceare increasedby3and8respectively,thenthe first fourterms form ageometric sequence. Find (i) thesum of the first fourterms ofA.P. (ii) second term of the G.P. [2.5+2.5] Q.7(a) Let x = 3 1 or x = – 15 satisfies the equation, log8(kx2 + wx + f ) = 2. If k, w and f are relativelyprime 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 m, n and p 0. Find the value of p n . [2.5+2.5] Q.8 Line 1 8 y 6 x intersects the x and y axes at M and N respectively. If the coordinates of the point P 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 ofthe point P and (b) the radius of the circle escribed opposite to the angle N. [2.5+2.5] Q.9 Startingattheorigin,abeamoflighthitsamirror(intheformofaline)atthepointA(4,8)andisreflected at the point B(8, 12). Compute the slope of the mirror. [5] CLASS : XII (ABCD) TIME: 75 Min. DPP. NO.-52
- Q.10 Find the solution set of inequality, ) x x ( log 2 3 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 equationto the straight lines joining the origin to thepoints of intersection ofthe straight line 1 b y a x andthe circle 5(x2 +y2 +bx +ay)=9ab.Also find thelinearrelation betweena and bso that thesestraightlinesmaybeatright 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 c x 4 x 3 x b x 3 x 2 x a x 2 x 1 x = 0 then all lines represented by ax + by + c = 0 pass through a fixed point. Findthe coordinates ofthat fixedpoint. [5] Q.15 If S1, S2, S3, ... Sn,.... are the sums of infinite geometric series whose first terms are1, 2, 3, ... n, ... and whose common ratios are 2 1 , 3 1 , 4 1 , ...., 1 n 1 , ... respectively, then find the value of 1 n 2 1 r 2 r S . [5] Q.16 In anytriangle if tan 2 A = 6 5 and tan 2 B = 37 20 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 Findtheequationofacircle passingthroughtheorigin ifthelinepair, xy– 3x +2y– 6 =0is 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 Findthelocus ofthecentres of thecircles whichbisects the circumferenceofthe 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 internallyat thepoint (–1, – 1). [5] Q.21 Find the equation of the line such that its distance from the lines 3x – 2y– 6 = 0 and 6x – 4y– 3 = 0 is equal. [5] Q.22 Findtherangeofthevariablex satisfyingthequadraticequation, x2 + (2 cos )x – sin2 = 0 R. [5] Q.23 If tan 2 y 4 = tan3 2 x 4 then prove that sin y = x sin 3 1 ) x sin 3 ( x sin 2 2 . [5]
- Revision Dpp on Permutation & combination Select the correct alternative. (Only one is correct) Q.1 Number of natural numbers between 100 and 1000 such that at least one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D) none Q.2 The number of ways in which 100 persons may be seated at 2 round tables T1 and T2 , 50 persons being seated at each is : (A) 99 25 ! (B) 100 50 ! (C) 100 2 ! (D) 100C50 Q.3 There are six periods in each working day of a school. Number of ways in which 5 subjects can be arranged if each subject is allotted at least one period and no period remains vacant is (A) 210 (B) 1800 (C) 360 (D) 120 Q.4 The number of ways in which 4 boys & 4 girls can stand in a circle so that each boy and each girl is one after the other is : (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 ! . 4 ! Q.5 If letters of the word "PARKAR" are written down in all possible manner as theyare in a dictionary, then the rank of the word "PARKAR" is : (A) 98 (B) 99 (C) 100 (D) 101 Q.6 The number of different words of three letters which can be formed from the word "PROPOSAL", if a vowel is always in the middle are : (A) 53 (B) 52 (C) 63 (D) 32 Q.7 Consider8verticesofaregularoctagonanditscentre.IfTdenotesthenumberoftrianglesandSdenotes thenumberofstraight linesthat canbeformedwiththese9points thenT –S hasthevalueequalto (A) 44 (B) 48 (C) 52 (D) 56 Q.8 A polygon has 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C) 20 (D) 25 Q.9 The number of ways in which a mixed double tennis game can be arranged from amongst 9 married couple if no husband & wife plays in the same game is : (A) 756 (B) 1512 (C) 3024 (D) 4536 Q.10 4 normal distinguishable dice are rolled once . The number of possible outcomes in which atleast one die shows up 2 is : (A) 216 (B) 648 (C) 625 (D) 671 Q.11 n r n r n r r n C C C 1 0 1 is equal to : (A) n 2 (B) n n n ( ) ( ) 1 2 1 (C) n1 2 (D) n n ( ) 1 2 Q.12 There are counters available in x different colours. The counters are all alike except for the colour. The total number of arrangements consisting of ycounters, assuming sufficient number of counters of each colour, if no arrangement consists of all counters of the same colour is : (A) xy x (B) xy y (C) yx x (D) yx y Q.13 In a plane a set of 8 parallel lines intersects a set of n parallel lines, that goes in another direction, forming a total of 1260 parallelograms. The value of n is : (A) 6 (B) 8 (C) 10 (D) 12 CLASS : XII (ABCD) TIME: 50 Min. DPP. NO.-53
- Q.14 A team of 8 students goes on an excursion, intwo cars, of which one can seat 5 and the other only4. If internal arrangement insidethecardoes notmatter then thenumber ofways in which theycantravel, is (A) 91 (B) 126 (C) 182 (D) 3920 Q.15 In a conference 10 speakers are present . If S1 wants to speak before S2 & S2 wants to speak after S3, then the number of ways all the 10 speakers can give their speeches with the above restriction if the remaining seven speakers have no objection to speak at any number is (A) 10C3 (B) 10P8 (C) 10P3 (D) 10 3 ! Q.16 There are 8 different consonants and 6 different vowels. Number of different words of 7 letters which can be formed, if they are to contain 4 consonants and 3 vowels if the three vowels are to occupyeven placesis (A) 8P4 . 6P3 (B) 8P4 . 6C3 (C) 8P4 . 7P3 (D) 6P3 . 7C3 . 8P4 Q.17 Number of ways in which 5 different books can be tied up in three bundles is (A) 5 (B) 10 (C) 25 (D) 50 Q.18 How many words can be made with the letters of the words "GENIUS" if each word neither begins with G nor ends in S is : (A) 24 (B) 240 (C) 480 (D) 504 Q.19 Number of numbers greater than 1000 which can be formed using only the digits 1, 1, 2, 3, 4, 0 taken four at a time is (A) 332 (B) 159 (C) 123 (D) 112 Select the correct alternative. (More than one are correct) Q.20 Identifythe correct statement(s). (A) Number of naughts standing at the end of is 30. (B) A telegraph has 10 arms and each arm is capable of 9 distinct positions excluding the position of rest. The number of signals that can be transmitted is 1010 1. (C) In a table tennis tournament, every player plays with every other player. If the number of games played is 5050 then the number of players in the tournament is 100. (D) Number of numbers greater than 4 lacs which can be formed by using only the digits 0, 2, 2, 4, 4 and 5 is 90. Q.21 n + 1C6 + nC4 > n + 2C5 nC5 for all ' n ' greater than : (A) 8 (B) 9 (C) 10 (D) 11 Q.22 The number of ways in which 200 different things can be divided into groups of 100 pairs is : (A) 2(1 . 3 . 5..199) (B) 101 2 102 2 103 2 .... 200 2 (C) 200 2 100 100 ! ( ) ! (D) 200 2100 ! Q.23 The continued product, 2 . 6 . 10 . 14 ...... to n factors is equal to : (A) 2nPn (B) 2nCn (C) (n + 1) (n + 2) (n + 3) ...... (n + n) (D) 2n · (1 · 3 · 5 ....... 2n – 1) Q.24 The Number of ways in which five different books to be distributed among 3 persons so that each person gets at least one book, is equal to the number of ways in which (A) 5 persons are allotted 3 different residential flats so that and each person is alloted at most one flat and no two persons are alloted the same flat. (B) number of parallelograms (some of which may be overlapping) formed by one set of 6 parallel lines and other set of 5 parallel lines that goes in other direction. (C) 5 different toys are to be distributed among 3 children, so that each child gets at least one toy. (D) 3 mathematics professors are assigned five different lecturers to be delivered, so that each professor gets at least one lecturer.
- Select the correct alternative. (Only one is correct) Comprehension (4 questions together) Consider the circle S: x2 + y2 – 4x – 1 = 0 and the line L: y= 3x – 1. If the line Lcuts the circle at A and B then Q.1 Length of the chord AB equal (A) 5 2 (B) 5 (C) 2 5 (D) 10 Q.2 The angle subtended by the chord AB in the minor arc of S is (A) 4 3 (B) 6 5 (C) 3 2 (D) 4 Q.3 Acute angle between the line L and the circle S is (A) 2 (B) 3 (C) 4 (D) 6 Q.4 If the equation of the circle on AB as diameter is of the form x2 + y2 + ax + by + c = 0 then the magnitude of the vector k̂ c j ˆ b î a V has the value equal to (A) 8 (B) 6 (C) 9 (D) 10 Q.5 How manybaseball nines can be chosen from 13 candidates ifA, B, C, D are the onlycandidates for two positions and can playat no otherposition? Q.6 Thevalues of a, for which oneof the roots oftheequation 2x2 – 2(2a +1)x +a(a+1) = 0is greater than a andtheother is smaller, is (A) , 2 1 (B) (0, 1) (C) (– , –3)(0, ) (D) (– , –1)(0, ) Q.7 If and are the roots of a(x2 – 1)+ 2bx = 0then, which one of thefollowing are the roots of the same equation? (A) + , – (B) 1 2 , 1 2 (C) 1 , 1 (D) 2 1 , 2 1 Q.8 The solutions of the equation, (1+ cos x) 2 x tan – 2 + sin x = 2 cos x are identical to the solutions of theequation (A) sin x = 1 (B) cos x = 0 (C) sin 2x = 0 (D) sec (x/2) = 2 Q.9 The solution of the equation ) x 2 3 ( log 2 x cos < ) 1 x 2 ( log 2 x cos is (A) (1/2, 1) (B) (– , 1) (C) (1/2, 3) (D) (1, ) – n 2 , n N Q.10 In ABC if B = 2 , s – a = 3; s – c = 2, then (A) r = 5/2 (B) = 12 (C) r1 = 2 (D) R = 3 SUBJECTIVE: Q.11 Find the least value of a, for which (5x + 1 + 51 – x); 2 a 3 1 and (25x + 25–x) and the successive terms of anA.P. for every x R. Q.12 Consider the quadratic polynomial f (x) = x2 – px + q where f (x) = 0 has prime roots. If p + q = 11 and a = p2 + q2 then find the value of f (a) where a is an odd positive integer. CLASS : XII (ABCD) TIME: 40 Min. DPP. NO.-54
- Revision DPP on Straight line and Circle Q.1 Theendsofaquadrantofacirclehavethecoordinates(1,3)and(3,1)thenthecentreofthesuchacircleis (A) (1, 1) (B) (2, 2) (C) (2, 6) (D) (4, 4) Q.2 Centroid of the triangle, the equations of whose sides are 12x2 – 20xy + 7y2 = 0 and 2x – 3y+ 4=0 is (A) (3, 3) (B) 3 8 , 3 8 (C) 3 8 , 3 (D) 3 , 3 8 Q.3 ThroughapointAonthex-axisastraight lineisdrawnparallelto y-axis soasto meet thepairofstraight lines ax2 + 2hxy + by2 = 0 in B and C. IfAB = BC then (A) h2 = 4ab (B) 8h2 = 9ab (C) 9h2 = 8ab (D) 4h2 = ab Q.4 A, B and C are points in the xy plane such thatA(1, 2) ; B (5, 6) andAC = 3BC. Then (A)ABC isauniquetriangle (B) Therecan be onlytwo such triangles. (C) No suchtriangle is possible (D)Therecan beinfinitenumberofsuch triangles. Q.5 IfA(1, p2) ; B (0, 1) and C (p, 0) are the coordinates of three points then the value of p for which the areaofthetriangleABCisminimum,is (A) 3 1 (B) – 3 1 (C) 3 1 or – 3 1 (D) none Q.6 m, nare integerwith 0< n< m.Ais thepoint(m, n) on the cartesian plane. Bis thereflection ofAin the liney=x.CisthereflectionofBinthey-axis,Dis thereflection ofCinthex-axis andEisthereflection of Din the y-axis.The area of the pentagonABCDE is (A) 2m(m + n) (B) m(m + 3n) (C) m(2m + 3n) (D) 2m(m + 3n) Q.7 The angle between the two tangents from the origin to the circle (x7)2 + (y+1)2 = 25 equals (A) /4 (B) /3 (C) /2 (D) none Q.8 The equation ofthe pair of bisectors of the angles between twostraight lines is, 12x2 7xy 12y2 = 0. If the equation of one line is 2y x = 0 then the equation of the other line is : (A) 41x 38y = 0 (B) 11x + 2y = 0 (C) 38x + 41y = 0 (D) 11x – 2y = 0 Q.9 Consider a quadratic equation in Zwith parameters x and yas Z2 – xZ + (x – y)2 = 0 The parameters x and y are the co-ordinates of a variable point P w.r.t. an orthonormal co-ordinate system ina plane. If thequadratic equation has equal roots then the locus of P is (A) a circle (B) a linepair through the originof co-ordinates with slope 1/2 and 2/3 (C) a linepair through the originof co-ordinates with slope 3/2 and 2 (D) a linepair through the originof co-ordinates with slope3/2 and 1/2 Q.10 Vertices of a parallelogram ABCD are A(3, 1), B(13, 6), C(13, 21) and D(3, 16). If a line passing throughtheorigindivides theparallelogramintotwocongruent partsthen theslopeofthelineis (A) 12 11 (B) 8 11 (C) 8 25 (D) 8 13 Q.11 The distance between the chords of contact of tangents to the circle ; x2+y2 +2gx+2fy+ c =0 from the origin & the point (g, f) is : (A) g f 2 2 (B) g f c 2 2 2 (C) g f c g f 2 2 2 2 2 (D) g f c g f 2 2 2 2 2 CLASS : XII (ABCD) TIME: 2 sitting of 75 minutes DPP. NO.-55
- Q.12 The locus of the center of the circles such that the point (2 , 3) is the mid point of the chord 5x + 2y = 16 is : (A) 2x 5y + 11 = 0 (B) 2x + 5y 11 = 0 (C) 2x + 5y+ 11 = 0 (D) none Q.13 The distance between the two parallel lines is 1 unit .Apoint 'A' is chosen to lie betweenthe lines at a distance'd'fromoneofthem.TriangleABCis equilateralwithB ononelineand Contheotherparallel line.Thelengthofthesideoftheequilateraltriangleis (A) 1 d d 3 2 2 (B) 3 1 d d 2 2 (C) 1 d d 2 2 (D) 1 d d2 Q.14 GivenA(0,0) and B(x, y)with x (0, 1) and y> 0. Let the slopeof the lineAB equals m1. Point C lies on the linex = 1 such that the slope ofBC equals m2 where 0 < m2 < m1. Ifthe area of thetriangleABC can be expressed as (m1 – m2) f (x), then the largest possible value of f (x) is (A) 1 (B) 1/2 (C) 1/4 (D) 1/8 Q.15 P lies on the line y = x and Q lies on y = 2x. The equation for the locus of the mid point of PQ, if | PQ | = 4, is (A) 25x2 + 36xy + 13y2 = 4 (B) 25x2 – 36xy + 13y2 = 4 (C) 25x2 – 36xy – 13y2 = 4 (D) 25x2 + 36xy – 13y2 = 4 Q.16 If the vertices Pand Q of a triangle PQR are given by(2, 5) and (4, –11) respectively, and the point R moves alongthe line N: 9x + 7y+ 4 = 0, then the locus of the centroid of the triangle PQRis a straight lineparallelto (A) PQ (B) QR (C) RP (D) N Q.17 The angle at which the circles (x – 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is (A) 6 (B) 4 (C) 3 (D) 2 Q.18 The value of 'c' for which the set, {(x, y)x2 + y2 + 2x 1} {(x, y)x y + c 0} contains only onepoint incommonis: (A) (, 1] [3, ) (B) {1, 3} (C) {3} (D) { 1 } Q.19 P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the co-ordinate axes cut at right angles, then : (A) a2 6ab + b2 = 0 (B) a2 + 2ab b2 = 0 (C) a2 4ab + b2 = 0 (D) a2 8ab + b2 = 0 Q.20 In a triangleABC , ifA(2, – 1) and 7x – 10y+ 1 = 0 and 3x – 2y + 5 = 0 are equations of an altitude and an angle bisector respectivelydrawnfrom B, then equation of BC is (A) x + y + 1 = 0 (B) 5x + y + 17 = 0 (C) 4x + 9y + 30 = 0 (D) x – 5y – 7 = 0 Q.21 A tangent at a point on the circle x2 + y2 = a2 intersectsa concentriccircle C at two points P and Q. The tangents to the circle C at P and Q meet at a point on the circle x2 + y2 = b2 then the equation of circle 'C' is (A) x2 + y2 = ab (B) x2 + y2 = (a – b)2 (C) x2 + y2 = (a + b)2 (D) x2 + y2 = a2 + b2 Q.22 AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle (otherthanAor B) andS is the centre ofthecircle inscribed intotriangleABC,thenmeasureof (A) angleASB changes as C moves on k. (B)angleASBis thesameforall positionsofCbutitcannotbedeterminedwithoutknowingtheradius. (C) angleASB = 135° for all C. (D) angleASB = 150° for all C.
- Q.23 Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles, x2 + y2 ( + 6)x + (8 2) y 3 = 0 . being the variable . The locus of the point of intersection of thesetangentsis : (A) 2x y+ 10 = 0 (B) x + 2y 10 = 0 (C) x 2y+ 10 = 0 (D) 2x + y 10 = 0 Q.24 Triangle formed by the lines x + y = 0 , x – y = 0 and lx + my = 1. If l and m vary subject to the condition l 2 + m2 = 1 then the locus of its circumcentre is (A) (x2 – y2)2 = x2 + y2 (B) (x2 + y2)2 = (x2 – y2) (C) (x2 + y2) = 4x2 y2 (D) (x2 – y2)2 = (x2 + y2)2 Q.25 ABCDisasquare ofunitarea.Acircle istangent to two sidesofABCD and passesthrough exactlyone of its vertices. The radius of the circle is (A) 2 2 (B) 1 2 (C) 1/2 (D) 2 / 1 Q.26 Aparallelogramhas 3 ofits vertices as (1, 2),(3, 8)and(4, 1).Thesum ofallpossible x-coordinates for the 4th vertex is (A) 11 (B) 8 (C) 7 (D) 6 Q.27 The image of the pair of lines represented by ax2 + 2h xy + by2 = 0 by the line mirror y= 0 is (A) ax2 2h xy by2 = 0 (B) bx2 2h xy + ay2 = 0 (C) bx2 + 2h xy + ay2 = 0 (D) ax2 2h xy + by2 = 0 Q.28 Two circles are drawn through the points (1, 0) and (2, 1) to touch the axis of y.Theyintersect at an angle: (A) cot–1 3 4 (B) cos 1 4 5 (C) 2 (D) tan1 1 Q.29 A is a point on either of two lines y + 3 x= 2 at a distance of 4 3 units from their point of intersection.The co-ordinates of the foot of perpendicular fromAon the bisector oftheangle between themare (A) 2 3 2 , (B) (0, 0) (C) 2 3 2 , (D) (0, 4) Q.30 A circle of constant radius 'a' passes through origin 'O' and cuts the axes of coordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is : (A) (x2 + y2) 1 1 2 2 x y = 4 a2 (B) (x2 + y2)2 1 1 2 2 x y = a2 (C) (x2 + y2)2 1 1 2 2 x y = 4 a2 (D) (x2 + y2) 1 1 2 2 x y = a2 Q.31 If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes atAand B then the locus ofthe centroid of thetriangleOAB is (A) x2 + y2 = (2k)2 (B) x2 + y2 = (3k)2 (C) x2 + y2 = (4k)2 (D) x2 + y2 = (6k)2 Q.32 Chords ofthe curve 4x2 + y2 x +4y=0 which subtend aright angleat theorigin passthrough afixed point whose co-ordinates are : (A) 1 5 4 5 , (B) 1 5 4 5 , (C) 1 5 4 5 , (D) 1 5 4 5 ,
- Q.33 Let x&ybetherealnumberssatisfyingtheequationx2 4x+y2 +3=0.Ifthemaximumandminimum values of x2 + y2 are M & m respectively, then the numerical value of M m is : (A) 2 (B) 8 (C) 15 (D) none of these Q.34 Ifthestraightlines joiningtheoriginand thepointsofintersectionofthecurve 5x2 + 12xy 6y2 + 4x 2y + 3 = 0 and x + ky 1 = 0 are equallyinclined to the co-ordinate axes then the value of k : (A) is equal to 1 (B) is equal to 1 (C) is equal to 2 (D) does not exist in the set of real numbers . Q.35 A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d1 & d2 arethe distances of thetangent to thecircleat the originOfrom thepointsAand B respectively, the diameterofthecircleis : (A) 2 d d 2 2 1 (B) 2 d 2 d 2 1 (C) d1 + d2 (D) 2 1 2 1 d d d d Q.36 If the line y= mx bisects the angle between the lines ax2 + 2h xy+ by2 = 0 then m is a root of the quadraticequation: (A) hx2 + (a b) x h = 0 (B) x2 + h (a b) x 1 = 0 (C) (a b) x2 + hx (a b) = 0 (D) (a b) x2 hx (a b) = 0 Q.37 Tangents are drawn from anypoint on the circle x2 + y2 = R2 to the circle x2 + y2 = r2. Ifthe line joining thepoints ofintersection of these tangents with thefirst circle also touch thesecond, then R equals (A) 2 r (B) 2r (C) 2 2 3 r (D) 4 3 5 r Q.38 Avariable circleC has theequation x2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter. If the power of point P(a,b) w.r.t. the circle C is constant then the ordered pair (a, b) is (A) 10 1 , 10 1 (B) 10 1 , 10 1 (C) 10 1 , 10 1 (D) 10 1 , 10 1 Q.39 Let C bea circle with twodiameters intersecting at an angle of 30 degrees.Acircle S is tangent to both the diameters andto C, and has radius unity.The largest radius of C is (A) 1 + 2 6 (B) 1 + 2 6 (C) 2 6 – 1 (D) none of these Q.40 Avariable circleC has theequation x2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter. The locus of the centre of the circle is (A) a parabola (B)anellipse (C) a hyperbola (D)pairofstraightlines Q.41 Letaandbrepresent thelengthofarighttriangle's legs. Ifdisthediameter of a circle inscribed into the triangle, and D is the diameter of a circle superscribed onthe triangle, then d + D equals (A) a + b (B) 2(a + b) (C) 2 1 (a + b) (D) 2 2 b a
- Select the correct alternatives : (More than one are correct) Q.42 The area of triangleABC is 20 cm2. The coordinates of vertex Aare (5, 0) and B are (3, 0). The vertex C lies on the line, x y= 2 . The coordinates of C are (A) (5, 3) (B) ( 3, 5) (C) ( 5, 7) (D) (7, 5) Q.43 Two vertices of the ABC are at the pointsA(1, 1) and B(4, 5) and the third vertex lines on the straight line y= 5(x 3) . Ifthe area of the is 19/2 then thepossible coordinates of thevertex C are: (A) (5, 10) (B) (3, 0) (C) (2, 5) (D) (5, 4) Q.44 A circle passes through the points (1, 1) , (0, 6) and (5, 5) . The point(s) on this circle, the tangent(s) at whichis/areparallel tothestraight linejoiningtheorigintoits centreis/are: (A) (1, 5) (B) (5, 1) (C) ( 5, 1) (D) ( 1, 5) Q.45 Line x a y b = 1 cuts the coordinate axes at A(a, 0) & B (0, b) & the line x a y b = 1 at A (a, 0) & B(0, b). If the points A, B, A, B are concyclic then the orthocentre of the triangle ABA is: (A) (0, 0) (B) (0, b') (C) 0 , aa b (D) 0 , ' bb a Q.46 Point M moved alongthecircle (x 4)2 + (y8)2 = 20. Then it broke awayfromit and moving along a tangent to the circle, cuts the xaxis at the point (2, 0) . The coordinates of the point on the circle at which the movingpoint broke awaycan be : (A) 3 5 46 5 , (B) 2 5 44 5 , (C) (6, 4) (D) (3, 5) Q.47 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x 3 y= 0 then the co-ordinates of the third vertex are : (A) (0, a) (B) 3 2 2 a a , (C) (0, a) (D) 3 2 2 a a , Q.48 The circles x2 + y2 + 2x + 4y 20 = 0 & x2 + y2 + 6x 8y + 10 = 0 (A) are such that the number of common tangents onthem is 2 (B) are not orthogonal (C) are such that the length of their common tangent is 5(12/5)1/4 (D) are such that the lengthof their common chord is 5 3 2 . Q.49 The centre(s) of the circle(s) passing through the points (0, 0) , (1, 0) and touching the circle x2 + y2=9 is/are : (A) 3 2 1 2 , (B) 1 2 3 2 , (C) 1 2 21 2 , / (D) 1 2 21 2 , / Q.50 The circles x2 + y2 2x 4y + 1 = 0 and x2 + y2 + 4x + 4y 1 = 0 (A) touchinternally (B) touchexternally (C) have 3x + 4y 1 = 0 as the common tangent at the point of contact. (D) have 3x + 4y+ 1 = 0 as the common tangent at the point of contact. Q.51 Three vertices of a triangle are A(4, 3) ; B(1, 1) and C(7, k) . Value(s) of k for which centroid, orthocentre,incentre andcircumcentre of the ABC lieon the same straightline is/are : (A) 7 (B) 1 (C) 19/8 (D) none
- Q.1 A Q.2 B Q.3 B Q.4 D Q.5 D Q.6 B Q.7 C Q.8 A Q.9 D Q.10 B Q.11 C Q.12 A Q.13 B Q.14 D Q.15 B Q.16 D Q.17 B Q.18 D Q.19 C Q.20 B Q.21 A Q.22 C Q.23 A Q.24 A Q.25 A Q.26 B Q.27 D Q.28 A Q.29 B Q.30 C Q.31 A Q.32 A Q.33 B Q.34 B Q.35 C Q.36 A Q.37 B Q.38 B Q.39 A Q.40 A Q.41 A Select the correct alternatives : (More than one are correct) Q.42 B, D Q.43 A, B Q.44 B, D Q.45 B, C Q.46 B, C Q.47 A, B, C, D Q.48 A, C, D Q.49 C, D Q.50 B, C Q.51 B, C Answer Key
- PART-A Select the correct alternative. (Only one is correct) [15 × 3 = 45] There is NEGATIVE marking and 1 mark will be deducted for each wrong answer. Q.1 A triangle with sides a = 15, b = 28 andc = 41. The length of the altitude from the vertex Bon the side AC is (A) 6 (B) 7 (C) 9 (D) 16 Q.2 If sides a, b and c of triangleABC satisfy c b a c b a 3 3 3 = c2 then tan 4 C has the value equal to (A) 2 – 1 (B) 2 – 3 (C) 1/ 3 (D) 2 + 3 Q.3 In a triangleABC, ABC = 45°, point D is onBC so that2BD =CD and DAB= 75°. ACBequals (A) 15° (B) 60° (C) 30° (D) 75° Q.4 Thefirst term of an infinitegeometric seriesis2 and itssum bedenoted by S. If |S– 2 |< 1/10 then the trueset of therangeof common ratio of theseriesis (A) 5 1 , 10 1 (B) 2 1 , 2 1 – {0} (C) 20 1 , 19 1 – {0} (D) 21 1 , 19 1 – {0} Q.5 Number of solution satisfyingthe equation, tan22x = 2 tan 2x · tan 3x + 1 in [0, 2] is (A) 0 (B) 1 (C) 2 (D) 4 Q.6 Numerical value of 12 cos 4 cos 12 5 sin + 12 sin 4 sin 12 5 cos , is (A) 2 1 (B) 2 3 (C) 2 + 3 (D) 2 3 1 Q.7 Two circles both touching the coordinate axes and pass through the point (6, 3). The radii of the two circles are the roots of the equation (A) t2 – 12t + 20 = 0 (B) t2 – 15t + 36 = 0 (C) t2 – 18t + 45 = 0 (D) t2 – 14t + 48 = 0 Q.8 Let 'a' and 'b' are the roots of the equation x2 – mx + 2 = 0. Suppose that b 1 a and a 1 b are the roots of the equation x2 – px + q = 0. If p = 2q then the value of m is equal to (A) 4 (B) 6 (C) 8 (D) 9 Q.9 The value of the determinant y x 1 x y x 1 1 x 1 0 1 depends on (A)onlyx (B)onlyy (C) both x and y (D)neitherx nor y Q.10 The sum of all the positive integers greater than 1 and less than 1000, which leave a remainderof one CLASS : XII (ABCD) TIME: 90 Min. DPP. NO.-56
- when divided by2, 3, 4, 5 and 6, is (A) 8176 (B) 7936 (C) 8167 (D) none Direction forQ.11 and Q.12 (2 questions together) Consider the digits 1, 2, 2, 3, 3, 3 and answer the following Q.11 If all the 6 digit numbers using these digits only are formed and arranged in ascending order of their magnitudethen29th numberwillbe (A) 213332 (B) 233321 (C) 233312 (D) none Q.12 LetMdenotesthenumberofsixdigitnumbersusingonlythegivendigitsifnotallthe2'saretogetherand N denotes the corresponding figure if no 3's are together then M – N equals (A) 16 (B) 28 (C) 54 (D) 36 Q.13 Number of selections that can be madeof 6 letters from the word "COMMITTEE" is (A) 20 (B) 17 (C) 34 (D) 35 Q.14 Acircle of radius r touches the lines given bythe equation 4x2 – 4xy+ y2 – 18x + 9y– 36 = 0.Area of thecircleinsquareunitsis (A) 45 (B) 75 (C) 45/2 (D) 45/4 Q.15 Ifthemaximum andminimumvalueoftheexpression 6 x 3 x 2 2 x 2 (x R)are Mand mrespectively then the value of m 1 M 1 equals to (A) – 13 (B) – 10 (C) 10 (D) 16 Select the correct alternatives. (more than one are correct) [5 × 5 = 25] ThereisNO NEGATIVE marking. Q.16 If sin (x + 20°) = 2 sin x cos 40° where x 2 , 0 then which of the following hold good (A) sec 2 x = 2 6 (B) cot 2 x =(2 + 3 ) (C) tan 4x = 3 (D) cosec 4x = 2 Q.17 If the vertices of an equilateral triangleABC are (1, 1); (–1, –1) and (a, b) then (A) a2 + b2 must be equals to 6 (B) a + b must be equals to zero (C) a + b can be equal to 3 2 (D)lengthofits median is 6 Q.18 The sides ofa right triangleT1 are 20, xand hypotenuse y.Thesides ofanotherright triangleT2 are30, x – 5 and hypotenuse y+ 5. If P1 and P2 are the radii of the circles inscribed and 1 and 2 arethe areas ofthetriangles T1 and T2 respectivelythenwhichofthefollowinghold good? (A) 61 = 52 (B) 81 = 72 (C) P1 = P2 (D) 2P1 = P2 Q.19 ABCD is aquadrilateral co-ordinates of whosevertices areA(1, 0),B(–1, 0), C(3,4) andD(–3, 4) then (A)Thediagonals of the quadrilateral are equal but not at right angle (B)Areaofthe quadrilateral is 16 (C) Circlepassingthrough anythreepoints of this quadrilateralalso passes through thefourthpoint (D)ThequadrilateralABCDisanequilateral trapezium Q.20 LetA (1, 2); B (3, 4) and C (x, y) be any point satisfying (x – 1)(x – 3) + (y– 2)(y – 4) = 0 then whichofthefollowingholdgood? (A) Maximum possible area of thetriangleABC is 2 squareunits (B) MaximumnumberofpositionsofCintheXYplanefortheareaofthetriangleABCtobeunity,is4 (C) Least radius of the circle passing throughAand B is 2 (D) If'O'is theoriginthen the orthocentre aswell as circumcentre ofthe triangle OABlies outside this triangle PART-B
- MATCH THE COLUMN [3 × 8 = 24] There is NEGATIVE marking. 0.5 Marks will be deducted for each wrong match. INSTRUCTIONS: Column-Iand column-IIcontainsfour entries each. Entries ofcolumn-Iareto bematchedwith some entriesofcolumn-II.Oneormorethanoneentriesofcolumn-Imayhavethematchingwiththesameentries ofcolumn-IIandoneentryofcolumn-Imayhaveoneormorethanonematchingwithentriesofcolumn-II. Q.1 ColumnI ColumnII (A) Numberofincreasingpermutationsof msymbols (P) nm are there from the n set numbers {a1, a2, , an} where theorder amongthenumbersis given by a1 < a2 < a3 < an–1 < an is (B) There are m men and n monkeys. Number of ways (Q) mCn in whicheverymonkeyhas a master, ifa man can haveanynumberofmonkeys (C) Number of ways in which n red balls and (m – 1) green (R) nCm balls canbe arranged in a line, so that no two red balls are together, is (balls of the same colour are alike) (D) Number of ways in which 'm'different toys can be (S) mn distributed in 'n'childrenifeverychildmayreceive anynumberof toys, is Q.2 ColumnI ColumnII (A) If the lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 (P) ArithmeticProgression and cx + 4y+ 1 = 0 passes through the same point, then a, b, c are in (B) Let a, b, c be distinct non-negative numbers. (Q) GeometricProgression If the lines ax + ay + c = 0, x + 1 = 0 and cx + cy + b = 0 passes through the same point, then a, b, c are in (C) If the lines ax + amy + 1 = 0, bx + (m + 1)by + 1 = 0 (R) HarmonicProgression and cx + (m + 2)cy + 1 = 0, where m 0 are concurrent then a, b, c are in (D) If the roots of the equation (S) None x2 – 2(a + b)x + a(a + 2b + c) = 0 be equal then a, b, c are in Q.3 ColumnI ColumnII (A) n 1 n n 2 n n 2 C Lim equals (P) 0 (B) Let the roots of f (x) = 0 are 2, 3, 5, 7 and 9 (Q) 1 and the roots of g (x) = 0 are – 1, 3, 5, 7 and 8. Number of solutions of the equation ) x ( ) x ( g f =0 is (C) Let y = x cos x sin3 + x sin x cos3 where 0 < x < 2 , (R) 3/2 thentheminimumvalueofyis
- (D) Acircle passesthrough vertex D of the squareABCD, (S) 2 and is tangent to the sidesAB and BC. IfAB = 1, the radius of the circle can be expressed as p + 2 q , then p + q has the value equal to PART-C SUBJECTIVE: [4 × 9 = 36] ThereisNO NEGATIVE marking. Q.1 If x1 and x2 arethe twosolutions oftheequation 3 16 2 3 3 9 log x log 3 1 log x 12 3 , then find thevalue of 2 2 2 1 x x . Q.2 A circle with center in thefirst quadrant is tangent to y = x + 10, y= x – 6, and the y-axis. Let (h, k) be the center of the circle. If the value of (h + k) = a + a b where a is a surd, find the value of a + b. Q.3 Suppose that there are 5 red points and 4 blue points on a circle. Find the number of convex polygons whose vertices areamongthe 9 points and having at least one blue vertex. Q.4 TriangleABC lies intheCartesianplaneand has an areaof 70sq. units.The coordinates of Band C are (12, 19)and(23, 20) respectivelyand the coordinates ofAare (p, q).The line containing themedian to the side BC has slope –5. Find the largest possible value of (p + q).