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TargeT MaTheMaTics by- agyaT gupTa Q.1 Q.2 Q.3 Q.4 Q.5 1-Marks T If A is a square matrix satisfying A A I . Write the value of A . Find the value of a for which the function f ( x) = x 2 2 ax 6 is increasing when x > 0. If the value of third order determinant is 12, than find the value of the determinant formed by its cofactors . Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13 Q.14 Q.15 Q.17 Q.18 Q.19 Q.20 Q.21 sum of 2 i 4 j 5 k and m i 2 j 3 k is equal to unity. Given f(x) = sin x check if function f is one-one for (i) (0,π) (ii) , 2 2 How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all ? Justify your answer . Write the number of all possible matrices of order 2 2 with each entry 0 or 1 and whose determinant is positive. Find when the projection of iˆ ˆj kˆ on iˆ ˆj is 2 units. x , find f f (2) If f : R R is defined by f ( x ) 2 x 1 Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis. If f : R R be given by f ( x ) ( 3 x 3 ) 1 / 3 , then fof ( x ) . The operation * define below on Q is binary operation a * b = ab+1. Check the existence of identity element and the inverse of all elements in Q . 3 2 sin x is 2 sin x 1 In the interval / 2 x , find the value of x for which the matrix singular. Let a and b be two vectors such that Q.16 Find the scalar m, such that the scalar product of i j k with the unit vector parallel to the a 3 ; b = 3 Then what is the angle between a b ? 2 and a b is a unit vector. If * be the binary operation on the set Q of rational numbers defined by a * b = identity element and inverse . ab . Find 4 Write total number of injective functions from set A and set B, where A = {1, 2, 3}, B = {a, b, c, d}. If A is a square matrix such that If a b a.b 225 and a 5 , 2 2 A2 I , 3 3 then find the simplified value of A I A I 7A . then write the value of b . Let set A = {3, 5, 6} and set B = {1 ,4 }. A relation R from set A to set B is defined as R = {(a,b) A×B: a–b is an even number}. List the elements of relation R. Let a 2 i j , b i 2 j and c 4 i 3 j . Find the values of x and y such that c x a y b . 1 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.22 Q.23 TargeT MaTheMaTics by- agyaT gupTa Prove that if E and F are independent events, then the events E and F are also independent. If a and b are two vectors such that | a b || a | , then prove that vector 2a b is _ perpendicular to vector b . Q.24 Q.25 Q.26 Q.27 Q.28 Q.29 Q.30 Q.31 Q.32 Q.33 Q.34 Q.35 Let * be binary operation on N defined by a*b=HCF of a and b. Does there exist identity for this binary operation on N? a dx 2 , (i) If 0 then find K . (ii) If 3 x dx 8 , write the value of 'a'. 2 16 2 8x 0 k If (3 x 1 2 0 2 x k ) dx 0, find the value of k. If f x sin2x cos2x ] find f ' 6 . Find the sum of the degree and the order for the following differential equation d d2y 4 [( ) ] 0. dx dx 2 In the following matrix equation use elementary operation R2 R2 R1 and write the equation thus obtained. Determine the value of the constant ‘k’ so that the function continuous at x 0 . If 3 A 3 and A 1 5 3 1 2 , 3 x If y 3 , then find 2 f(x)= Q.3 Q.4 x 0 is x 0 dy . dx dy at x 1 . dx d 2 y dy Write the order of the differential equation: log 2 x . dx dx 3 Find the acute angle which the line with direction cosines 2-Marks 1 3 , 1 6 ,n makes with positive direction of z- Find the value of m and n, where m and n are order and degree of differential equation d2 y 4 2 3 dx d y x 2 1. d3 y dx 3 dx 3 3 Q.2 kx , if lxl 3, if then write the adj A. 2 2 If f ' ( x) 2 x 1andy f ( x ), then find axis. Q.1 2 3 1 0 8 3 1 4 2 1 9 4 cos sin n , then for any natural number n, find the value of Det ( A ) . sin cos If A If x change from 4 to 4.01, then find the approximate change in logex. Prove that: a b c a 2 b 3 c a b c . 2 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13 TargeT MaTheMaTics by- agyaT gupTa 1 1 4 Evaluate: sin cos . 5 2 A bag contains 4 green and 6 white balls. Two balls are drawn one by one without replacement. If the second ball drawn is white, what is the probability that the first ball drawn is also white? Find the differential equation for all the straight lines, which are at a unit distance from the origin. Evaluate x 1 sin dx. 4 The income I of Dr. Rastogi is given by I ( x) Rs. ( x 3 3x 2 5x) . Can an insurance agent ensure him for the growth of his income? Prove that : tan 1 2 tan 1 3 Q.15 Q.16 Q.17 Q.18 Q.19 Q.20 Q.21 Q.22 Q.23 Q.24 Q.25 Q.26 Q.27 2 2x 2x 1 1 x 1 3 sin 4 cos 2 tan 1 x2 1 x2 1 x2 3 Solve the following equation : 1 . Two cards are drawn without replacement from a well shuffled pack of 52 cards. Find the probability that one is a king and other is a queen of opposite color. Find the integrating factor for the linear differential equation : ( y 2 1) 2xy Q.14 3 . 4 dy 2 dy . dx y 2 1 dx If â, b̂ and ĉ are mutually perpendicular unit vectors, then find the value of | 2â b̂ ĉ | . A pair of fair dice is thrown. Find the probability that the sum is 10 or greater, if 5 appears on the first die. Find the differential equation of all the lines in the xy-plane. 4 3 If A 2 5 , find x and y such that A 2 xA yI 0 . Evaluate : dx 3 sin x sin( x a ) 1 . A man is walking at the rate of 6.5 km/hr towards the foot of a tower 120 m high. At what rate is he approaching the top of the tower when he is 50 m away from the tower ? Solve : cos 1 sin cos 1 x . 3 Find the value of c in Rolle’s theorem for the function f x x3 3x in Let 2 1 5 2 2 5 A ,B ,C 3 4 7 4 3 8 If f : R R be defined as f ( x) If xy e xy , then show that 3,0 . , Find a matrix D such that CD-AB=O. 3x 7 , then find f 9 dy y x 1 . dx x y 1 1 ( x) . 3 1 1 1 3 sin 1 4 tan ( 3 ) 5 cos cos 1 1. Find the value of 2 Show that f : R R , given by f ( x) x x , is neither one-one nor onto, where x denotes greatest integer less than or equal to x . If a,b and c are mutually perpendicular vectors of equal magnitudes, find the angles which 3 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.28 Q.29 Q.30 Q.31 Q.32 Q.33 Q.34 Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42 Q.43 Q.44 Q.45 Q.46 TargeT MaTheMaTics by- agyaT gupTa the vector 2 a b 2 c makes with the vector a,b and c . If A aij is a matrix of order 2 2 , such that A 15 and C ij represents the cofactor of aij then find a 21 c 21 a 22 c 22 . Let * be a binary operation on the set N, of rational numbers defined by a*b =a+b+10, for a, b N. Find the identity elements for * if exists. Find λ , if 2i 26 j 27k i j k 0 . For what value of k, the matrix 2k 3 4 0 4 5 6 6 2k 3 5 is skew symmetric ? Let * be a binary operation Q+ , the set of all positive integers, defined by a*b= Q . Find the inverse of 0.1. Write total number of bijective functions from set A and set B, where A = {1, 2, 3}, B = {a, b, c, d}. ab , for a, b 100 Prove that if E and F are independent events, then the events E and F are also independent. _ Form the differential equation of the family of circle in the second quadrant and touching the coordinate axes. 1 Evaluate 1 x x dx. For what values of x is the rate of increase of x 3 5x 2 5x 8 is twice the rate of increase of x? a 2b 1 1 1 a tan cos 1 Prove that : tan cos . b b a 4 2 4 2 If the vectors p aiˆ ˆj k,ˆ q ˆi bjˆ kˆ and r ˆi ˆj CK are coplanar, then for 1 1 1 1 1 a 1 b 1 c a,b,c 1 show that A die whose faces are marked 1,2,3 in red and 4,5,6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find if A and B are independent events. Solve the following equation for x: 3 cos tan1 x sin cot 1 4 1 1 1 x 1 xyz yz zx xy . Using properties of determinants show that 1 1 y 1 z 1 1 If y sin 1 6 x 1 9 x 2 , If a ,b, c 1 3 2 x 1 3 2 , then find dy . dx are three vectors such that a b c 0 , then prove that a b b c c a, and hence show that a b c 0 . 2 at t 3 when x=10 (t-sin t) and y=12 (1-cos t). Find the sum of the degree and the order for the following differential equation Find dy dx 4 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 TargeT MaTheMaTics by- agyaT gupTa d d2y 4 [( ) ] 0. dx dx2 Q.47 Q.48 Q.49 Q.50 Q.51 Q.52 Q.53 Q.54 Q.55 Q.56 Q.57 Q.58 Q.59 Q.60 Q.61 Q.62 Q.63 Q.64 Q.65 2x 0 1 0 , find the value of x. and A 1 x x 1 2 If A Evaluate : 2 22 x 2 2 x dx. 2x 2 3 The equation of tangent at (2, 3) on the curve y ax b is y 4x 5 Find the value of a and b. Solve: tan 1 1 x 1 1 tan x ; x 0 . 1 x 2 If m and n are the order and degree, respectively of the differential equation 2 3 2 dy 3 d y y x 2 xy sin x , then write the value of m + n. dx dx 2x If A x Evaluate: 0 1 0 and A 1 , find the value of x. x 1 2 sin x x cos x dx x ( x sin x ) 3 2 Slope of the normal to the curve x 8a y, a 0 at the point in the first quadrant is 2 . Find the point, 3 In a school, there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in Class XII given that the chosen student is a girl ? Form the differential equation of all circles which touch the x-axis at the origin. 0 2 If matrix A and f(x)=1+x+x2 + x4 + x8 +x16 , find f(A). 0 0 Evaluate: 1 x ( x 1) 3 / 5 3 5 dx . ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a 2 i 2 j 3 k , b i 2 j k c 3 i j a b If and are such that is perpendicular to c , then find the value of . Find the approximate change in the value of . 1 , when x changes from x = 2 to x = 2.002 x2 1 x 1 then cos 1 x cos 1 x Prove that if 2 2 . 3 Let A = Z × Z and * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd). Find the identity element for * in the set A. Find: Find x x 4 2 sin 2 x sec 2 x 1 x2 x4 1 x5 3 3x 2 2 dx dx If A and B are two independent events, prove that A' and B are independent. 5 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 TargeT MaTheMaTics by- agyaT gupTa Q.66 Q.67 Q.68 Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13 4-Marks d 2 y dy 2 dy . Show that Ax By 1 is a solution of the differential equation x y 2 y dx dx dx 2 2 x a If y tan 1 log Evaluate : sin xa dy 2ax 2 : prove that . 4 xa dx x a 4 sin x dx x cos 3 x 3 Evaluate: (i) 0 sin 1 x 1 x x 1 x 2 dx . (ii) 1 Evaluate: (i) x x dx . (ii) sin 2 2 3 1 sin x cos x dx sin 2 x /6 /3 x cos x dx 2 3 Find the equation of the normal’s to the curve y = x 2 x 6 which are parallel to the line x + 14y + 4 = 0. Why are the fruits good for health? Find whether the following function is differentiable at x = 1 and x = 2 or not : x, x 1 f (x) 2 x, 1 x 2. 2 3 x x 2 , x2 The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009? For the curve y 4 x 2 x , find all point at which the tangent passes through origin. 3 Evaluate: (i) /2 x2 1 5x /2 5 dx (ii) 2 0 dx 1 e sin x . An unbiased coin is tossed 'n' times. Let the random variable X denote the number of times the head occurs. If P ( X 1), P ( X 2 ) and P ( X 3 ) are in AP, find the value of n. Water is running into a conical vessel ,15 cm deep and 5 cm in radius , at the rate of 0.1 cm 3 / sec . When the water is 6cm deep, find at what rate is (i) the water level rising ? (ii) the water –surface area increasing ? (iii) the wetted surface of the vessel increasing ? 6 For what value of K is the following function continuous at x ? 6 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.14 Q.15 Q.16 f ( x) Q.18 3 sin x cos x x k 6 Q.20 Q.21 Q.22 Q.23 x 6 6 . At what point on the curve x y 2x 3 0 , is the tangent parallel to X-axis. 2 y/x (i)From the differential equation of the family of curves given by (a bx)e x . (ii)Solve the differential equation, (1 y x 2 y ) dx ( x x 3 ) dy 0 where y = o when x = 1 If x cos(a+y)= d 2 dx y 2 cos y then sin 2 ( a y ) sec x dx Find : 1 cos ecx prove dy 0 . dx that 2 dy cos a y . dx sina Hence show that 3 2 Find the inverse of the matrix . Hence, find the matrix P satisfying the 5 3 3 Q.19 x 2 sin a Q.17 TargeT MaTheMaTics by- agyaT gupTa matrix equation P 5 2 1 2 . 3 2 1 Find the value of p for when the curves x 2 9p(9 y ) and x 2 p( y 1) cut each other at right angles. by c z a If a x c z 0 , then using properties of determinants, find the value of ax by c b a b c x, y,z 0 x y z , where 3 1 cosx 1 cosx x 1 Prove that : tan 1 cosx 1 cosx 4 2 , where x 2 Find: e x 1 sin 2 x dx . 1 cos 2 x x a 2, 0 x /4 f ( x ) 2 x cot x b, /4 x /2 Let a cos 2 x b sin x, / 2 x is continuous function on 0 x . Then determine the values of ‘a’ and ‘b’ . What are your views about ‘’learning ‘ ? Is ‘learning’ a continuous process? Q.24 Q.25 Q.26 Form the differential equation of the family of curve y ae be ce ; where a, b, c are some arbitrary constants. dy 1x Find the particular solution of the differential equation (1 x 2 ) (e m tan y ) , given that y = 1 when x dx = 0. x 2x 3x Find the point on the curve 9y 2 x 3 , where the normal to the curve makes equal intercepts on the axes. 7 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.27 Q.28 Q.29 Q.30 Q.31 TargeT MaTheMaTics by- agyaT gupTa 2 dy x 3 d y x x y . If y x log then, prove that 2 dx dx a bx 2 Vectors a, b, c are of the same magnitude and taken pairwise in order form equal angles. If a ˆi ˆj and b ˆj kˆ find c . Show that the equation of a plane, which meets the axes in A, B and C and the given controid of the triangle ABC is the point , , , is x y z 3. If 3p is distance of plane from origin, show that p . 2 2 2 2 Find the image of the line Evaluate : /4 0 sec x dx 1 2 sin 2 x x 1 y 3 z 4 0 1 7 in the plane 2 x y z 3 0. . Q.32 Solve the differential equation Q.33 2 1 1 If y cot ( cos x ) tan ( cos x ) Prove that sin y tan Q.34 Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42 Q.43 Q.44 y y y y x cos y sin ydx y sin x cos xdy . x x x x If yx , x d 2 y 1 dy y 0. prove that dx 2 y dx x 2 x . 2 2 Slope of the normal to the curve x3 8a2 y,a 0 at the point in the first quadrant is . 3 Find the point. Is f (x) = x 1 x 2 continuous and differentiable at x = 1,2 . Bag I contains 5 red and 4 white balls and bag II contains 3 red and 3 white balls. Two balls are transferred from bag I to bag II and then one ball is drawn from bag II. The balls so drawn are both found to be red. Find the probability that the transferred ball is 1 red and 1 white. Find the equation of tangent to the curve y cos x y , 2 x 0 , that is parallel to the line x 2y 0 . Solve the differential equation : x dy y x xy cot x 0;x 0. dx Determine the values of ‘a’ and ‘b’ such that the following function is continuous at x= 0: x sin x sin a 1 x ,if x 0 2, if x 0 . sin b x e 1 2 ,ifx 0 bx Find the intervals in which the function f ( x) 3log(1 x) 4 log(2 x) 4 2 x is strictly increasing or strictly decreasing. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 times the radius of the base. 1 sin x 1 sin x 1 ; x , Prove that: cot 2 1 sin x 1 sin x 2 2 x If the vectors a 2iˆ ˆj kˆ, b iˆ 2 ˆj 3kˆ and c 3iˆ ˆj 5kˆ are coplanar, find the value of . 8 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.45 TargeT MaTheMaTics by- agyaT gupTa x 1 x2 x 1 Evaluate: 3 dx Q.46 Find the value (s) of , such that the volume of a parallelepiped whose adjacent edges are Q.47 Evaluate :eku Kkr dhft, % Q.48 Q.49 Q.50 Q.51 Q.52 ˆ 3jˆ k, ˆ 4iˆ 4kˆ is 40 cu units. represented by vectors ˆi 2j, 2x 2 2 . Differentiate tan 1 1 x 1 x with respect to sin 1 2 1 x2 1 x2 1 x If a Q.54 Q.55 Q.56 Q.57 Q.58 Q.59 Q.60 are three vectors such that a b c 0 , then prove that a b b c c a, ,b , c and hence show that a b c 0 . 2 , then shown that x x 1 y 2 ( x 1) 2 y 1 2 . x 1 y 1 z and Find the equation of the plane containing two parallel lines 2 1 3 x y 2 z 1 4 2 6 . Also, find if the plane thus obtained contains the line x 2 y 1 z 2 3 1 5 or not. Check whether the following differential equation is homogeneous or not If y log x2 Q.53 x3 x dx . x4 9 1 x x 2 x dy xy 1 cos ,x 0 dx y Find the general solution of the differential equation using substitution y=vx. Evaluate Evaluate : 2 0 (3 x 5x 2 7 x 2 ) dx as limit of sums. dx . x 5 x 2 16 4 Check whether the following differential equation is homogeneous or not x cos , x ≠ 0 . Find the general solution of the differential equation . Find , if y = e − xy = 1 + Obtain the differential equation of all circles of radius r. If the curves 2tan . y aex and y= be x cut orthogonally, find the relation between a and b. x 2 ax b , The function f(x) is defined as f(x) 3 x 2 , 2 ax 5 b , 0 x 2 2 x 4 . If f(x) is continuous on 4 x 8 [0, 8], find the value of a and b. How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%? 9 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.61 Q.62 Q.63 Q.64 TargeT MaTheMaTics by- agyaT gupTa ( + + ) − If A+B+C = , then find the value of . ( + ) − Find the equation of the line drawn through point (1, 0, 2) to meet at right angles the line Q.66 Q.67 Q.68 Q.69 Q.70 Q.71 Q.72 Fine the equation of the plane through the point (4,-3,2) and perpendicular to the line of intersection of the planes x-y+2z-3=0 and 2x-y-3z = 0. Find the point of intersection of the line r i 2 j k ( i 3 j 9k) and the plane obtained above. Evaluate : /4 0 sin x cos x dx 1 sin 4 x . Find the foot of the perpendicular from P(1, 2, 3) on the line Given that vectors triangle is 5 a, b, c from a triangle such that Evaluate (e3 x 7x 5)dx as a limit of sums. 2 a b c ˆ ˆ ˆ ˆ ˆ ˆ 6 where a p i q j rk, b s i 3 j 4k . Find p, q, r, s such that area of ˆ ˆ ˆ and c 3i j 2k. 1 Suppose every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates and the corresponding values for rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs 5 per kg and rice Rs 20. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gm and 200 gm respectively. In what quantities should wheat and rice be mixed in the daily diet to provide the minimum daily requirements of proteins and carbohydrates, if both the items are to be taken up in each diet ? A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white? Using properties of determinates, show that ABC is isosceles if: 1 1 1 1 cosB 1 cosC cos2 A cos A cos2 B cosB cos2 C cosC 0 Given that vectors a , b , c from a triangle such that a b c . Find p, q, r, s such that Q.74 x 6 y 7 z 7 .Also 3 2 2 obtain the equation of the plane containing the line and the point (1, 2, 3). 2 2 2 2 1 a 1 b 1 a .b a b (a b ) . For any two vectors a & b , show that 1 cos A Q.73 . A man is known to speak truth 5 out of 6 times. He draws a ball from the bag containing 4 white and 6 black balls and reports that it is white. Find the probability that it is actually white? Do you think that speaking truth is always good? Q.65 x 1 y 2 z 1 3 2 1 area of triangle is 5 6 where a p i q j r k b si 3 j 4 k & c 3i j 2 k . If the following function is differentiable at x=2, then find the values of a and b. x 2 ,ifx 2 f x . ax b,ifx 2 Q.75 Determine the intervals in which the function f ( x ) x 4 8 x 3 2 2 x 2 2 4 x 2 1 is Q.76 Show that the equation of normal at any point t on the curve x 3 co s t cos 3 t strictly increasing or strictly decreasing. 10 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 and TargeT MaTheMaTics by- agyaT gupTa Q.77 Q.78 Q.79 Q.80 Q.81 Q.82 Q.83 Q.84 Q.85 Q.86 Q.87 Q.88 Q.89 Q.90 Q.91 Q.92 Q.93 y 3 s in t sin 3 t is 4 y co s 3 t x sin 3 t 3 sin 4 t . Evaluate : 2 /4 0 d2 y dy 2x 3 x 2 2 dx dx dx cos 3 x 2 sin 2 x x cos x . Then find Let y log X X x dy dx . 3 Find the equation (s) of the tangent (s) to the curve y x 1 x 2 at the points where the curve interests the x-axis. Find the equation of tangent to the curve x acos a sin , y a sin a cos at any point of the curve. Also show that at any point of the curve the normal is at a constant distance from origin. 2 Find the area of the region x , y : x y x . d2y dy cot x y sin 2 x 0 . If y cos(cos x) ;Prove that 2 dx dx Prove that the function f :0, R Given by f (x) = 9x2 + 6x – 5 is not invertible. Modify the codomain of the function f to make it invertible, and hence find f 1 . 1 Find the value of : sin 2 tan 1 cos tan 1 2 2 . Find: x x4 1 x 2 1 2 4 dx . Check whether the relation R in the set R of real numbers, defined by R= {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive. If y log 1 2t 2 t 4 , x tan 1 t, find Evaluate: x 1 1 x | x | 1 dx . 2 | x | 1 d2 y dx 2 2 Find the particular solution of the following differential equation cos y dx 1 2e x sin y dy 0; y 0 Find the general solution of the differential equation: dx y tan y x tan y xy dy y tan y . . 4 ˆ find a vector of magnitude 5 3 units perpendicular to the vector q and If P ˆi ˆj kˆ and q ˆi 2jˆ k, coplanar with vectors p and q . Q.94 2 2 1 If y sec x , then shown that x x 1 11 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 TargeT MaTheMaTics by- agyaT gupTa Q.95 Q.96 Q.97 Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 6-Marks Find the vector equation of the line parallel to the line x 1 2 y z 3 2 3 4 and passing through the point ( 2 , 4, 5 ) . Also find the distance between two lines Find the equation of the plane passing through the line of intersection of the planes x - 2y +z = 1 and 2x + y + z = 8 and parallel to the line with direction ratio 1,2,1. Also find the distance of P(1,2,-2) from this plane measured along a line parallel to r = t (i – 2j – 5k ) . Find the equation of the line passing through the point (-4, 3, 1), parallel to the plane y 3 x 1 z 2 x 2 y z 0 and intersecting the line . 3 2 1 Using integration, find the area bounded by the curves y = |x – 1| & y 3 x . Use product 1 1 2 2 0 1 0 2 3 9 2 3 To 3 2 4 6 1 2 solve the system of equations x+3z=9, -x+2y-2z = 4, 2 - 3 y+4z =-3. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit? A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each rand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many ags of each brand should be used? What is the minimum amount of nitrogen added in the garden? Kg per bag Brand P Brand Q 12 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Nitrogen TargeT MaTheMaTics by- agyaT gupTa Phosphoric Chlorine Q.9 Q.10 Q.11 3.5 3 1.2 1 acid Potash Q.8 3 2 1.5 2 A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of three types of seeds are 45%, 60% and 35%. Calculate the probability (a) of a randomly chosen seed to germinate.(b) that it is of the type A2, given that a randomly chosen seed does not germinate. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit? 6 2 4 3 6 9 , find A-1 and hence solve the system For A = 10 5 20 2 3 10 4 x y z 1 If a a2 a 6 9 20 4 6 5 2 1& x y z x y z ; a2 . 1 4 , then using properties of determinants, find the value of a a2 1 a3 1 0 a a4 0 a a4 a3 1 . a a4 a3 1 0 Q.12 Q.13 Q.14 Find the foot of the perpendicular from P(1, 2, 3) on the line x6 y7 z 7 .Also 3 2 2 obtain the equation of the plane containing the line and the point (1, 2, 3). If 1 1 0 A 2 5 3 , 0 2 1 (i)Using 1 2x find A 1 using elementary row transformation. the properties x 1 x(x 1) x x(x 1) 3x(1 x) x(x 1)(x 2) x(x 1)(x 1) (ii)Prove , using a(b2 c2 a2) 2b3 2a3 b(c2 a2 b2) Q.15 of equations . 2a3 2b3 2c3 2c3 of 6x2 (1 x2 ) properties c(a2 b2 c2) determinants: . of abc (a2 b2 c2)3 determinants: . Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and newspaper’s envelope as carry bags. It is found that the shopkeepers A, B, C are using (20, 13 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 TargeT MaTheMaTics by- agyaT gupTa 30, 40), (30, 40, 20), (40, 20, 30) polythene, handmade bags and newspapers envelopes respectively. The shopkeepers A, B, C spent Rs. 250, Rs. 270& Rs. 200 on these carry bags respectively. Find the cost of each carry bags using matrices. Keeping in the mind the social & environmental conditions, which shopkeeper is better? Why? Q.17 Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one students is chosen at random from the school and he was found to have an A grade. What is the probability that the students has 100% attendance? Is regularity required only in school? Using integration, find the area of the triangle bounded by the lines x + 2y = 2, y – x = 1 and Q.18 Using the method of integration find the area of the region enclosed between the circles Q.16 Q.19 2x + y = 7. + = 1 and − + = 1. p q p q p 0,q 0and q r q r 0 If , then, using properties of determinants, p q q r 0 prove that at least one of the following statements is true (a) p,q,r, are in G.P.,(b) is a root of Q.20 2 the equation px 2qx r 0 . An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table: Distance in (km) From/To A B E 6 4 D 7 F Q.21 Q.22 Q.23 3 Q.25 2 Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost? Using integration, find the area of the region {( x , y ) : x 2 y 2 1 x y ; x, y R } 2 An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit? Using integration find the area of the region bounded by the parabola y 4 x and the 2 circle 4x 4 y 9 . 2 Q.24 3 2 Find the area cut off the parabola 4 y 3x 2 by the straight line 2 y 3x 12 . There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of 14 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.26 TargeT MaTheMaTics by- agyaT gupTa phosphoric acid for her crop. If F1 cost Rs 6/kg and F2 costs Rs 5/ kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost? Sketch the graph y x 3 . Evaluate 0 6 Q.27 Q.28 Q.29 Q.30 Q.31 Q.32 Q.33 x 3 dx . what dose this integral represent on the graph? Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary operation on a defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) A. Then find (i) The identity element of * in A. (ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and 1 . ,4 2 Two trusts A & B receive Rs. 70000 and 55000 respectively from central government to award prize to persons of a district in 3 fields agriculture, education and social services. Trust a awarded 10, 5 and 15 persons in the field of agriculture, education and social services respectively while trust B awarded 15, 10 and 5 persons in the field of agriculture, education and social services respectively. If all three prizes together amount to Rs. 6000, them find amount of each prize by matrix method. What field you prefer most for award for development of society? Give answer with justifications. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society? A company produces two different products. One of them needs 1/4 of an hour of assembly work per unit, 1/8 of an hour in quality control work and Rs1.2 in raw materials. The other product requires 1/3 of an hour of assembly work per unit, 1/3 of an hour in quality control work and Rs 0.9 in raw materials. Given the current availability of staff in the company, each day there is at most a total of 90 hours available for assembly and 80 hours for quality control. The first product described has a market value (sale price) of Rs 9 per unit and the second product described has a market value (sale price) of Rs 8 per unit. In addition, the maximum amount of daily sales for the first product is estimated to be 200 units, without there being a maximum limit of daily sales for the second product. Formulate and solve graphically the LPP and find the maximum profit. Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle which the plane x + y = 3. Also find the equation of the 4 plane. A dietician wants to develop a special diet using two foods X and Y. Each packet (contains 30g) of food X contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Y contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. the diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. Make an LPP to find how many packets of each food should be used to minimize the amount vitamin A in the diet, and solve it graphically. 2 Let A 1 0 1 0 2 1 1 1 ,find the inverse of A using elementary row transformations and hence solve the following matrix equation XA 1 0 1 15 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89 Q.34 Q.35 Q.36 Q.37 Q.38 Q.39 TargeT MaTheMaTics by- agyaT gupTa Find the vector equation of the plane through the line of intersection of the planes x + y +z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line x2 y3 z 5 4 5 Let X be a non – empty set. P(x) be its power set. Let ‘* be an operation defined on element of P(x) by A*B = A B A, B P(X) Then, (i) Prove that * is a binary operation in P(X). (ii) Is* commutative ? (iii) Is* associative? (iv)find the identity element in P(X) w.r.t * . (v) find the all the invertible element of P(X) (vi) if O is another binary operation defined on P(X) as A O B = A B then verify that O distribution itself over *. Consider f : R 4 R 4 , given by f x 3 3 inverse of f and hence find f 1 0 and 4x 3 . Show that f is bijective. Find the 3x 4 x such that f 1 x 2. A cuboidal shaped godown with square base is to be constructed. Three times as much cost per square meter is incurred for constructing the roof as compared to the walls. Find the dimensions of the godown if it is to enclose a given volume and minimize the cost of constructing the roof and the walls. Find the area bounded by the curves y x , 2y 3 x and x – axis. Find the equation of the plane through the line x 1 y 4 z 4 3 2 2 x 1 1 y z 2 . Hence, find the shortest distance between the lines. 2 4 1 and parallel to the line Q.40 Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is coplanar with the Q.41 The members of a consulting firm rent cars from three rental agencies: 50% from agency X, 30% from agency Y and 20% from agency Z. Form past experience it is known that 9% of the cars from agency X need a service and tuning before renting, 12% of the cars from agency Y need a service and tuning before renting and 10% of the cars from agency Z need a service and turning before renting. If the rental car delivered to the firm needs service and tuning, find the probability that agency Z is not to blamed. Q.42 Q.43 line x 1 y 1 z 1 . Also find the equation of the plane containing them. 1 2 3 3 1 2 If A 3 2 3 , find A-1. 2 0 1 Hence, solve the system of equations: 3x + 3y + 2z = 1 X + 2y = 4 2x – 3y – z = 5 2 1 3 Find the inverse of the following matrix using elementary transformations. 5 3 1 3 2 3 **************** 16 P.T.O. Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com. TMC/D/79/89