Friday, 24 February 2023
area
Dry sketch graph showing the area of the reason bounded by the parabola Y is equal to x square is equals to calculate its area bounded by y square is equal to 12x find its area
COMPACT REVISION (MAXIMUM AND MINIMUM)
1) State the condition for maximum and minimum of a function y= f(x) at a point where d²y/dx² ≠ 0. find the maximum minimum values of x³-9x²+ 24x -12. 8,4
2) Find the maximum and minimum values of f(x) when f(x)= (x²-7x+6)/(x -10). Also obtained the values of x for which f(x) becomes maximum or minimum. MX at 4 mx.v is 1, mn at 16 mn v is 25
3) For what a value of x will (x-1)(3-x) have its maximum ? 2
4) Show that the maximum value of the function x+ 1/x is less than its minimum value.
5) Find the maximum value of f(x)¹⁾ˣ. e¹⁾ᵉ
6) Find the maximum minimum value of x³ + 1/x³. -2, 2
7) find the value of x for who is the function x⁴- 8x³+ 22x² - 24x+5 is maximum and minimum. Find also the maximum and the minimum value of the function. MX at 2 MX.v is -3, mn at 1 mn.v is -4
8) Find so that the function has neither maximum prove that the greatest rectangle inscribed in a circle is a square so that of all the tang given area the square as the lift perimeter obtain the maximum minimum value of the function in the interval the perimeter of the triangle is 8 cm find the length of the other side so that the area of the triangle is 100 cm is the area language A particle moving in straight line starts from the rest is the acceleration the particle time t or abr positive to constant then prove that the maximum velocity of the particle age what will be the radius of the base of a solid cylinder of volume 16 Pi Peru is a total surface area will be the smallest so that the all rectangles of a given perimeter the square as the greatest area give an xy is 4 and find the maximum minimum values of 4x + 9 Y divide 10 into two parts such that the product of a maximum find the maximum value of the product up to number the space is describing time T by a particle moving in a state line is given by find the minimum value of acceleration straight line its distance x from a fixed point over anytime trees given by the relation find the co-ordinate of a point on the parabola which is nearest to the straight line the parabola why find the point of least distance from the straight line the particle was the state line described distance x cm from a fixed point of the line at MTC can we wear find its acceleration in environment acceleration find the point on the straight line which is closest to the origin the coordinate of the required points 12 13 18 13 for what value of the function 2 sin x Cos 2x at else maximum and minimum values if a log has its extreme values find the value of PNB prove that the extreme Sur minimum a line is drawn rectangular maximum area is inscrib in a triangle is base is an altitude age find the area of the rectangle rectangle of British area is inscrib in a semicircle of radius a find its dimension so that the radius of the height right circular cylinder of the greatest car surface which can be describing a given cone is half that at the cone
COMPACT REVISION (TANGENT AND NORMAL)
1) Write down the equation of the tangent at the point (4,3) on the ellipse 9x²+ 16y²=288. 3x+ 4y = 24
2) Write down the equation of the tangent of the parabola y²=4x at (1,2). x - y +1= 0.
3) Find the equation the normal to the curve xy= c² at (ct, c/t). t³x -yt= c(t⁴-1)
4) Write down the equation of the normal to the parabola y²=25x at point (1,-5). 2x- 5y = 27
5) Write down the length of the tangent drawn from an external point (a, b) to the circle x²+ y²+ 2x =0. a²+b²+ 2a units
6) Write down the value of slope of the tangent to the parabola y²= 8(x -6) at the point (8,-4). -1
7) Determine the points on the curve y= x + 1/x, where are the tangent is parallel to the x-axis. (1,2) and (-1,-2)
8) Find the gradient of the tangent to the parabola y²= 4x at the point (1,2). 1
9) Show that the equation of the normal to the hyperbola x = a sec k, y = b tan k at the point (a sec k, b tan k) is ax cos k + by cot k = a²+ b².
10) Find the condition that the straight line lx + my = n touches is the ellipse x²/a² + y²/b² =1. a²l²+ b²m²= n²
11) Write down the equation of the tangent and normal of the parabola y²= 4ax at (0,0). x=0, y= 0
12) Obtain the equation of the normal to the hyperbola x²/b² - y²/b² =1 at (a sec k, b tan k). ax cos k+ by cot k = a²+ b²
13) If the straight line lx + my =1 is a normal to the parabola y²=4ax, then show that al³+ 2alm²= m².
14) Find the equation of the tangent to the hyperbola x²/a² - y²/b²= 1 at (a sec k, b tan k). x² - y²= 1
15) Find the equation of the tangent and normal to the ellipse 4x² +9y²= 72 at the point (3,2). 2x+ 3y = 6, 3x- 2y = 5
16) Find the equation of the tangent to the hyperabola y²= 4x+ 5, which is parallel to the straight line y= 2x +7. y= 2x+3
17) Find the equation of the normal to the curve y= x²- x at the point (3,6). Show that this normal touches the parabola x²+ 660y=0. x+ 5y = 33.
18) If the straight line lx + my = n is normal to the hyperbolas x²/a² - y²/b² = 1, then show that a²/l² - b²/m²= (a²+ b²)²/n²
19) If the straight line x cos k + y sin k = p touches the ellipse x²/a² + y²/b² = 1, then prove that a² cos²k + b² sin²k = p²
20) Prove that the straight line x+ y= 2+ √2 touches the circle x²+ y² - 2x - 2y +1= 0. Find the point of contact. (1+ 1/•2, 1+ 1/√2)
21) If the straight line lx+ my + n =0 touches the parabola y²= 4ax then prove that am² = nl.
22) Find the equation of the tangent and the normal to the curve y= x³- 3x at the point (2,2). 9x-y-16=0, x+ 9y -20= 0.
23) Show that the condition that the two curves ax²+ by² = 1 and a'x²+ b' y² =1 (ab'- a'b ≠ 0) intersects orthogonally is 1/a - 1/b = 1/a' - 1/b'.
24) Find the equation of the tangent and normal of the curve y(x -2)(x -3) - x +7 = 0 at its point of intersection with the x-axis. x- 20y-7=0; 20x + y=140.
COMPACT REVISION (RATE MEASURE)
1) A point moves on the parabola 3y = x² in such a way that, when x= 6. The abscissa is increasing at the rate of 3 cm/second. At what rate is the ordinate increasing at the point ? The ordinate increases at the rate of 12 cm/sec at that point.
2) if in the rectilinear motion of a particle s= at + 1/2 ft², when u and f are constants, then prove that velocity at time t is u + ft and acceleration is f.
3) Find the co-ordinates of the position of a particle moving along the paraboz y²= 4x at which the rate of increase of the abscissa is twice the rate of increase of the ordinate. (4,4)
4) Show that x/(1+ x) < log(1+ x) < x, when x> 0.
5) If f(x)= (x -1) eˣ +1, show that f(x) is positive for all positive values of x.
5) The time rate of change of radius of a sphere is 1/2π. When its radius is 5cm. Find the rate of change of the area of the surface of the sphere with time. 20cm²
6) Given y=(a sinx + b cosx)/(c sinx + d cosx), prove that
A) if a=1, b= 2, c= 3, d=4 then y decreases for all the value of x.
B) if a=2, b=1, c=3, d=4, then y increases for all values of x.
7) Find the range of value of x for which
A) the function x³ -9x²+ 24x - 16 increases as x increases. x<2 or x >4
B) the function 2x³ -9x²+ 12x - 3 decreases as x increases. 1< x < 2
8) A man 1.5 m tall walks away from a lampost 4.5 m at the rate of 4 km/hr.
A) how fast is shadow lenthening ?
B) how fast is the farther end of his shadow moving on the pavement ?
9) Water is poured into vertex into an inverted conical vessel of which the radius of the base is 2m and height 4 m at the rate of 77 litre/min . At what rate is the water-level rising at the instant when the depth is 70 cm ? 20cm/min
10) A spherical ice-ball is melting, the radius is decreasing at a constant rate of 0.1 cm per second. Find the amount of water formed in one second when the radius of the sphere is 7 cm. (Given π=22/7) sp. gr. of ice = 0.9). 55.44gm
11) the radius of a balloon is 7 cm. If an error of 0.1 cm is made as measuring the radius. Find the error in measuring the volume of the balloon. 6.16c.c
12) A spherical balloon is being inflated so that its volume increases uniformly at the rate of 40 cm³)min. How fast is its surface area increasing when the radius is 8 cm ? Find approximately, how much the radius will increase during the next 1/2 minute. 0.025. 0.025
13) Water is pumped out at a constant rate of 88 cubic metre per minute from a conical container held with its axis vertical. If the semivertical of this cone is 45°, find the rate of depression of the water level when the depth of the water level is 2 m. 7 m/min
13) find the set of all values 'a' for which the function f(x)= [{√(a+4)/(1- a)} -1] x⁵- 3x + log 5 decreases for all real x.
COMPACT REVISION (PROBABILITY)
1) If an unbiased coin is tossed. What is the probability of obtaining a head ? 1/2
2) Two unbiased coins are tossed. What is the probability of obtaining tail in both the coins ? 1/4
3) An unbiased die is rolled
A) What is the probability of getting 5 ?
B) What is the probability of getting an even number?
C) What is the probability of getting a number greater than 3? 1/6,1/2,1/2
4) Two unbiased dice are thrown together. Find the probability of obtaining
A) 5 in both the dice. 1/36
B) 3 in 1 and 4 in the other. 1/18
C) same number in both the dice. 1/6
5) A card is drawn at random from a pack of 52 cards. Find the probability of obtaining
A) a black card. 1/2
B) a diamond. 1/4
C) an Ace. 1/13
6) A card is drawn at random from a pack of 52 cards. what is the probability
A) the card is king. 1/13
B) the card is not king ? 12/13
7) An urn contains 5 red and 4 white balls. 3 balls are drawn at random from the urn. What is the probability that all the balls are red ? 5/42
8) A leap year is selected random. What is the probability that it will content 53 Mondays ? 2/7
9) A box contains 3 white and 5 black balls. A ball is drawn at a random.
A) What is the probability that the ball is black ?
B) find the odds in favour of the events
C) odd in against. 5/8, 5:3,3:5
10) A card is drawn at random from a pack of 52 cards. Find the probability that the card is a spade or an Ace? 4/13
11) Two unbiased dice are thrown.
A) What is the probability that the sum of the digits on the dice is 7. 1/6
B) Find also the odds in favour of this event. 1:5
22) 8 men in a company of 25 are graduates. If three men are selected from 25 men at random, What is the probability that
A) they are all graduates. 14/575
B) at least one of them is graduate. 81/115
23) There are 4 white and 3 black balls in a bag. If four balls are drawn at random then what is the probability that 2 of them are white and two or black ? 18/35
24) three unbiased coins are and tossed. What is the probability that all of them are heads. 1/8
25) A card is drawn at random pack of 52 cards. What is the probability that
A) the card is not a club. 3/4, 1/2
B) the card is neither a spade no a heart.
26) Two unbiased dice are rolled. What is the probability that the product of the digits in the dice is 12. 1/9
27) In a family there are two childrens. Find the probability they will have different birthdays. 364/365
28) A pair of dice is thrown. Find the probability that the sum of the two numbers is neither 8 nor 10. 7/9
29) A box contains 6 green balls and 4 yellow balls. 3 balls are drawn from the box at random. What is the probability that out of 3 balls 2 are green and one is yellow. 1/2
30) 4 unbiased dies are thrown at random. Find the probability of getting different digits in the four dice. 5/18
31) 6 unbiased coins are tossed together. What is the probability
A) exactly 4 heads. 15/64
B) at least 4 heads. 11/32
32) 5 unbiased coins are tossed together. Find the probability of obtaining 3 heads & 2 tails. 5/14
33) 3 unbiased coins are tossed together. Find the sample space in connection with it. Find the probability
A) at least one head. 7/8
B) exactly one tail. 3/8
34) 10 balls are distributed at random in 3 boxes. What is the probability of keeping 3 balls in the first box. 5120/19683
35) If 20 dates are named to random. What is the probability that 5 of them will be Mondays ? 15505 x6¹⁵/7²⁰
36) Two unbiased dies are thrown. Find the probability that the sun of the faces equals or exceeds 10. 1/6
37) If for the two events A and B, P(A)= 3/8 , P(B)=5/8 and P(A U B)=3/4. Then find the values of
A) P(A/B). 2/5
B) P(B/A). 2/3
Also examine whether A and B are independent or not.
38) There are two identical urns. One of them contains 4 white and 3 red balls and the other contains 3 white and 7 red balls. An urn is choosen at a random and a ball is drawn from it. Find the probability that the ball is white. If the ball is white then find the probability that it is taken from the first urn. 61/140, 40/61
39) if A and B are two independent events and P(A)=1/5, P(B)=2/3 then find the value of P(A U B). 11/15
40) if A and B are two events such that P(A)= 1/3 , P(B)=1/4 and P(A UB)=1/2 then find the value of
A) P(A ∩B'). 1/4
B) P(A/B'). 1/3
41) If two events are mutually independent then prove that their complementary events are also mutually independent.
42) There are 4 white, 3 red and 3 blue balls in a box and 5 white, 4 red and 3 blue balls in another box. If a ball is drawn at random from each of the boxes then find the probability that both of them are the same colour. 41/120
43) There are 3 red and 4 white balls in a bag. Two balls are drawn at random one after another, without replacement.
A) what is the probability that the ball drawn in the second time is white ?
B) under the condition that the second ball is white, what is the probability that the first ball is white ? 4/7, 1/2
44) A and B are two events such that P(A UB)= 7/8, P(A ∩B) = 1/4, P(B) =1/4. Find
A) P(A) . 3/8
B) P(B). 3/4
C) P(A ∩B'). 1/8
45) A box contains 4 red and 3 blue balls. Two balls are drawn at a time twice from that box. Find the probability that the first two balls are red and the next two balls are blue when first two balls are
A) replaced. 2/49
B) not replace before drawing the next two. 3/35
46) The first bag contains 5 white and 4 Black balls. The second Back contains 3 white and seven black balls. A ball is drawn at random from the first bag and is kept in a second bag. Now a ball is drawn at random from the second bag. What is the probability that a ball is white ? 32/99
47) if P(A) = 2/3 , P(B) = 1/2 and P(A UB) = 1 then find the value of
A)P(A/B). 1/3
B) P(A/B'). 1
C) P(A' ∩B'). 0
Are the events A and B mutually exclusive ? No
48) Suppose that all the four possible outcomes e₁ e₂ , e₃ and e₄ of an experiment are equally likely. If A={e₁,e₂}, B{e₂ , e₃}, C= e₃,e₄} then prove that A and B are independent, B and C are independent but A and C are not independent.
49) In an examination 30% students failed in mathematics, 20% students failed in chemistry and 10% students failed in both mathematics and Chemistry. A student is selected at random. What is the probability that
A) the student may fall in mathematics if it is known that the student has failed in chemistry. 1/2
B) the student may fail in mathematics or chemistry ? 2/5
50) The chance of solving a problem by 3 students are 2/7, 3/8 and 1/2 respectively. If each of them try independently. Find the chance that the problem is solved. 87/112
51) if A and B be two independent events and P(A)=2/3, P(B) =3/5 then find the values of
A) P(AU ∩B). 2/5
B) P(A∩B). 13/15
52) A candidate is selected for interview for 3 posts. For the first post there are three candidates, for the second post there are 4 candidates and for the third post there are two candidates. What is the chance of his getting at least one post ? 3/4
53) There are three identical boxes containing red and blue balls. In the first box there are 3 red and 2 blue balls, in the second there are 4 red and 5 blue balls and in the third there are two red and 4 blue balls. A box is chosen at random and a ball is drawn from it. If the ball is drawn be red then What is the probability that it has been drawn from the second box ? 10/31
54) an integer is choose at random from the first 200 positive integers. find the probability that chosen integer is divisible by 6 or 8. 1/4
55) The probability of winning of a player is 3/10 if the path of the running is fast, and the probability of winning is 2/5. if the path is slow. At a particular day the probability that path is fast is 7/10 and the probability that the path is slow Is 3/10. Find the probability of winning of that player on that day. 33/100
Thursday, 23 February 2023
Last time Revision (X) 2022/23
1) Yash opens a recurring deposit account with the bank of Rajasthan and deposits ₹600 per month for 20 months. Calculate the maturity value of this account, if the bank pays intrest at the rate of 10% per month. 13050
2) A recurring deposit account of ₹1200 per month has a maturity value of ₹12440. If the rate of interest is 8% and the intrest is calculated at the end of every month, find the time of this recurring deposit account. 10 months
3) Sujata deposited, a certain sum of money, every month, for 2 and half years (5/2 yrs) in a cumulative time deposit account. At the time of maturity she collected ₹4965. If the rate of interest was 8% p.a. find the monthly deposit. 150
4) Sumit paid ₹300 per month in a cumulative time deposit account for 2 yrs. He received ₹7875 as the maturity amount. Find the rate of interest. 9%
5) On depositing ₹200, every month in a cumulative time deposit account, paying 9% p.a. intrest, a person collected ₹117 as intrest. Find the period. 12 months
1) Find the range of values of x, which satisfy the inequality -1/5 ≤ 3x/10 +1 < 2/5, x belongs to R. Graph the solution set on the number line. 4≤x <-2
2) Solve the following inequation, and graph the solution set on the number line: 2y - 3 < y+ 2 ≤ 3y+ 5, y(-R). The solution set={y: y belongs to R, -3/2 ≤ y < 5}
3) 5x/4 - (4x -1)/3 > 1, x belongs to R. Show in number line. {x:x<-8, x belongs to R}
4) 2x -1 ≥ x + (7- x)/3 > 2.
5) If P is the solution set of -3x +4< 2x -3, x belongs to N, Q is the solution set of 4x -5 < 12, x belongs to W, find
A) P ∩ Q
B) P - Q
C) P' ∩ Q
1) (x-3)/(x+3) + (x+3)/(x-3) = 5/2, x≠- 3, x ≠ 3. -9,9
2) 2x -3 =√(2x² - 2x +21). 6
3) a/(ax -1) + b/(bx -1)= a+ b, a+ b≠ 0, ab≠ 0. (a+ b)/ab, 2/(a+ b)
4) 2x²- 9x +10= 0, when
A) x belongs to N. 2
B) x belongs to Q . 2, 5/2
5) 3x²- x - 7 correct upto two decimal places. 1.70 or -1.37
6) 2/(x -1) + 3/(x+1) = 4/(x+2). Correct to 2 significant figures. 0.23 or -8.77
7) A train covers a distance of 600 km at x km/hr. Had the speed been (x+20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in terms of x and solve it to evaluate x. 40
8) The cost of 2x articles is ₹(5x+54) while the cost of (x +2) articles is ₹ (10x -4). Find x. 6
9) The difference of the square of two numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers. 9,6
10) A year ago the father was 8 times as old his son. Now his age is the square of his son's age. Find their present ages. 49,7 yrs
11) The length of the hypotenuse of a right angled triangle exceeds the length of the base by 2cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle. 8, 15,27
12) The sum of the numerator and denominator of a fraction is 8. If 1 is added to both the numerator and denominator, the fraction is increased by 1/15. Find the fraction. 3/5
1) A: B =1/4 : 1/5 and B : C = 1/7: 1/6, find A: B: C. 15:12:14
2) Find the number which bears the same ratio to 7/33 that 8/21 does to 4/9. 2/11
3) A bag contains ₹142 in the form of one-rupee, 50-paise and 20-paise coins in the ratio 3:5:8. Find the number of coins of each type. 60,100,160
1) Find the remainder when 3x³+ 5x²-11x -4 is divided by 3x+1. 1/9
2) Find the values of a and b, if x-2 and x+3 are both factors of x³+ ax²+ bx - 12. 3,-4
3) Find the value of a if the division of x³+ 5x²- ax +6 is divided by x-1 leaves the remainder 2a. 4
4) If 3x -2 is the factor of 3x³ - kx² +21x - 10, then find k. 11
5) Prove that x- 5 is a factor of 2x²- x -45. Hence, factorise completely. (x-5)(2x+9)
6) y³ - 13y -12 factorise completely. (y+1)(y-4)(y+3)
1) a -2 = 2 c
b 7 3 2c+ d then find a,b,c,d. 2,3,-2, 11
2) If A= 3 -4 & B= 0 2
0 1 3 -1 find the Matrix X if 2A + 3X = 5B.
3) If A= 1 2 B= -2 -1 C= 0 3
-2 3 1 2 2 -1 find A+ 2B - 3C.
4) If x+ 3y = 1 2 and 2x + y= 5 0
-1 5 -3 3 find the Matrix x and y.
5) If A= 1 1
8 3 evaluate A² - 4A.
6) If X= 4 1
-1 2 show that 6X - X² = 9I, where I is unit Matrix.
7) If B= 4 -5 & C= 10 -11
-2 1 12 -15 find the Matrix A such that AB= C.
1) The point P(3,4) is reflected to P' in the x-axis and O' is the image of O(origin) when reflected in the line PP'. Using graph paper, give:
A) The coordinates of P' and O'
B) The length of the segment PP' and OO'.
C) The perimeter of the quadrilateral POP'Q'.
D) The geometrical name of the figure POP'Q'. (-3,4), 6, 20, Rhombus
2) Use a graph paper for this question. Plot the points P(3,2) and Q(-3,-2). From P and Q, draw perpendiculars PM and QN on the x-axis.
A) name the image of P on reflection in the origin.
B) Assign the special name to the geometrical figure PMQN and find its area.
C) Write the coordinates of the point to which M is mapped on reflection in (i) x-axis (ii) y-axis (iii) origin. Q, parallelogram, 18, (3,0),(-3,0),(-3,0)
3) A point P is reflected in the origin. Coordinates of its image are (2,-5). Find
A) the coordinates of P. (-2,5)
B) the coordinates of the image of P in the x-axis. (-2,-5)
4) The point A(2,3), B(4,5) and C(7,2) are the vertices of ∆ABC.
A) Write down the coordinates of A', B', C' if ∆A'B'C', is the image of ∆ABC when reflected in the origin. (-2,-3),(-4,-5),(-7,-2)
B) Write down the coordinates of A", B", C" if A"B"C" is the image of ∆ABC when reflected in the x-axis. (2,-3),(4,-5),(7,-2)
C) Assign the special name to the quadrilateral BCC"B" and find its area. Isosceles trapezium, 33 sq units
5) A) point P(a,b) is reflected in the x-axis to P'(5,-2). Write down the values of a, b. 5,2
B) P" is the image of P when reflected in the y-axis. Write down the coordinates of P". (-5,2)
C) Name a single transformation that maps P' to P". Reflection in the origin
1) Find the coordinates of the point C which divides the join of A(4,-3) and B(9,7) in the ratio 3:2. (7,3)
2) Find a point P on the line segment joining A(14,-5) and B(-4,4), which is twice as far from A as from B. (2,1)
3) The midpoint of the line joining (a,2) and (3,6) is (2,b). Find a, b. 1,4
4) The midpoint of the line joining (2a,4) and (-2,3b) is (1, 2a+1). Find a, b. 2,2
5) The line segment joining A(2,3) and B(6,-5) is intersected by x-axis at a point k. Write down the ordinate of the point k. Hence, find the ratio in which k divides AB. 0, 3:5
6) Calculate the ratio in which the line segment joining (3,4) and (-2,1) is divided by the y-axis. 3:2
7) Find the coordinates of the vertices of the triangle, the middle point of whose sides are (0,1/2), (1/2,1/2),(1/2,0). (0,0),(1,0),(0,1)
8) If (0,b),(-a/2, b/2),(a/2, b/2) are the midpoints of the sides of a triangle, find the coordinates of its centroid. (0,2b/3)
1) Find the value of p, given that the line y/2 = x - p passes through the point (-4,4). -6
2) The equation of the line PQ is 3y - 3x +7= 0.
A) find the slope of PQ. 1
B) calculate the angle that the line PQ makes with the positive direction of x-axis. 45°
3) The graph of the equation y= mx + c passes through the points (1,4) and (-2,-5). Find the value of m and c. 3,1
4) Find the equation of the line passing through the point (2,-5) and making an intercept of -3 on the y-axis. x+y+3= 0
5) Find the equation of a straight line passing through (-1,2) and slope is 2/5. 2x- 5y+ 12= 0
6) Find the equation of a straight line passing through the origin and through the point of intersection the lines 5x+ 7y -3= 0 and 2x - y= 7. x+2y= 0
7) Calculated the coordinates of the point of intersection of the lines by x+y -6= 0 and 3x - y = 2. (2,4)
8) The vertices of a triangle ABC are A(2,-11), B(2,13) and C(-12,1). Find the equation of its sides. x-2= 0, 6x - 7y+ 79= 0, 6x+ 7y+ 65= 0
9)A(2,-4) B(3,3) and C (-1,5) are the vertices of a ∆ ABC . Find the equation of the median of the triangle through A. 8x+y -12= 0
1) On a map drawn to a scale 1: 25000, a rectangular plot of land ABCD has the following measurements AB= 12cm, BC= 16cm, Angles A, B, C, D are all 90° each. Calculate
A) the diagonal distance of the plot in km. 5 km
B) the area of the plot in km². 12km²
2) The model of a ship is made to a scale of 1: 250. Find
A) the length of the shii, if the length of its model is 1,2m. 300m
B) the area of the deck of the ship, if the area of the deck of its model is 1.6m². 100000 m²
C) the volume of its model, when the volume of the ship is 1 cubic kilometre. 64 m³
1) Find the length of the tangent drawn to a circle of radius 4cm from a point 5cm away from the centre of the circle. 3 cm
2) Three circles with centres X,Y and Z touch each other externally. If XY= 6cm, YZ= 9cm, and XZ= 7.5cm. find the radii of the circles. 2.25,3.75,5.25cm
3) Two circles of radii 6cm and 14cm have their centres 30cm apart. Find the radii of the smallest circle that can be drawn to touch them and encloses the smaller. 11cm
4) The length of the direct common tangent to two circles of radii 8cm and 6cm is 15cm. Calculate the distance between their centres. 15.33 cm
5) If AB and CD are two chords of a circle intersecting at a point P inside the circle, such that AB= 12cm, AP= 2cm and DP= 4cm. Find PC. 5cm
6) If AB and CD are two chords of a circle intersecting at a point P outside the circle when produced, such that PA= 10cm, PC= 8cm and PB= 4cm. Find PD. 5cm
1) A girl fills a cylindrical bucket 32cm in height and 18 cm in radius with sand. She emptied the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24cm.
A) find its radius. 36cm
B) its slant height. √1872
( leave your answer in square root from)
2) Water flows at the rate of 10 m per minute through a cylindrical by 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm ? 51min 12sec
3) An exhibition tent is in the form of the cylinder surmounted by a cone. The height of the tent above the ground is 85 m and the height of the cylindrical part is 50m. If the diameter of the base is 168m, find the quantity of canvas required to make the tent. Allows 20% extra foe folds and for stitching, give your answer to the nearest m². 60509 m²
4) From a solid cylinder whose height is 8 cm and radius 6cm, a conical cavity of height 8 cm and base radius 6 cm is hollowed out. Find the volume of the remaining solid correct to 4 significant figures also find the total surface area of the remaining solid spherical metallic solid. 603.2, 603.2
5) A spherical metallic ball of radius 3 cm is melted and recast into three spherical balls. The radii of 2 of these balls are 2.5 cm and 2 cm respectively. Find the radius of third ball. 1.5cm
6) A cylindrical can whose base is horizontal and of radius 3.5cm contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can. calculate
A) the total surface area of the can in contact with water when the sphere is in it. 385/2
B) the depth of water in the can before the sphere was put into the can take π to be 22/7. 7/2
7) the internal and external radii of a hollow sphere are 3cm and 5 cm respectively. the sphere is melted to form a solid cylinder of 8/3 cm. Find the diameter and curved surface area of the cylinder. 14, 352/3
1) (cos³x + sin³x)/(cosx + sinx) + (cos³x - sin³x)/(cosx- sinx)
2) cosx/(1- tanx) + sinx/(1- cotx)
3) sinx/(cotx + cosecx) = 2+ sinx/(cotx - cosecx)
4) √{1+ cosx)/(1- cosx)}= cosecx + cotx
5) 1/(sinx + cosx) + 1/(sinx - cosx) = 2 sinx/(1- 2 cos²x)
6) sec²x + cosec²x = sec²x cosec²x.
7) sin⁴x + sin²x cos²x = sin²x
8) sin⁴x cosec²x + cos⁴x sec²x =1
9) tan⁴x + tan²x = sec⁴x - sec²x.
10) sin²x/(sinx - cosx) + cosx/(1- tanx) = sinx + cosx.
11) sin⁶x + cos⁶x = 1- 3 sin²x cos²x.
1) An aeroplane is flying horizontally 4000 m above the ground and is going away from an observer on the level ground. At a certain instant the observer finds that the angle of elevation of the plane is 45°. After 15 seconds, its elevation from the same point changes to 30°. Find the speed of the aeroplane in km/h. 702.72 km/h
2) At the foot of a mountain, the elevation of its summit is 45°. After ascending 1000m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain. 1396.86m
3) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 50m away from the bank, he finds the angle of elevation to be 30°. Calculate
A) the width of the river. 25m
B) the height of the tree. 43.3m
4) A kite is flying at a height of 75m from the level ground attached to a string inclined at 60° to the horizontal. Find the length of the string to the nearest metre. 87m
1) Construct a histogram for the following frequency distribution.
Class-interval frequency
05-12 4
13-20 12
21-28 26
29-36 15
37-44 6
45-52 18
2) Draw an Ogive for the following distribution.
Marks no of students
00-10 6
10-20 9
20-30 11
30-40 23
40-50 28
50-60 32
60-70 21
70-80 14
80-90 4
90-100 2
3) Draw a more than cumulative frequency curve for the following data:
Class-interval frequency
00-10 4
10-20 5
20-30 11
30-40 14
40-50 11
50-60 10
60-70 6
4) If 7 is the mean of 5, 3, 0.5, 4.5, b, 8.5, 9.5. find b. 18
5) Find the mean of following distribution:
Class-interval frequency
00-50 4
50-100 8
100-150 16
150-200 13
200-250 6
250-300 3 143
6) Given below are the weekly wages of 200 workers in a small factory:
Weekly wages no of workers
80-100 20
100-120 30
120-140 20
140-160 40
160-180 90
Calculate the mean weekly wages of the workers. 145
7) Calculate the mean, the median and the mode of the following:
3,1,5,6,3,4,5,3,7,2. 3.9,3.5,3
8) The marks scored by 19 students in a test are given below:
31,22,36,27,25,26,33,24,37,32,29,28,36, 27,35,35,32,26,28. Find
A) median. 29
B) lower quartile. 26
C) upper quartile. 35
D) inter quartile. 9
9) From the following frequency distribution, find median , lower quartile, upper quartile, semi-inter- quartile
Variate: 13 15 18 20 22 24 25
Frequ: 6 4 11 9 16 12 2
21,18,22,2
10) Marks obtained by 120 students in a Mathematics test are given below:
Marks. No of students
00-10 5
10-20 9
20-30 16
30-40 22
40-50 26
50-60 18
60-70 11
70-80 6
80-90 4
90-100 3
Draw the Ogive for the given distribution. Use a suitable scale for your Ogive. Use your Ogive to estimate
A) the median. 43.5
B) the lower quartile. 30
C) the number of students who obtained more than 75% in the test. 10
D) the number of students who did not pass in the test if the pass percentage was 40. 52
11) The daily profits in rupees of 100 shops in a departmental store are distributed as follows:
Profit per shop(in ₹) no of shops
000-100 12
100-200 18
200-300 27
300-400 20
400-500 17
500-600 6
Draw a histogram of the above data on a graph paper and hence estimate the mode. 255
Tuesday, 21 February 2023
COMPACT REVISION - DIFFERENTIATION (1st Order)
A) Find dy/dx:
1) x + 1/x. 1 - 1/x²
2) tan⁻¹ x. 1/(1+ x²)
3) ₑ√x. 1/2√x .ₑ√x
4) log(sin x). cot x
5) cos⁻¹(ₑ√(tanx).
6) (x²- 4x+3)/(x²+ 2x -3). at x=1. 3/8
7) 3x⁵+ 7x⁴ - 2x² - x +6. 15x⁴+28x³-4x-1
8) 1+ x + x²/2! + x³/3! + x⁴/4!. 1+ x + x²/2! + x³/3!
9) √x + 2√x² + 3√x³ + 4√x⁴ + 5 √x⁵. 1/2√x + 2 + 9/2√x + 8x + 25/2. √x³
10) 5x³/⁵√x² - 3x/³√x⁴ + 7x/⁷√x² + 12 ⁴√x/³√x.
11) logₐx + logxᵃ + eˡᵒᵍ ˣ + log eˣ + e¹⁺ˣ. 1/x logₐe + a/x + 2+ e¹⁺ˣ.
12) x² log x. x + 2x log x
13) 10ˣ. x¹⁰. 10ˣ(10x⁹+ x¹⁰ log 10)
14) (x²+7)(x³+10). 5x⁴+ 20x+ 21x²
15) √x . eˣ.secx. eˣ.secx(1+ 2x + 2x tanx)/2√x
16) x secx log(xeˣ). secx[1+ x + +1+ x tanx)(x + log x)]
17) (1+ sinx)/(1- sinx). 2cosx/(1- sinx)²
18) (sinx + cosx)/√(1+ sin2x). 0
19) (cosx - cisèx)/(1- cosx). -2sinx
20) (x³- 2+ 1/x³)/(x -2+ 1/x). 2(x +1 - 1/x² - 1/x³)
21) (eˣ + e³ˣ)/(eˣ + e⁻ˣ). e²ˣ/(1+ 2x)
22) y= tanx/x . Log(eˣ/xˣ). tanx/x + (1- log x) sec²x
23) √(2x) - √(2/x) +(x+4)/(4- x) at x= 2. 11/4
24) y= 1/(1+ xᵇ⁻ᵃ + xᶜ⁻ᵃ) + 1/(1+ xᵃ⁻ᵇ + xᶜ⁻ᵇ) + 1/(1+ xᵃ⁻ᶜ + xᵇ⁻ᶜ). 0
25) {¢(x)}ⁿ . n{{¢(x)}ⁿ⁻¹ ¢'(x)}
26) √(log x). 1/{2x√(logx)}
27) tan⁵x. 5 tan⁴x sec²x
28) (tan⁻¹x)². 2(tan⁻¹x)/(1+ x²)
29) ₑ(ax²+ bx+ c). (2ax+ b)ₑ(ax²+ bx+ c).
30) ₑ√(x+1) - ₑ√(x -1). ₑ√(x+1)/2√+x+1) - ₑ√(x -1)/2√(x -1).
31) ₇(x²+ 2x). 2(x+1)logₑ7. ₇(x²+ 2x)
32) log(ax²+ bx+ c). (2ax+b)/(ax²+ bx + c)
33) logₛᵢₙₓx . (log sinx - x cotx log x)/x(log sinx)²
34) logₑ(x + √(x²± a²)). 1/√(x²± a²)
35) f(x) = sin ¢(x). cos¢(x)¢'(x)
36) cos(ax + b). - a sin(ax + b)
37) sin x°. π/180 cosx°
38) sinx sin 2x sin 3x. 1/2 (cos2x + 2 cos4x - 3 cos6x)
39) sin⁻¹(x/a). 1/√(a²- x²)
40 cot⁻¹(cosecx + cot x). 1/2
41) log(cos x²). 2x tanx²
42) cot ⁻¹{√(1+ x²) - x}. 1/2(1+x²)
43) cos⁻¹{(1- x²)/(1+ x²)}. 2/(1+ x²)
44) sin⁻¹{2x/(1+ x²)}. 2/(1+ x²)
45) tan⁻¹{2x/(1- x²)}. 2/(1+ x²)
46) tan⁻¹{1/(x² -1)}. -1/{x√(x²-1)}
47) tan⁻¹{(3x - x³)/(1- 3x²)}. 3/2
48) tan⁻¹{cosx/(1- sinx)}. 1/2
49) cos(sin⁻¹x) + tan(cot⁻¹x). -[x/√(1- x²) + 1/x²]
50) sin(cos⁻¹x) + 1/2 sin⁻¹{2x/(1+ x²)}. 1/(1+ x²) - x/√(1- x²)
51) Find the derivative of f(logx) where f(x)= log x. 1/(x logx)
52) tan⁻¹{cosx/(1+ sinx)} + sin(eˣ). eˣcos(eˣ) - 1/2
53) logₓ(tanx). (2x logₑx cosec2x - logₑ(tanx))/{xlogₑx)²}
54) logₑ √{(1- cosx)/(1+ cosx)} + aˣ. Cosecx + aˣ logₑa
55) sinx/(1+ cosx). 1/2 sec²(x/2)
56) tan⁻¹ √{(1- x)/(1+ x)}. -1/{2√(1- x²)}
57) cos[2 sin⁻¹(cosx)]. 2 sin2x
58) x√(x²+ a²)+ a²logₑ{x + √(x²+ a²)} at f'(0). 2a
59) eᵃˣ sin bx. eᵃˣ(a sin bx + b cos bx)
60) log₁₀√{(1- cosx)/(1+ cosx)}. log₁₀e cosecx
61) tan⁻¹√{(1+ cos2x)/(1- cos2x)}. -1
62) ₑ√cotx. ₑ√cotxcosec²x/2√cotx
63) ₃√(1+ x+ x²). (1+ 2x) log 3. ₃√(1+ x+ x²)/2√(1+ x + x²)
64) log sec(ax + b)³. 3a(ax+ b)² tan(ax + b)³
65) log[2x + 4+ √(4x²+ 16x -12). 1/√(x²+4x -3)
66) tan log sinₑx². 2x ₑx²cot(ₑx²) sec²(log sinₑx²)
67) xˣ. xˣ(1+ logx)
68) ₓ(1+ x + x²). ₓ(1+ x + x²)[x +1+ 1/x + (2x+1)log x)
69) (tanx)ˢᶦⁿ ˣ at x=π/4 √2
70) xᶜᵒˢˣ + sin(logₑx). xᶜᵒˢˣ[cosx/x - sinx logₑx]+ 1/x cos(logₑx)
71) sin⁻¹ {2x/(1+ x²)} w.r.t. cos⁻¹{(2- x²)/(1+ x²)}. 1
72) ₓsin⁻¹x w.r.t. sin⁻¹x. ₓsin⁻¹x [√(1- x²)/x sin⁻¹x + log x]
73) tan⁻¹x/(1+ tan⁻¹x) w.r.t. tan⁻¹x. 1/(1+ tan⁻¹x)²
74) cos⁻¹{(1- x²)/(1+ x²) w r t. tan⁻¹{2x/(1- x²)}. 1
75) cos⁻¹{(1- x²)/(1+x²)} w.r.t.sin⁻¹{2x/(1+ x²)}
76) tan⁻¹{2x/(1- x²)} w.r.t sin⁻¹ {2x/(1+ x²)}.
77) log[eˣ {(x-1)/(x+1)³⁾²}]. (x²+2)/(x²-1)
78) {x³ √(x²-12)}/³√(20- 3x) at x=4. 120
79) [x/{1+ √(1- x²)}]ⁿ.
80) cos⁻¹(8x⁴- 8x²+1). -4/√(1- x²)
81) 1/{√(x+a) + √(x+ b)}.
82) log(x²+ x +1)/(x²- x +1). 2(1- x²)/(1+ x²+ x⁴)
83) tan⁻¹ (cosx - sinx)/(cosx + sinx). -1
84) sin⁻¹{2ax √(1- a²x²)}. 2a/√(1- a²x²)
85) cos⁻¹{(3+ 5 cosx)/(5+ 3 cosx)}. 4/(5+ 3cosx)
86) log√{(a cosx - b sinx)/(a cosx + b sinx)}. -ab/(a² cos²x - b² sin²x)
87) sin[2tan⁻¹√{(1- x)/(1+x)}]. -x/√(1- x²)
88) tan⁻¹[√{(a- b)/(a+ b)} tan(x/2]. √(a²- b²)/2(a+ b cosx)
89) y= x+ [1/{x + 1/(x + 1/x)}].
90) 3x⁴- x²y + 2y³= 0. 2x(6x²-y)/(x²- 6y²)
91) 3x³+ y³= 3axy. (x²- ay)/(ax - y²)
92) eˣʸ - 4xy = 2. -y/x
93) ₓ cos⁻¹x.
94) xy = cos(xy) when x=π/2, y= 0. 0
95) xʸ . yˣ = 1.
95) (x+2)/{(x-1)(x+5)}.
96) ₓcos²x.
97) x= ylogₑ(xy). y(x-y)/x(x+y)
98) xʸ+ yˣ = 1.
99) ₐx².
100) (sinx)ᶜᵒˢˣ + e³ˣ.
101) (sinx)ᶜᵒˢˣ +(cosx)ˢᶦⁿˣ.
102) logₑ(xy)= eˣ⁺ʸ + 2.
103) (siny)ˣ = (cosx)ʸ.
104) xᵖ yᑫ = (x+y)ᵖ⁺ᑫ. y/x
105) ax²+ by²+ 2hxy+ 2gx + 2fy + c = 0.
106) x= at², y= 2at.
107) x= a cos t, y= b sin t.
108) x= sin²t, y= tan t.
109) x= a(2t+ sin 2t), y= a(1- cos 2t)
110) tany = 2t/(1- t²), sinx = 2t/(1+ t²)
111) y= tan⁻¹t/√(1- t²)) , y= sec⁻¹(1/(2t²-1).
112) y=sin(3t - 4t³), x= sec(1/(1- 2t²)).
113) y= ₑsin⁻¹t, x= ₑ-cos⁻¹t.
114) y= a sin³t, x= a cos³t.
115) x=a(cos t+ t sin t) and y= a(sint - t cos t) at t=3π/4. -1
116) tany= 2t/(1- t²), cosx = (1- t²)/(1+ t²). 1
117) x=3at/(1+ t³), y= 3at²/(1+ t³).
118) {(a+ x)/(1+ x)}ᵃ⁺¹⁺²ˣ find f'(0).
119) [(tan)ᵗᵃⁿˣ]ᵗᵃⁿˣ.
120) tan⁻¹[{√(1+t²)+ √(1- t²)}/{√(1+ t² - √(1- t²)}]. -t/√(1- t⁴)
121) sin{π/6 eˣʸ} at x=0. √3 π/24
122)
1) If y= 2 tan⁻¹√{(x - a)/(b - x)} then show that (dy/dx)²+ 1/{(x-a)(x - b)}= 0
2) If cos y= x cos(a+ y), prove dy/dx= {cos²(a+ y)}/sin a
3) If sin y= x sin(a+ y), prove dy/dx= {sin²(a+ y)}/sin a= sina/{1- 2x cos a + x²}
4) xʸ = eˣ⁻ ʸ then prove dy/dx = log x/(1+ logx)²= log x/(logex)²
5) If √(1- x²)+ √(1- y²)= a+x - y) then show dy/dx = √{(1- y²)/(1- x²)}
5) {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ then show that f'(0)= (2 log(a/b) + (b²- a²)/ab)(a/b)ᵃ⁺ᵇ
6) If x= a sin 2t(1+ cos2t) and y=a cos2t(1- cos2t), then show 1 +(dy/dx)² = sec²t.
Monday, 20 February 2023
COMPACT REVISION MATRIX
1) If A= 1 2 & B= 1 0
0 1 0 1 then Find
A) A+ B. 2 2
0 2
B) AB. 1 2
0 1
C) BA. 1 2
0 1
2) If x+ y y - z t - x z - t
5 - t 7 + x = z - y x+ z + t 1,2,3,4
3) If A= 1 2 3 & B= 0 1 2
5 4 6 3 4 8
7 8 9 5 3 6
Then Find
A) 2A+ 3B. 2 7 12
19 20 36
29 25 45
B) 3A - 4B. 3 2 1
3 -4 -14
1 12 -9
4) If A+ B= 2 2 & 2A+ 3B = 5 4
0 2 0 5 then Find the Matrix A and B. 1 2 & 1 0
0 1 0 1
5) If 2 -1
1 3 then show A²- 5A+ 7= 0.
6) If A= 1 -1 0 & B= 2 2 -4
2 3 4 -4 2 -4
0 1 2 2 -1 5 then prove that AC = CA = 6.
7) If A + I= 1 3 4
-1 1 3
-2 -3 1 then Find the value
A) A+ I. 0 3 4
-1 0 3
-2 -3 -1
B) A - I -1 3 4
-1 -2 3
-2 -3 -1
8) If A= 0 2 & B= 0 -1
1 1 1 0 then show that (A+ B)(A - B)≠ A²- B².
9) If A= 1 2 2
2 1 2
2 2 1 then Prove that A² - 4A - 5I= 0 Where I is a unit Matrix of order 3x3.
10) If A= 1 2 3 & B= 1 2
3 -2 1 2 0
-1 1 then Prove (AB)'= B' A'.
11) Show that 2 -3 -4
-1 3 4
1 -2 -3 is an idempotent Matrix.
12) 1/√2 1/√2
-1√2 1/√2 show that it is an orthogonal Matrix.
13) A= 1 0 0
-1 -2 -1
-2 3 -2 Express as the sum of a symmetric and a skew-symmetric Matrix.
14) If A= 1 2
3 5 then prove that,
A. adj A/|A| = adj A/|A| . A = I
15) A= 1 0 -1
1 2 3
0 -1 2 then Find Inverse of A. 7/8 1/8 1/4
-1/4 1/4 -1/2
-1/8 1/8 1/4
16)
B) Solve by Matrix OR Martin's Rule:
1) x+ y+ z= 4, 2x- y+ 3z= 1; 3x+ 2y- z= 1.
COMPACT REVISION (DETERMINANT)
A) Without Expanding Prove:
1) 17 38 97
19 60 99 = 0
18 59 98
2) 8 5 3
9 7 2 = 0
6 1 5
3) 1 a b+ c
1 b c+a = 0
1 c a+ b
4) a d 3a- 4d
b e 3b- 4e = 0
c f 3c - 4f
5) x a b
a x b =(x -a)(x -b)(x+a+b)
a b x
6) a+1 2 3
3 a+2 4 prove a -1 is the factor
4 4 a+3
7) 0 a b
-a 0 c = 0
-b -c 0
8) a b c
c a b = a³+b³+c³- 3abc
b c a
9) 1+a 1 1
1 1+b 1 =abc(1/a +1/b +1/c)
1 1 1+ c
10) a-b-c 2a 2a
2b b-c-a 2b = (a+ b+ c)³
2c 2c c-a-b
11) a+ b +2c a b
c b+c+ 2a b =2(a+b+c)³
c a c+ a + 2b
12) a+ b + c -c -b
-c a+ b + c -a
-b -a a+ b + c
= 2(b+c)(c+a)(a+b)
13) a²+10 ab ac
ab b²+10 bc is divisible by 100
ac bc c²+10
14) y+z z+x x+y 2x 2y 2z
z+x x+y y+z = 2y 2z 2x
x+y y+z z+x 2z 2x 2y
15) 1 a a²-bc
1 b b²-ca = 0
1 c c²-ab
16) a b c
a² b² c²
b+c c+a a+b
=(a+ b+ c)(a- b)(b - c)(c - a)
17) y²+ z² xy zx
xy z²+ x² yz = 4x²y²z²
zx yz y²+ z²
18) x²+y²+ 1 x²+2y²+3 x²+ 3y²+4
y² +2 2y²+ 6 3y²+8 =x²y²
y²+ 1 2y² + 3 3y²+4
19) -bc bc+ b² bc+ c²
ca +a² -ca ca+ c²= (ab+bc+ca)²
ab +a² ab+ b² -ab
B) Solve by Cramer's Rule:
1) 2x - y = 1; 3x + 2y = 5. 1,1
2) x+ 2y - z = 9; 2x - y +3z = -2 ; 3x+ 2y +3z = 9. 2,3,-1
3) -x+ 2z+1 = 0 ; 2x - y -4z =2 ; y +2z = 4. 5,0,2
Thursday, 9 February 2023
LAST TIME REVISION (XII) (2024/25)
Set - I
2) Prove that the relation R in the set A={ 1, 2, 3, 4, 5} given R={(a, b): |a - b| is even}, is an equivalence relation.
3) Let f: N-->N be defined by
(n + 1 )/2 , if n is odd
f(x)= n/2, if n is even
Find whether the function f is bisective. f is not bijective, f is not one-one
1) let * be binary operation on the set of rational numbers given as a* b = (2a- b)², a, b ∈ Q. Find
A) 3*5. 1
B) 5*3. 49
C) Is 3*5=5*3? No
1) Show that the relation S in the set R of real numbers, defined as A={(a, b) ∈ R and a ≤ b³} is neither reflexive, nor symmetric, nor transitive.
2) Let f: R---> R be defined as f(x)= 10x+7. Find the function g: R---> R such that g o f = f o g = Iᴿ⁺ . (x-7)/10
1) Find the value of
A) cot(π/2 - 2 cot⁻¹√3). √3
B) tan⁻¹{2sin(2 cos⁻¹(√3/2))}. π/3
C) tan[2 tan⁻¹(1/2)- cot⁻¹3]. 9/13
2) Solve:
A) sin{cot⁻¹(x+1)}= cos(tan⁻¹x) -1/2
B) tan⁻¹{(x-2)/(x-4)} + tan⁻¹{(x+2)/(x+4)}= π/4. ±√2
C) cos⁻¹x + sin⁻¹(x/2)=π/6. 1
3) Prove:
A) tan⁻¹(3/4) + tan⁻¹(3/5) - tan⁻¹(8/19)=π/4.
B) sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36)
C) tan⁻¹(1/4) + tan⁻¹(2/9) =1/2 sin⁻¹(4/5)
4) Find the principal value of tan⁻¹√3 - sec⁻¹(-2). -π/3
5) If cos⁻¹x + cos⁻¹y + cos⁻¹z=π then show that x² + y²+ z²+ 2xyz = 1.
1) A= 3 1
7 5 find x and y so that A² + xI = yA. Hence, find A⁻¹. 8,8 & 5/8 -1/8
-7/8 3/8
2) Find the adjoint of 1 0 -1
3 4 5
0 -6 -7 and Hence find A⁻¹.
2 6 4 & 1/10 3/10 1/5
21 -7. -8. 21/20 -7/20 -2/5
18 6 4 -9/10 3/10 1/5
3) solve by Martin Rule:
2/x +3/y. +10/z = 4 ; 4/x -6/y +5/z. = 1; 6/x + 9/y - 20/z = 2. 2,3,5
4) Find A⁻¹ if A= 0 1 2
1 2 3
3 1 1 Hence, solve the system equation y+ 2x +8= 0, x+ 2y +3z +14= 0, 3x+ y +z +8= 0.
1/2 -1/2 1/2
-4 3 -1
5/2 -3/2 1 and -1,- 2, -3
5) Solve: x+ y+ z = 9, 2x+ 5y +7z = 52, 2x+ y -z = 0. 1, 3, 5
6) If A=1 3 B= y 0 & C= 5 6
0 x 1 2 1 8 With the relation 2A + B = C, then find the value of (x+y). 6
7) Write a 2 x 2 Matrix which is both symmetric and skew-symmetric. 0 0
0 0
8) Given A= 0 2b -2
3 1 3
3a 3 -1 is symmetric, find the value of a and b. 2/3,3/2
9) A= 1 3 -2 1 -2 -3
-3 0 -2 -2 4 7
2 1 0 find A⁻¹. -3 5 9
10) If A= -5 1 3 & B= 1 1 2
7 1 -5 3 2 1
1 -1 1 2 1 3 find AB. Then solve the following equations by Matrix method. x+ y+ 2z = 1, 3x+ 2y +z = 7, 2x+ y +3z = 2. 2, 1, -1
1) without expanding prove :
A) a+b b+ c c+a
c a b = 0
1 1 1
2) a b c
a² b² c²
b+ c c+a a+ b
= (a- b)(b - c)(c- a) (a+b + c)
3) 1+ a²- b² 2ab -2b
2ab 1- a²+ b² 2a
2b -2a 1- a² - b²
= (1+ a²+ b²)³
4) - a² ab ac
ba - b² bc = 4a²b²c²
ca cb - c²
5) 1+ a² ab ac
ba b²+1 bc = 1+ a²+ b²+ c²
ca cb c²+1
6) a b - c c+ b
a+ c b c- a
a- b b+ a c
= (a+ b+ c)(a²+ b²+c²)
1) If the function f(x) given by:
2ax + b, if x > 1
f(x) = 11, if x= 1
5ax -2b, if x< 1 is continuous at x= 1, then find the value of a and b. 3,2
2) Find the value of k, so that the function f Defined by:
k cos x/(π - 2b), if x ≠ π/2
f(x) = 3, if x= π/2
is continuous at x=π/3. 6
3) Find the value of k, so that the function f Defined by:
kx + 1, if x ≤ π
f(x) = cosx, if x> π
5ax -2b, if x< 1 is continuous at x =π. -2/π
1) 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+3x -x², x> 2. f(x) is not differentiable at x=1, f(x) is differentiable at x= 2
2) For what value for k for the function defined by f(x) = k(x²+2), if x≤0
4x +6, if x> 0 is continuous at x= 0 ? Hence check the differentiability of f(x) at x= 0. k= 3, f is not differentiable at x= 0
1) DIFFERENTIATE THE FOLLOWING:
A) tan⁻¹√{(1+ cosx)/(1- cosx)}. -1/2
B) tan⁻¹(secx + tanx). 1/2
C) tan⁻¹√{(a- x)/(a+x)}. -a/2√(a²- x²)
D) log (secx + tanx). secx
E) sin⁻¹(x³). 3x²/√(1- x⁶)
F) log tan(π/4 + x/2). Secx
G) sin⁻¹{(2x/(1+ x²)} w.r.t. tan⁻¹x. 2
H) If tan⁻¹{√(1- x²)/x} w.r.t. cos⁻¹(2x√(1- x²). -1/2
2) If y= [x +√(x²+ a²]ⁿ then show dy/dx= ny/√(x² + a²).
3) If y= log [x +√(x²+ a²] then show (x²+ a²) d²y/dx²+ x dy/dx = 0.
4) If xʸ = eˣ⁻ʸ, show dy/dx = log x/(1+ log x)².
5) If y= (sin⁻¹x)², show (1- x²) d²y/dx² - x dy/dx - 2 = 0.
6) If eʸ(x +1)= 1, show that d²y/dx² = (dy/dx)².
7) If √(1- x²) + √(1- y²)= a(x - y), show that dy/dx =√{(1- y²)/(1- x²)}.
8) If y= [x +√(1+ x²]ⁿ then show (1+ x²)d²y/dx²+ x dy/dx = n²y.
9) If xᵐ yⁿ = (x+y)ᵐ⁺ⁿ, then show dy/dx = y/x.
10) If y= xˣ then show d²y/dx²- 1/y (dy/dx)²- y/x = 0.
11) If x√(1+ y)+ y√(1+ x)=0(x≠y), then show dy/dx = -1/(1+ x)².
1) lim ₓ→₀ (sinx - x)/x³. -1/6
2) lim ₓ→₀ (x - tan x)/x³. -1/3
3) lim ₓ→₀ (1 - tan x)/cos 2x. 1
4) lim ₓ→₀ (1 - logx - x)/(1- 2x+x²). -1/2
5) lim ₓ→₀ (x - tan⁻¹x)/(x - sinx). 2
6) lim ₓ→π/2 (tan5x)/tanx. 1/5
7) lim ₓ→₀ (cosecx - 1/x) 0
8) lim ₓ→π/2 (x tanx - π/2 secx). -1
9) lim ₓ→π/2. (Cosx log tanx) 0
10) lim ₓ→₀ (1+ sinx)ᶜᵒᵗˣ e
1) Verify Rolle's theorem for the following:
A) f(x)= x²- 5x +6 on [2,3]. c=2.5
B) f(x)= log{(x²+ ab)/x(a+ b)} on (a,b), where 0 < a < b. c=√(ab)
C) f(x)= sin x - sin 2x on [0,π]. c= 32°32' , 126° 23'
2) Verify Lagrange 's mean value theorem for the following:
A) f(x)= (x-3)(x-6)(x-9) on [3,5]. 4.8
B) y=√(x-2) on [2,3], where the tangent is parallel to the chord joining the end points of the curve. (9/4,1/2)
1) The side of an equilateral triangle is increasing at the rate of 2cm/s. At what rate is its area increasing, when the side of the triangle is 20 cm? 20√3 cm²/s
2) Find intervals in which the function f(x)= 3x⁴- 4x³- 12x²+5 is
A) strictly increasing. (-1,0)U(2, ∞)
B) strictly decreasing. (-∞,-1) & (0,2)
3) Find the intervals in which the function f(x)= (x-1)(x+2)² is
A) increasing. (-∞,-2)U(0,∞)
B) decreasing. (-2,0)
4) Find intervals in which the function f(x)= sinx+cos x, 0≤ x≤ 2π is
A) increasing. (0,π/4) &(5π/4,2π)
B) decreasing. (π/4,5π/4)
1) Using differentials, find the approximate value of √49.5. 7.035
2) The area of a circle of radius r increases at the rate of 5 cm²/s.
A) Find the rate at which the radius increases. 5/2πr cm/s
B) Also, find the value of this rate when the circumference is 10 cm. 0.5 cm/s
1) Find the slope of the tangent to the curve y= 3x²- 4x at the point whose x-coordinate is 2. 8
2) Find the points on the curve y= x³- 11x²+5 at which the equation of the tangent is y= x -11. (2,-9) and (-2,19)
3) Find the equation of the tangent to the curve 3= x²+ 3y which is parallel to the line y - 4x +5= 0. 4x - y +13= 0
4) Find the equation of the normal to the curve y= x³+ 2x+6, which is parallel to the line 14y +x +4= 0. x + 14y= 254
5) Find the equation of the tangent and normal to the curve x= 1- cos k, y= k - sin k at k=π/4. 4√2 x +(8- 4√2)y =π(2 -√2)
1) An open box with a square base is to made out of a given quantity of cupboard of area C² square units. Show that the maximum volume of the box is C³/6√3 cubic units.
2) If the sum of lengths of the hypotenuse and a side of a right angled triangle is given. Show that the area of a triangle is maximum, when the angle between them is π/3.
3) If the length of three sides of a trapezium other than the base is 10cm each then find the area of the trapezium when it is maximum. 75√3cm²
4) Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r cm. 4πr³/3√3
5) Show that the semivertical angle of a right circular cone of maximum volume and given slant height is tan⁻¹√2.
6) Find the point on the curve y²= 2x which is nearest to the point (1,4). (2,2)
7) Three numbers are given whose sum is 180 and the ratio of the first two is 1:2. If the product of the numbers is greatest, find the number. 40,80,60
8) Prove that the Area of a right angled triangle of a given hypotenuse is maximum when the triangle is isosceles.
9) If the sum of the length of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.
10) A closed right circular cylinder has volume 539/2 cubic units. Find the radius and the height of the cylinder, so that the total surface area is minimum. 7/2, 7
11) Find the maximum volume of the cylinder which can be inscribed in a sphere of 3√3cm. radius. (Leave the answer in terms of π). 108π cm³
12) Find the maximum and minimum values of:
A) -(x -1)²+ 10. 10, nil
B) sin 2x +5. 6,4
C) sinx - cos x. .√2, -√2
1) ∫ tan(2 log x)/x dx. 1- 1/2 log|cos(2logx)|
2) ∫ cosecx/log(tan(x/2)) dx. Log|log(tan(x/2))|
3) ∫ cos x/(sinx √sinx) dx. 2log|√sinx +1|
4) ∫e²ˣ/(2+ eˣ) dx. eˣ - 2 log(2+ eˣ)
5) ∫secx log|secx + tan x| dx. 1/2 (log|secx + tanx|)²
6) ∫x³ log x dx. x⁴/16 (4 log x -1)
7) ∫x tan⁻¹(x²) dx. x²/2 tan⁻¹x² 1/4 log(1+ x⁴)
8) ∫ (sin⁻¹x)² dx. x(sin⁻¹x)² + 2√(1- x²) sin⁻¹x - 2x
9) ∫ (cos⁻¹x)/x². (cos⁻¹x)/x + log|1+ √(1- x²)/x |
10) ∫ (x -1)eˣ/(x+1)³ dx. eˣ/(x+1)²
11) ∫ logx/(1+ log x)² dx. x/(1+ log x)
12) ∫ eˣ{(2+ sin2x)/(1+ cos2x)} dx. eˣ tan x
13) ∫ eˣ{(1- sin x)/(1- cosx)} dx. - eˣcot(x/2)
14) (sin⁻¹x)² dx. x(sin⁻¹x)² - 2{-√(1-x²) ∫sin⁻¹x + x}
15) ∫ x/(x²+ x +1) dx. 1/2 log|x²+ x+1| - 1/√3 tan⁻¹{(3x+1)√3}
16) ∫ (x⁴+1)/(x²+1) dx. x³/3 - x + 2 tan⁻¹x
17) ∫ x²/(x⁴+1)dx. 1/2√2 tan⁻¹{(x²-1/x√2} + 1/4√2 log|(x²- x√2+1)/(x²+ x√2+1)|
18) ∫ dx/(sin⁴x+cos⁴x)dx. 1/√2 tan⁻¹{(tan²x -1)/√2 tanx}
19) ) ∫ x²/(x⁴²- 4x+3)dx. x + 9/2 log|x -3| - 1/2 log|x-1|
20) ) ∫ (x²-5x-1)/(x⁴+x²+1)dx. 1/2 log|{(x²-x+1)/(x²+x+1)} - 5/√3 tan⁻¹{(2x²+1/√3}
21) ) ∫ (4x+5)/(2x²+ x-3)dx. 2 √(2x²+ x -3) + 2√2 log|(x+ 1/4 + √(x²+ x/2- 3/2)|
22) ¹₀∫ sin⁻¹{2x/(1+ x²)} dx. π/2 - log2
23) ∫ (3 sinx + 4 cos x)/(sinx + cos x) dx at (π/2,0). 7π/4
24) ¹₀∫ x tan⁻¹x dx. π/4 -1/2
25) ²₁∫ 2/(4x²-1) dx. 1/2 Log (9/5)
26) ∫ 4x sinx/(1+ cos x) dx at (π, 0). π²
27) ³₁∫ [|x -1| + |x -2|+ |x -3|] dx. 5
28) ∫ x sin x/(1+ cos²x) dx at (π,0). π²/4
29) Prove: x tan x/(secx + tanx) dx at π,0) =π/2 (π -2)
30) ²₁∫ (x²+5x) dx. 59/6
31) ³₋₃∫|x+2| dx. 13
32) ∫sin2x log tanx dx at (π/2,0). 0
33) ∫ log tan x dx at (π/2,0). 0
34) ∫ √sinx/(√sinx + √cosx) at π/2,0). π/4
Solve:
1) cos dy/dx + cos 2x = cos 3x. y= sin2x - 2sin x - x + log|secx + tan x|
2) log(dy/dx)= 2x - 3y. 2e³ʸ= 3e²ˣ + c₂ where c₂ = 6c₁, c₂ is an arbitrary constant.
3) 2 dy/dx = y/x + y²/x². 2 log |(y-x)/y| = log |x|
4) x dy/dx - y = √(x²+ y²). y+ √(x²+ y²) = C x²
5) dy/dx + sin(x + y)= sin (x - y). Log|cosy - cot y| = -2 sinx
6) x dy/dx = y - x tan(y/x). x sin(y/x) = C
7) Find the particular solution of the differential equation (3xy + y²)dx + (x²+ xy) dy = 0, from x= 1 and y= 1. y² x² + 2yx³ = 3.
8) y dx -(x + 2y²) dy = 0. x/y = 2y+ c
9) y log x dy/dx + y = 2/x log x, x> 0. y log x = -2/x (1+ cosx) + c, x> 0
10) (1+ y + x²y) dx + (x + x³) dy= 0. xy = - tan⁻¹x + c
11) cos²x dy/dx + y = tan x. yeᵗᵃⁿˣ = e ᵗᵃⁿˣ (tanx -1)+ c
12) (x+1) dy/dx - y = e³ˣ(x +1)². y= 1/3 (x+1)e³ˣ + (x+1) c
13) dy/dx + y(1- 2x))x² = 1. y= x²(1+ e¹/ˣ.
7) x dy/dx + y = x cos x + sin x given y(π/2)= 1
1) Find the probability that a leap year, selected at random, will contain 53 Sundays. 2/7
2) A word consists of 9 letters of 4 consonants and 5 vowels. Three letters are chosen at random. Find the probability that more than one vowel is chosen. 25/42
3) If two balls are drawn from a bag containing three red balls and 4 blue balls , find the probability that
A) they are of the same colour. 3/7
B) they one of different colours. 4/7
4) Two horses are considered for a race. The probability of selection of the first horse is 1/4 and that of the second is 1/3. What is the probability that
A) both of them will be selected? 1/12
B) only one of them will be selected? 5/12
C) none of them will be selected? 1/2
5) Bag A contains 6 red and 5 blue balls and another bag B contains 5 red and 8 blue balls. A ball is drawn from bag A without seeing its colour and it is put into the bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour. 93/154
6) On answering a question of a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3, what is the probability that the students knows the answer, given that he answered it correctly? 9/11
7) Bag A contains 3 white and 4 black balls and bag B contains 4 white and 5 black balls. One ball is transferred from bag A to bag B and then a ball is drawn at random from bag B. If the ball so drawn is white, find the probability that the transferred ball is black. 16/31
8) A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. 3/8
9) Suppose that 5% of men and 0.25% of women have grey hair. A grey- haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females. 20/21
10) Find the mean and variance for the following probability distribution:
X: 0 1 2 3
P(x): 1/8 3/8 3/8 1/8 3/2,3/4
11) Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and standard deviation of the number of kings. 2/13, 0.373
12) If the mean and variance of a binomial distribution are 9 and 6 respectively, find the distribution. (2/3 + 1/3)²⁷
13) Find the binomial distribution when the sum of its mean and variance for five trials is 4.8. (1/5+ 4/5)⁵
14) A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die. 625/3x6⁵
1) Find a unit vector perpendicular to each of the vector a+ b and a - b where a= 3i+ 2j+ 2k and b= i+ 2j - 2k. 2i/3 -2j/3 -k/3
2) The vectors -2i+ 4j +4j and -4i-2k represent the diagonal BD and AC of a parallelogram ABCD. Find the area of the parallelogram. 6√5
3) For any three vectors a, b, c, prove that [a+ b, b+ c c+ a]= 2[a b c]
4) Prove by vector method that in any triangle ABC, a/sinA = b/sinB = c/sinC.
5) Find the length and the foot of the perpendicular drawn from the point (2,-1,5) to the line (x -11)/10 =(y+2)/-4 = (z+8)/-11. (1,2,3), √14
6) Find the equation of the line passing through the point (-1,3,-2) and perpendicular to the lines x/1= y/2 = z/3 and (x+2)/-3= (y-1)/2 = (z+1)/5. (x+1)/2 = (y-3)/-7 = (z+2)/4.
7) Find the shortest distance between the lines (x-1)/2= (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5. 1/√6
8) Find the shortest distance between the lines (x-8)/3= (y+9)/-16 = (z-10)/7. (x-15)/3 = (y- 29)/8 = (5-z)/5. Also, find the equation of this line. 14, (x-5)/2 = (y-7)/3 = (z-3)/6.
9) Find the equation of the plane passing through the intersection of the planes 2x+ 3y - z+1= 0 and x+ y - 2z+3= 0 and perpendicular to the plane 3x - y - 2z-4= 0. 7x+ 13y +4z-9= 0
10) Find the equation of the plane passing through the intersection of the planes 2x+ y +z-1= 0 and 2x+ 3y - z+4= 0 and parallel to the x-axis y -3z+6= 0
11) Find the distance of the point (1,-2,3) from the plane x- y +z-5= 0 and x+ y - 2z+3= 0 measured parallel to the line (x+1)/2 = (y+3)/3 = (z+1)/6. 1 unit
12) Find the equation of the plane through (1, 2, 3) and perpendicular to the plane x+ y + 2z- 3= 0 and 3x+ 2y +z= 4. 3x - 5y + z + 4= 0
13) Find the equation of the plane passing through (2,1, 0) and (5,0,1) and (4,1,1). x + y - 2z -3 = 0
1) Draw a rough sketch of the curve y²+1= x, x≤ 2. Find the area enclosed by the curve and the line x=2. 4/3
2) Find the area enclosed by the curve y²= x and the line y+ x=2. 9/2
3) Find the area enclosed by the curve y= x² and x=y². 1/3
4) Draw a rough sketch of the curve y= (x-1)² and y=|x -1|. Hence find the area enclosed by the curve. 1/3
5) Find the area of the region enclosed by the curve 4y - y²= x and the y-axis. 32/3
6) Find the area enclosed by the curve y= 2x - x² and x=y. 1/6
7) Find the area of the region in the first quadrant enclosed by the x-axis, the line y= x and the circle x²+ y²= 32. 4π
1) A television manufacturer finds that total cost for the production and marketing of x television sets is
C(x)= 300 x²+ 4200x+ 13500
Each product is sold for₹8400. Determine the break even points. 5,9
2) The average cost function associated with producing and marketing x units of an item is given by AC= 2x -11+ 50/x.
A) Find the total cost function and the marginal function. 2x²-11x+ 50, 4x-11
B) The range of values of the output x, for which AC is increasing. x> 5
3) The average cost function AC for a commodity is given by AC= x +5+ 36/x in terms of output x.
A) Find the total cost function and the marginal function. x²+5x+6, 2x+5
B) The range of values of the output x, for which AC is increasing. x> 6
4) Given that the total cost function AC for a commodity is given by C(x)= x³/3+ 3x²- 7x+ 6 in terms of output x.
A) Find the marginal cost(MC). x²+6x-7
B) find the average cost(AC). x²/3 + 3x -7+ 16/x
C) prove that marginal average cost (MAC) = x(MC)- C(x)/x².
5) The marginal cost function MC for a product is given by MC =2/√(2x+9) and the fixed cost is ₹14. Find the average cost for 8 units of output. ₹9/4
6) Find the total revenue and demand function if the marginal revenue function (MR)= 100- 9x². 100x - 3x³, p= 100- 3x²
1) Out of the following two regression lines, find the line of regression of y on x: 3x+12y=7; 3x+ 9y=46. 3x+12y=7
2) The equation of two lines of regression are 4x+3y= -7 and 3x+4y +8=0.
A) find the mean values x and y. -4/7, -11/7
B) find the regression coefficients bᵧₓ and bₓᵧ. -3/4, -3/4
C) Find the correlation coefficient between x and y. -3/4
3) The regression lines are represented by 4x+10y=9 and 6x+3y=4. Find the regression line of x on y and y on x. 4x+10y=9 and 6x+3y=4
4) Find the equation of the lines of regression for the data:
X: 1 2 3 4 5
Y: 7 6 5 4 3
And hence find an estimate of the variable y for x= 3.5 from the appropriate line of regression. Y on x is x+y=8, x on y is x+y=8, y=4.5
1) A mill owner buys two types of machines A and B for his mills. Machine A occupies 1000 sq.m of area and requires 12 men to operate, while Machine B occupies 1200 sq m of area and requires 8 men to operate. The owner has 7600 sq.m of area available and 72 men to operate the machines. If machine A produces 50 units and machine B produces 8 units daily, how many machine of each type should be buy to maximize the daily output? Use linear programming to find the situation. A=4; B= 3
2) A new cereal formed of a mixture of bran and rice, contains atleast 88g of protein and atleast 36mg of iron. Knowing that bran contains 89g of protein and 40mg of iron per kilogram and that rice contains 100g of protein and 30mg of iron per kilogram, find the minimum cost of producing a kilogram of this new cereal if bran costs ₹88 per kg and rice costs ₹25 per kg. ₹26.8 at the point (0.6,0.4)
SHORT ANSWER QUESTIONS
Set -2
1) Solve: cos(sin⁻¹x)= 1/7. ±4√3/7
2) ∫ sin³x cos²x dx. -1/3 cos³x + 1/5 cos⁵x + c
3) Use Matrix solve: 2x+3y= 10; x+ 6y= 4. 16/3,-2/9
4) Find dy/dx: y+ siny = x². x sec²(y/2)
5) Let f : X--> Y be a function defined on a relation R on X given by R={(a,b): f(a)= f(b). Show that R is an equivalence relation on X.
6) If f: R---> R is defined by f(x)= (3 - x)¹⁾³, then find f o f(x). x
7) If P(A)= 7/13, P(B)= 9/13 and P(A∩ B)= 4/13, find P(A/B). 4/9
8) Consider f: R ₊ -->[9,∞) given by f(x)= 5x²+ 6x -9. Prove that f is invertible with f⁻¹(y)= {√(54+5y) -3}/5.
9) The mean number of success of a binomial distribution (p+q)ⁿ is 240 where p is the probability of success. The standard deviation is 12. Calculate the values of n, p and q. 2/5,3/5,600
10) Find the value of tan(2tan⁻¹(1/5)). 5/12
11) The vector 2i+ j - k is perpendicular to i+ 4j - Mk, if M is equal to
A) 0 B) -1 C) -2 D) 2 -2
12) Solve the determinant x² x 1
0 2 1= 28
3 1 4
13) Find the Inverse of -2 5
3 4
14) Using Matrix, solve : 5x+7y =-2; 4x+6y =- 3. 9/2,-7/2
15) The probability density function y of a continuous variable x is given by y= k/x, 0≤ x ≤ 2 and y = 0 for all other values of x. Calculate the value of k and the probability that x < 1. 2, 1/4
16) Solve: cos⁻¹x+ sin⁻¹(x/2)= π/6. ±1
17) dy/dx of cos⁻¹√{(1- cosx)/2}. -1/2
19) If f is an invertible function defined as f(x)= (3x-4)/5, then find f⁻¹(x). (5x+4)/3
20) Find inverse of A= 1 2
3 7
21) The binary operation * R x R --> R is defined as a * b= 2a+ b. Find (2*3)*4. 18
22) Find the probability of throwing a total of 3 or 5 or 11 with two dice. 2/9
23) Find the magnitude of the vector ax b if a= 3i+ 4j and b= 5j +12k. 15√17
24) ¹₀∫ dx/(2x -3). -1/2 log 3
25) dy/dx of √{(1- cos2x)/(1+ cos 2x)}. sec²x
26) A company wants to lunch a new product. It invested ₹37500 as fixed cost and ₹200 per unit as the variable cost of production. The revenue function for the sale of x units is given by 4825x - 125x². Find the break even points. 12, 25
27) In a simultaneously throw of two dice, find the probability of getting a total of 7. 1/6
28) Does the Lagrange 's mean value theorem apply to f(x)= x¹⁾³ ; -1≤ x≤1? N
29) ∫ x²/(1- x) dx. -x²/2+ x + log|1- x|
30) Form the differential equation y= aeᵇˣ, where a and b are arbitrary constants. y(d²y/dx²)= (dy/dx)²
31) Find the value of M so that the vectors 2i+ 3j+ 4k and 4i - 2j - Mk may be perpendicular to each other. -1/2
32) If sin⁻¹x + sin⁻¹y = 2π/3 then find the value of cos⁻¹x + cos ⁻¹y. .π/3
33) Find the differential equations of the family of curves y= Aeˣ + Be⁻ˣ, where A and B are arbitrary constants. d²y/dx²= y
34) Find the value of p for which the vectors a= 3i+ 2j+ 9k and b= i+ pj + 3k are parallel. 2/3
35) You roll a pair of dice and record the sum of numbers on the top of these dice. Find the probability that the score is atleast 10. 1/6
36) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that it is neither an Ace nor a king.
37) ∫ 8/{(x+2)(x²+4)} dx log|x+2| - 1/2 log|x²+4)| + tan⁻¹(x/2)
38) Find the Cartesian equations of the line which passes through the point (-2,4,-5) and is parallel to the line (x-3)/3 = (y-4)/5 = (z+8)/6. (x+2)/3 = (y-4)/5 = (z+5)/6.
39) If a=3i+ j +2k and b= 2i - 2j+ 4k, find the magnitude of a x b. 8√3
40) if y= tan⁻¹[{√(1+ x²) - 1}/x] then find dy/dx. 1/2(1+ x²)
41) Determine the binomial distribution whose mean is 9 and whose standard deviations is 3/2. Find the probability of obtaining one success at the most.
43) if f(x)= 27x³ and g(x)= x¹⁾³, then find g o f(x). 3x
44) If f: R---> R defined by f(x)= (3x+5)/2 is an invertible function, then find f⁻¹(x). (2x-5)/3
45) What is the probability that a number selected from the numbers 1,2,3,4,...25 is a prime number? 9/25
46) ∫ x/{(x-2)(x -1)} dx. log|(x+2)²/(x-1)|
47) ᵅ₀∫ √x/{√x +√(a- x)} dx. 9/2
48) ∫ sec²x/cosec²x dx. tanx - x
49) Find the differential equations for y= a cos(x+ b), where a, b are parameter. d²y/dx² + y= 0
51) Find the area between the curve y= 2x² + 3x, the x-axis and the ordinate x= 3. 63)2
52) Without expanding the determinant, show that
1 a b+ c
1 b c+a = 0
1 c a+ b
54) ²₀ ∫ √x/{√x+ √(3- x)} dx. 1/2
55) Determine the order and degree of the differential equations d²y/dx² = cos 3x+ sin 3x. Also, state whether it is linear or bon-linear. 2, 1, linear
56) The regression equation of y on x is 3x - 5y =13 and the regression equation of x on y is 2x - y= 7. Estimate the value of x when y= 10. 8.5
57) If eˣ + eʸ = eˣ⁺ʸ prove that dy/dx= eʸ⁻ˣ = 0.
58) Find K if ᴷ₀∫ dx/(2+ 8x²)=π/16. 1/2
59) If y= sin⁻¹√{(1+ x²)/2} , find dy/dx. x/√(1- x⁴)
60) Using differentials, find the approximate value of √49.5. 7.035
61) Differentiate xˣ. xˣ(log x +1)
62) Find the local maxima and minima for the function x³- 12x. 2 at -16 and-2 16
63) ∫ dx/(1+ cosx). Cosecx - cot x
64) ∫ √(1+ cos 2x) dx at (π/2,0). √2
65) ¹₀∫ 2x/(1+ x²) dx. log 2
66) Solve: 1/sin⁻¹x dy/dx = 1. y= x sin⁻¹x + √(1- x²)
67) If y= x + tan x, prove cos²x d²y/dx² - 2y + 2x = 0.
68) lim ₓ→₃ (x⁴-81)/(x-3). 108
69) ∫ x sin⁻¹x/√(1- x²) dx. -√(1- x²) sin⁻¹x + x
70) If a= 3i + j + 2k and b= 2i- 2j + 4k, find the magnitude of a xb. 8√3
71) ∫ sin³x dx at (π/4,-π/4). 0
72) ⁵₂∫(3x²-5). 102
73) Solve: dy/dx= e³ˣ⁻²ʸ+ x²e⁻²ʸ. e²ʸ/2= e³ˣ/3 + x³/3.
74) find dy/dx if y tan x - y² cosx + 2x = 0. (y sec²x+ y² sinx+2)/(2y cosx - tanx)
75) A bag contains 5 white and 3 black balls and 4 balls are successively drawn out and not replaced. Find the probability that they are alternately of different colours. 1/7
76) Out of the following two regression equation lines, find the line of regression of y on x; 3x+ 12y = 9 ; 3y+ 9x = 46. 3x+ 12y = 9
77) Find the are enclosed between the line 3x - 2y +12= 0 and the parabola y = 3 x²/4. 27 units
78) Solve: dy/dx + 2y tan x = sinx. y= cos x + C cos²x.
79) Find a unit vector perpendicular to the plane a and b, where a= i - 2j + 3k and b= i + 2j - k. ±1/√3(-i + j+k)
80) ₁²∫ 2/(4x²-1) dx. 1/2 log(9/5)
81) If P(A)= 0.4, P(B)= 0.3 and P(A∩B)= 0.2, find
A)P(A/B). 2/3
B) P(B/A). 1/2
82) ∫ x/(1+ sinx) dx at (π,0). π
83) solve: 7x = 6y - 8 and 12y = 9+ 5x by Martin rule. -7/9,23/54
84) If A= 5 3
-1 -2 find A⁻¹.
85) Verify Rolle's theorem for the function f(x)= sinx in the interval (π/4, 3π)4).
86) ²₋₂∫ |x +1| dx. 5
87) Find the particular solution of the differential equations (x-y)(dx+ dy)= dx - dy, given that y=-1, when x=0. Log|x -y| = x + y+1
88) The probability of an event X occurring is 0.5 and that of Y is 0.3. if X and Y are mutually exclusive events, find the probability of neither X nor Y occuring. 0.2
89) Ten cards numbered 1 to 10 are placed in a box, and one card is drawn randomly. If it is known that the number on the card drawn is more than 3, what is the probability that it is an even number? 4/7
90) find the inverse Matrix of 1 2
3 7
91) The probability of solving a problem by three students A, B and C are 1/2, 1/3, and 1/4 respectively. The probability that the problem will be solved is:
a) 1/4 B) 1/2 C) 3/4. D) 1/3. 3/4
92) Find M if the vectors a= i+ 3j + k, b= 2i- j - k, and c= Mi+ 3k are coplanar. 7
93) The function f(x)= x⁴- 62x² + kx +9 attains its maximum value on the interval [0,2] at x=1. Find the value of k. 120
94) If y= 2 sinx + 3cosx, find the value of d²y/dx² + y. 0
95) Find the area of a parallelogram whose adjacent dides are given by a= 3i- 6j + 2k and b=2 i+ j -2k. 5√17
96) Using Martin rule, solve the following system of equation: 6x+ y - 3z=5, x+ 3y - 2z=5, 2x+ y +4z=8. 1,2,1
97) Without expanding prove the determinant. x+y y+z z+ x
z x y =0
1 1 1
98) If A= x 4 1 B= 2 1 2 & C= x
1 0 2 4
0 2 -4 -1
And the relation ABC= 0, find x. -2, or -1
99) dy/dx of cos⁻¹{(1- x²)/(1+ x²)}. 2/(1+ x²)
100) ∫ dx/√(4x²-9). 2 log|x + √(x² - 9/4)|
Subscribe to:
Posts (Atom)