5/5/22
Probability
1) A problem in mathematics is given to three students A, B, C and their respective chances of solving the problem is 1/2,1/3,1/4, then their probability that the problem is solved. 3/4
2) The probability that a leap year will have 53 Tuesday or Saturday. 4/7
3) P(A)=2/3 , P(B)=1/2 and P(AU B)=5/6, then the events A and B are
A) mutually exclusive
B) independent as well as mutually exclusive
C) independent.
D) none
4) A fair dice is thrown till we get 6, then the probability of obtaining 6 exactly in even number of turns is. 5/11
5) A and B are two events such that P(A U B)= 3/4 , P(A ∩ B)= 1/4, P(A')=2/3 find P(A' ∩ B). 5/12
6) If A and B are two events such that P(A U B)= 5/6 , P(A ∩ B)= 1/3, then which one is the following is not correct?
A) A and B are independent
B) A and B' are independent
C) A' and B are independent
D) A and B are dependent.
7) A coin and a six faced die, both unbiased are thrown simultaneously. The probability of obtaining a head on the coin and an odd number on the die. 2/3
8) A number is chosen at random among the first 120 natural numbers. What is the probability that the number chosen being a multiple of 5 or 15 ? 1/5
9) A die is thrown. If it shows a six, we draw a ball from a bag containing 2 black balls and 6 white balls. If it does not show a six then we toss a coin. Then the number of event points in the sample space of this experiment is. 18
10) If A and B are two events such that P(A)= 1/4, P(B/A)= 1/2 and P(A /B)= 1/4 then the value of P(A' B')= ? 3/4
11) The probability that a regularly scheduled flight departs on time is 0.9, the probability that it arrives on time is 0.8 and the probability that it departs and arrives on time is 0.7. then the probability that a plane arrives on time, given that it departs on time, is... 7/9
12) A sample of 4 items is drawn at random from a lot of 10 items, containing 3 defectives. If x denotes the number of defective items in the sample, then P(0< x < 3) is equal to... 4/5
13) If A and B are two independent events such that P(A)= 1/2, P(B)= 1/3, then the value of P(A' ∩ B'). 1/3
14) The probability that in a year of 22nd century chosen at random, there will be 53 Sundays. 5/28
15) The probability that in a family of 5 members, exactly 2 members have birthday on Sundays is. 10x6³/7⁵
16) A bag contains 5 white and 3 black balls and 4 balls are successively drawn out and not replaced. The probability that they are alternatively of different colours is .. 1/7
17) The probability that A speaks truth is 4/5, while this Probability for B is 3/4, then the probability that they will contradict each other when asked to speak on a fact, is ... 7/20
18) Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 is .. 4/1155
19) Find the probability of obtaining a total of 7 or 12 with two dice. 7/36
20) Five horses are in a race. Mr. A selected two of the horses at random and bets on them. The probability that Mr. A selected the winning horse. 2/5
21) For three events A, B, C, if P(B)= 3/4, P(A' ∩ B ∩ C')= 1/3 and P(A ∩ B ∩ C')= 1/3 , then the value of P( B ∩ C)= ? 1/12
22) In tossing a fair coin twice, let A and B denote the events of occurrence of head on first toss and second toss respectively; then the value of P(A U B) is. 3/4
23) A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen be white is. 8/15
24) A five digit number is formed by writing the digits 1, 2, 3, 4, 5 in a random order without repetition. Then the probability that the number is divisible by 4 is . 1/5
25) From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number on the card is divisible by 6 or 8 but not by 24 is.. 1/5
26) Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both the dice will get the prize. If A throws first then the probability of B winning the prize is.. 8/17
27) A fair coin is tossed n times. The probability of obtaining head atleast once greater than 0.8. then the least value of n is.... 3
28) A card is drawn from an ordinary pack of 52 cards and a gambler beta that either a spade or an ace is going to appear. Then the odds against his winning the prize are... 9:4
29) Out of 30 consecutive natural numbers, 2 are chosen at random. The probability that their sum is odd, is.. 15/29
30) The probability of having a king and a queen when two cards are drawn at random from a pack of 52 cards is. 8/663
31) A, B, C are mutually exclusive events such that P (A)= (3x+1)/3, P(B)= (1- x)/4 and P(C)= (1- 2x)/2; then the set of possible values of x are in the interval.... (1/3,1/2)
32) The probability of throwing a total of 7 or 12 with two dice is.. 5/36
33) If A and B are two events and P(A U B)= 5/6, P(A ∩ B)= 1/3 and P(B')= 1/2 , then A and B are
A) dependent B) independent.
C) mutually exclusive
D) none of these
34) If two fair coins are tossed together 5 times, then the probability of obtaining 5 heads and 5 tails is. 63/256
35) A fair coin is tossed 10 times. The probability of obtaining exactly 6 heads is. 105/512
36) A box contains 5 apples and 7 oranges and another box contains 4 apples and 8 oranges. One fruit is picked out from each box. Then the probability that the fruits are both apples or both oranges is. 19/36
37) Three numbers are selected at random from the first 20 natural numbers. The probability that their product is even, is.. 15/19
38) If P(A U B)= 0.8, P(A ∩ B)= 0.3 then the value of and P(A')+P(B')= ? 0.9
39) 12 balls are kept in 3 different boxes; then the probability that the first box will contain 3 ball is... (12C3. 2⁹)/3¹²
40) Four persons A, B, C, D throw an unbiased dice, turn by turn, in succession till one gets an even number and wins the game. If A starts then the probability that he wins the game is. 8/15
41) The probability of obtaining a total of atleast 6 in the simultaneously throw of 3 dice is. 103/108
42) 5 boys and 5 girls are sitting a row randomly. The probability that the boys and girls are sit alternatively is. 1/42
43) Bag A contains 4 green and 3 red balls and bag B contains 4 red and 3 green balls. One bag is taken at random and a ball is drawn and noted it is green. Then the probability that it comes from bag B
44) If A and B are two events such that P(A U B)' = 1/6, P(A ∩ B)= 1/4 and P(A')= 1/4 , where A' stands for the complement of event A. Then events A and B are
A) mutually exclusive and independent
B) independent but not equally likely.
C) equally likely and mutually exclusive
D) equally likely but not independent.
45)
X P(X)
1 0.15
2 0.23
3 0.12
4 0.10
5 0.20
6 0.08
7 0.07
8 0.05
Let the event A and B be defined as follows:
A: X is a prime number
B: (X< 4); then find the value of P(A U B) is. 0.77
46) A person puts three cards addressed to three different people in three envelopes with three different addresses without looking. Then the probability that the cards go into their respective envelopes is. 1/6
47) If birth of a male child and that of a female child are equal probable, then the probability of having atleast one of the three children born to a couple is male, is. 7/8
48) An ordinary coin is tossed 2n times. Then the chance that the number of times one gets head is not equal to the number of times one gets tail is. 1 - 2n!)(n!)² . 1/2^(2n)
Area
A) 1 B) 2/3. C) 4/3 D) 8/3
2) The area of the region bounded by the curve cos x= y and y = sinx and the ordinates x= 0, x=π/4 is
A) √2+ 1 B) √2 -1. C) 1/√2 D) (√2-1)/√2
3) 1) The area of the region bounded by the curve y²= 8x and its latus rectum is
A) 16/3 B) 25/3 C) 16√2/3 D) 32/3.
4) The area of the region bounded by the curve -3y²= x - 9 and the lines x= 0, y= 0 and y= 1 is
A) 8/3 B) 3/8 C) 8. D) 3
5) The area of the region bounded by the curve y²= 12x, the line x= 0 and y= 6 is
A) 12 B) 16 C) 3 D) 6.
6) The area of the region bounded by the curve x³= y and y = 2x² is
A) 2/3 B) 3/4 C) 4/3. D) 1/3
7) The ratio of the areas bounded by the curve cosx= y and y = cos2x between x= 0,x =π/3 and x-axis is
A) √2:1 B) 1:1 C) 1:2 D) 2:1.
8) The area in the first quadrant between y² = 4x, y²= 16x and the straight line x= 9 is
A) 36. B) 24 C) 18 D) 9
9) The area of the region bounded by the curve y= x²- 6x +10 and the lines x= 6, y= 2 is
A) 20/3. B) 16/3 C) 8 D) 32/3
10) The area of the region bounded by the curve x²= 72y and the lines y= k be 64√2 sq unit, then the value of k is
A) 2 B) 3 C) 4. D) 6
11) Two intersecting circles have their radii 1 metre and√3 metre. The distance between their centres is 2m. Then the overlapping area is
A) (19+ 6√3)/6 B) (5π+ 6√3)/6 C) π/6 D) (5π - 6√3)/6.
12) The area of the region bounded by the curve y= sinx between the ordinates x= 0, x= π and the x-axis is
A) 2. B) 4 C) 3 D) 6
13) The area above the x-axis bounded by the curve 2ᵏˣ= y and the lines x= 0, x= 2 be 3/log 2 sq unit, then the value of k is
A) 1. B) 1/2 C) -1 D) 2
14) The area of the common region between the two parabolas y² - ax= a² and y² + ax = a² is
A) 8a²/3. B) 16a²/3 C) 4a²/3 D) 32a²/3
15) The area of the region {(x,y): x²+ y² ≤ 1 ≤ x + y} is
A) π/4 B) π/2 C) π²/4 D) π/4 - 1/2.
16) The area of the region bounded by the curve secx= y, the x-axis and the lines x= 0 and x=π/4 is
A) log(√2 -1) B) log(√2 +1) . C) 1/2 log 2 D) √2
17) The area of the smaller segment cut off from the circle x² + y² = 9 by the line x= 1 is
A) 1/2(9 sec⁻¹3 - √8)
B) (9 sec⁻¹3 - √8).
C) √ 8 - 9 sec⁻¹3
D) (9 sec⁻¹3 + √8)
18) The area surrounded by the curve |x| + |y|= 1 is
A) 1 B) 4 C) 3 D) 2.
19) The area surrounded by the curve y= √x, the straight line x= 2y+3 in the first quadrant and the x-axis is
A) 9. B) 2√3 C) 18 D) 35/3
20) The area surrounded by the curve y= ax² and x = ay², a> 0 is 1 unit, then the value of a is
A) 1 B) 1/√3. C) 1/3 D) 1/√2
21) Which of the following definite integral represents the area included between the parabola 4y = 3x² and the straight line 2y = 3x+12?
A) ∫ 3x²/4 dx at (4,-2)
B) ∫ {(3x+12)/2 - 3x²/4}dx at (4,0)
C) ∫ (3x+12)/2 - 3x²/4}dx at (4,-2).
D) ∫(3x+12)/2 - 3x²/4}dx at (2,-2)
22) A function y= f(x) is defined as follows:
y= f(x)= x² when 0≤ x ≤ 1
= √x when x≥ 1
Then the area above the x-axis included between the curve y = f(x) and the line x= 4 is
A) 16/3 B) 4 C) 5. D) 6
23) the area of the region bounded by the curve 9x² + 4y²= 36 is
A) 3π. B) 9π C) 6π D) 4π
24) the area of the triangle bounded by the lines x + y = 0, y= 0 and x- 4= 0 is
A) 4 B) 8. C) 12 D) 16
25) If A is the area of the region bounded by the curve y= √(3x+4), x-axis and the lines x= -1 and x= 4 and B is the area bounded by the curve y² = 3x + 4 and the lines x=-1 and x= 4, then the value of A: B is
A) 1:2. B) 2:1 C) 2:3 D) 3:2
26) the curves y= sinx and, y= cosx intersect infinitely many times giving bounded regions of equal areas. The area of one such region is..
A) 4√ 2 B) 3√2. C) √2 D) 2√2
27) The area of the region enclosed between the curve y= log(x+e) and the coordinate axes is
A) 3 B) 4 C) 1. D) 2
28) The area bounded by the curve y= |x| - 1, and y = -|x |+ 1 is
A) 1 B) 2. C) 2√2 D) 4
29) The area bounded by the curve 16y= x², y-axis and it's latus rectum is
A) 32/3 B) 64/3. C) 128/3 D)16/3
30) The area bounded by the curve f(x)= 4 - |x|, and the x-axis is
A) 16. B) 32 C) 12 D) 24
31) If the area bounded by the curve y= x - x² and the line y= mx is 9/2 sq unit, then the value of m is
A) 1 B) 2 C) 3 D) 4.
32) The area bounded by the curve y= 2 - x² and the line x + y= 0 is A sq unit, then the value of A is
A) 9/2. B) 2/9 C) 1/3 D) 7/3
33) the area of the region bounded by the curve y= |x -2|, x-axis and the ordinates x= 1, x= 3 is
A) 4 B) 3 C) 2 D) 1.
34) the area of the region bounded by the curve 2y²= x, 3y² = x + 1 and y= 0 is
A) 4 B) 4/2 C) 2/3. D) 2
35) the area of the region bounded by three curves y= (x -1)², y = (x + 1)² and y = 1/4, is
A) 1/6 B) 1/3. C) 2/3 D) 5/6
36) the area between the curvey y= xeˣ, y= xe⁻ˣ, and the line x= 1 is
A) 2(e+ 1/e) B) 2e C) 2(e - 1/e) D) 2/e.
37) the area enclosed between the curve y= x³, and y² = x is
A) 5/12. B) 5/3 C) 5/4 D) 12/5
38) the area in the first quadrant bounded by the curve y² + x² = 8 x and its tangent at (2,2) is
A) 8-2π. B) 8+2π C) 2π-8 D) 4+2π
39) the area bounded by the curve y= - x²+ 2x+3, and y = 0 is
A) 32 B) 32/3. C) 5/32 D) 1/3
40) the area bounded by the curve y² = 2x+ 1 and the line x - y = 1 is
A) 16/3. B) 8/3. C) 24/3 D) 18/5
41) the area bounded by the curve y= - x² and y = x³ is
A) 1/3 B) 1/4 C) 1/6 D) 1/12.
42) the area enclosed between the curve y²= 4a(x+a) is (a, b > 0)
A) 1/3 (a+b) √(ab)
B) 2/3 (a+b) √(ab)
C) 4/3 (a+b) √(ab)
D) 8/3 (a+b) √(ab).
Tangent and Normal
1) The equation of the tangent to the curve (1+ x²)y = 2 - x where it crosses the x-axis is
A) x + 5y = 2. B) x - 5y = 2
C) 5x - y = 2 D) 5x + y = 2
2) The equation of the tangent and the normal drawn at the point (6,0) to the curve x²/36 + y²/9 = 1 respectively are
A) x= 6, y= 0. B) x +y = 6, y-x+6= 0
C) x = 0, y= 3 D) x = -6, y= 0
3) The straight line x+ y= a will be a tangent to the ellipse x²/9+ y²/16 = 1 if the value of a is
A) 8 B) ±10 C) ±5. D) ±6
4) The equation of the tangent to the parabola y² = 8x which is perpendicular to the line x - 3y+8= 0 is
A) 3x + y = -2 B) 3x - y = 1
C) 9x - 3y = -2 D) 9x + 3y = -2.
5) If the slope of the normal to the curve x³ = 8a²y at P is (-2/3), then the coordinates of P are
A) (2a,a). B) (a,a) C) (2a,-a) D) n
6) If the curves y= aˣ and y= bˣ intersect at an angle p, then the value of tan p is
A) (a- b)/(1+ ab)
B) (log a- log b)/(1+ loga logb) .
C) (a+ b)/(1- ab)
D) (loga+ logb)/(1+ log a log b)
7) If the straight line y = 4x - 5 touches the curve y² = px³+ q at (2,3), then the values of p and q are..
A) 2,-7 B) 2,7 C) -2,-7 D) -2,7
8) The equation of the normal to the parabola y² = 4ax at the point (at², 2at) is
A) tx +y= 2at + at³.
B) x + ty= 2at + at³
C) tx - y= at + 2at³
D) x - ty= at + at³
9) if the slope of the normal to the parabola 3y²+ 4y+2 = x at a point on it is 8, then the co-ordinate of the point are---
A) (1,-1) B) (6,2). C) (9,1) D) (2,0)
10) if the line lx + my + n= 0 is a tangent to the parabola y² = 4ax, then..
A) an² = ml B) al² = mn
C) am² = nl. D) a²m = nl
11) the equations of the tangents to the hyperbola 3x² - 4y² = 12 which are inclined at an angle 60° to the x-axis are ..
A) y= √3 x ±12 B) y= √3 x ±10
C) y= √3 x ±15 D) y= √3 x ±13.
12) the equation of the tangent to the curve xy²= 4(4- x) where it meets the line y = x is..
A) y= - x - 4 B) y= x + 4.
C) y= x - 2 D) y= x + 2
13) The normal to the curve x = 3 cos a - cos³a, y= 3 sin a - sin³a at a= π/4....
A) is at a distance of 2 unit from the origin
B) is at a distance of 4 unit from the origin
C) passes through the origin.
D) passes through the point (2,3)
14) the point on the curve x² + 2y = 10 at which the tangent to the curve is perpendicular to the line 2x - 4y = 7 is..
A) (2,3). B) (-2,3) C) (4,-3) D) (-4,-3)
15) if the gradient of the tangent at any point (x,y) of a curve which passes through the point (1,π/4) is {y/x - sin²(y/x)}, then the equation of the curve is...
A) y= cot⁻¹(log x)
B) y= cot⁻¹{log (x/e)}
C) y= x cot⁻¹(log xe).
D) y= cot⁻¹(log(e/x))
16) The number of tangents that can be drawn from the point (6,2) on the hyperbola x²/9 - y²/4 = 1 is .
A) 0. B) 1 C) 2 D) 3
17) The equation of the tangent to the curve x²⁾³+ y ²⁾³= a ²⁾³ at the point (a cos³k, a sin³k) is .
A) x cos k + y sin k= a sink cosk
B) x cos k - y sin k= a sin 2k
C) x sin k - y cos k= a sin2k
D) x sin k + y cos k= a sink cosk.
18) the equation of the tangent to the curve y= be⁻ˣ⁾ᵃ at the point where it crosses the y-axis is..
A) bx + ay = ab. B) ax + by = 1
C) bx - ay = ab D) ax - by = 1
19) The equation of the two common tangents to the circle x² + y² = 2a² and the parabola y² = 8ax are
A) x= ±(y + 2a) B) y= ±(x + 2a).
C) x= ±(y + a) D) y= ±(x + a)
20) the equation of the normal to the ellipse x²/a² + y²/b² = 1 at the point (a cosk, b sink) on it is..
A) ax sink - by cosk = a² - b²
B) ax sink + by cosk = a² - b²
C) ax cosk - by sink = (a² - b²) sink cosk
D) ax sink - by cosk = (a² - b²) sink cosk
21) the point on the curve √x + √y= √a, the normal at which is parallel to the x-axis is...
A) (0,0) B)(a,0) C)(0,a). D)a/4,a/4)
22) the slope of the tangent to the curve x= 3t²+1, y= t³ -2 at x= 1 is .
A) 1/2 B)0 C) )-2 D) undefined
23) if the line x+ y= a is a tangent to the parabola y² - y + x= 0, then the point of contact is ..
A) (0,1). B)(a,0) B)(1,1) C)(-1,0)
24) the angle between the Curve y = Sin x and y= cos x is..
A) tan⁻¹(5√2) B)tan⁻¹(3√3)
C) tan⁻¹(3√2) D) tan⁻¹(2√2).
25) the point on the curve y²= x, the tangent at which makes an angle 45° with the x-axis is
A) (0,9) B)(1/4,1/2). C)(1/2,1/4) D)(2,4)
26) if the straight line joining the point (0,3) and (5,-2) is a tangent to the curve y(x+1)= c, then the value of c will be..
A) 3 B)-3 C) 4. D) -4
27) the equation of the normal to the hyperbola x= a sec k, y= b tank at the point (a seck, b tank) is .
A) ax cosk + by cotk = a²+ b².
B) ax cosk + by tan k = a²+ b²
C) ax sin k - by cot k = a²- b²
D) ax cosk - by tan k = a²- b²
28) if the straight line lx + my = 1 is a normal to the parabola y² = 4ax, then ...
A) al² + 2 km = m²
B) al³ - 2 alm = m²
C) al³ + 2 alm = m².
D) al² + 2 am = m²
29) the equation of a tangent to the hyperbola x² - 2y¹ = 2 parallel to the line 2x - 2y+5= 0 is..
A) y= 2x+1 B) y= 2x - 1
C) x = y +1. D) x+ y +1= 0
30) the slope of the normal in the hyperbola x²/a² - y²/b² = 2 at the point (a sec k, b tan k) is .
A) b/a sin k B) - a/b sin k.
C) a/b sin k D) - b/a sin k
Maximum and minimum
1) The minimum value of f(x)= x² + 250/x. 75
2) The maximum value of f(x)= 1/(4x² + 2x+1) 4/3
3) If f(x)= 2x² - 3x² -12x +4 has
A) no maxima and minima
B) one maxima and one minima.
C) two maxima
D) two minima
4) maximum value of (log x)/x in (0, ∞). 1/e
5) Let the function f: R--> R be defined by f(x)= 2x + cosx; then f(x) is
A) has maximum value at x= 0
B) has minimum value at x=π
C) is a decreasing function
D) is an increasing function.
6) Let x and y be two variables and x > 0, xy= 1, then the minimum value of x+ y is. 2
7) The function y =a(1- cosx) is maximum when x is.. π
8) If minimum value of f(x)= x² + 2bx+ 2c² is greater than maximum value of g(x)= x² - 2cx + b², then for real value of x is. |c| > √|b|
9) Let f(x)= x³ + bx² + cx +d, 0 < b²< c, then f(x)..
A) has a local maximum
B) has a local minimum
C) is strictly decreasing
D) is strictly increasing.
10) If x+ y= 60, x, y > 0, then the maximum value of xy³ is. 15.(45)³
11) If the function f(x)=2x³ - 9ax² +12a²x+1, where a> 0 attains its maximum and minimum at x= p and x= q respectively, such that p²= q, then the value of a is. 2
12) The maximum value of the function f(x)= 3 cosx - 4 sinx is. 3
13) The minimum value of f(x)=2x²+ x -1 is. -9/8
14) If M and m are the maximum and minimum values respectively of the function f(x)= x+ 1/x, then the value of M + m is. -4
15) The maximum value of xʸ when x + 2y= 8 is. 8
16) The greatest value of the function f(x)= x² log(1/x). 1/2e
17) The minimum value of 4e²ˣ + 9e⁻²ˣ is.. 12
18) The maximum value of (x²- x+1)/(x² + x +1) is. 3
19) The difference between the maximum and minimum value of the function f(x)= x³/3 - 2x² + 3x +1 is. 4/3
Increasing and decreasing
1) f(x)= kx³- 9x² + 9x+ 4 is an increasing function then
A) k< 3 B) k≤ 3 C) k> 3. D) k is indeterminate
2) If f(x)= 1/(x+ 1) - log(1+x), x> 0, then f(x) is
A) a decreasing function.
B) an increasing function
C) neither a increasing nor decreasing function
D) increasing when x> 1
3) The function f(x)=1 - x³ - x⁵ is decreasing for. All real values of x
4) Let f(x)= x³ + 6x² + px +2, if the largest possible interval in which f(x) is a decreasing function is (-3,-1), then the value of p is. 9
5) The function f(x)= x³+ 3x² + 2x +7 is increasing for. x> 0
6) The function f(x)=2x³ - 3x²+ 90x +174 is increasing in the interval. (-∞ < x<∞)
7) The interval in which the function f(x)= 2x² - log x (x≠0) is increasing. -1/2< x< o or x>1/2
8) If f(x)= x³ - 6x²+ 9x +3 be a decreasing function, then x lies in. (1,3)
9) The value of x for which the polynomial 2x³ - 9x²+12x+4 is a decreasing function of x, is. 1<x<2
10) The function f(x)=x + 2 + (x -2) eˣ is positive. x> 0
Differentiation
1) Find dy/dx
2) x= ₑ tan⁻¹{(y- x²)/x². 2x(1+ tan(log x))+ x sec²(log x)
3) If x= a cos⁴ t, y= a sin⁴ t at t= 3π/4. -1
4) xˣ. xˣ(1+ logx)
5) ₑx³ w r t. Log x. 3x³ ₑx³
6) √[x+ √{x+ √(x +.......∞. 2/(2y -1)
7) sin⁻¹x + sin⁻¹y =π/2. -x/y
8) √{sin√x}. Cos√(x/4√x√{sin√x})
9) If f(x)= cos⁻¹{(1- (logx)²)/(1+ (logx)²)} then f'(e) is. 1/e
10) ₓeˣ. y eˣ(logx + 1/x)
11) If 2ˣ + 2ʸ =2ˣ⁺ʸ, then the value of dy/dx at x= y= 1 is. -1
12) siny + e ⁻ˣ ᶜᵒˢʸ = e at (1,π). e
13) If x= 2 cost + cos 2t and y= 2 Sint - sin 2t, at t=π/4. 1 -√2
14) If y= x+ x²+ x³+...... Where |x|< 2, then for |y|< 1 the value of dx/dy is... 1 - 2y+ 3y² - .......
15) sec⁻¹{1/(2x² -1) w r t. √(1- x²) at x= 1/2. 4
16) log₅(log₇x) (x>7). 1/(x log5 log 7 log₇x)
17) 2y= (x - a) √(2ax - x²) + a² sin⁻¹{(x-a)/a}. √(2ax - x²)
18) (secx + tanx)/(secx - tanx). 2 secx (secx + tanx)²
19) Tan⁻¹[{√x(3-x)}/(1- 3x)]. 3/{2(1+x)√x}
20) y= logₐx+ logₓa + logₓx + logₐa. 1/(x loga) - loga/{x(logx)²}
21) (x+y) ᵐ⁺ⁿ = xᵐ yⁿ. y/x
22) sin(π/6 eˣʸ) at x= 0. √3 π/24
23) y= log (tan x/2)+ sin⁻¹(cosx). Cosecx +1
24) sin² cot⁻¹[√{(1-x)/(1+x)}]. 1
25) x= sin⁻¹(3t - 4t³) and y = cos⁻¹√(1- t²). 1/3
26) Cosec⁻¹{(x+1)/(x-1)} + Tan⁻¹{(x-1)/(x+1)}. π
27) Y= Tan⁻¹[{√(1+x²) -1}/x] and z= Tan⁻¹{2x/(1- x²)} then find dy/dz. 1/4
28) sin⁻¹x w.r.t. cos⁻¹√(1- x²). 1
29) x= a(t+ Sint), y= a(1- cost). tan(t/2)
30) x√(1- y²)+ y√(1- x²)= k at x= 0. - √(1- k²)
31) tan⁻¹[√(1+ x²)- √(2- x²)}/{√(1+x²) + √(1- x²)}. x/√(1- x⁴)
32) tan⁻¹[{2x √(1- x²)}/(1- 2x²)] w.r.t. tan⁻¹{√(1+x²) -1}/x. 4
33) If y= Log(x+y)= 2xy then the value of y'(0) is. 1
34) y= sin⁻¹(x² √(1- x)) - √x √(1- x⁴).
35) sec{(x²-y²)/(x²+ y²)}= eᵃ. y/x
36) If f(x)= √(1+ cos²(x²)) then the value of f'(√π/2) is. - √π/√6
Find d²y/dx²
36) a sin³t w.r.t a cos³t at t=π/4. 0
38) sinx + eˣ. (Sinx - eˣ)/(cosx +eˣ)
39) a cos mx - b sin MX. -m²y
40) f(x)= sin3x cos4x at f"(π/2). 25
41) y= aˣ b²ˣ⁻¹. y(log ab²)²
42) y= 1/(1+ x+ x²+ x³) at x= 0. 0
43) If y= sinx log(tan x/2) then the value of d²y/dx² + y is. tanx
44) x= t²+ 2t, y= t³ - 3t at t= 1. 3/8
45) If eʸ + xy = e² at x= 0. 0
46) x= a cot t and y= 1/(x²+ a²) at t= π/6. 1/4a⁴
47) If dx/dy= u and d²x/dy²= v, then the value of d²y/dx² is. v/u²
48) If y= (x + √(1+ x²))ⁿ then (1+ x²) d²y/dx² + x dy/dx =? n²y
49) If log x= z, then find the value of x² d²y/dx². d²y/dx² - dy/dz
50) The value of x, at which the first derivative of x + 1/x w r t x is 3/4 is. ±2
51) If y= x³ then the value of d²y/dx²/{1+ (dy/dx)²}³⁾² at the point (1,1). 3/5√10
52) If x= 1/z, y= f(x) and d²y/dx² = kz³ dy/dx + z⁴ d²y/dx², then the value of k is. 2
53) If xy = ax² + b/x then the value of x² d²y/dx² + 2x dy/dx. 2y
54) If y= sinx° and z= log x then the value of dy/dz is. x°sinx°/ log e
55) If x= sin t, y= cos pt prove (1- x²) d²y/dx² - x dy/dx + p²y= 0
56) x= eᵗsin t and y= eᵗcos t, prove (x+y)² d²y/dx² - 2x dy/dx is. -2y
57) If y= f(x²) and f'(x)=√(3x²+1) then the value of dy/dx at x= 2. 28
58) If x= sec t - cos t, y= secⁿt - cosⁿ t and (x²+4)(dy/dx)²= k(y² +4) then the value of k is. n²
59) If y² = 4ax, then the value of d²y/dx². d²x/dy². -2a/y³
60) f(x)= logₓ(logₑx), then the value of f'(e) is. 1/e
61) If (cos⁻¹x)², then prove (1- x²) d²y/dx² - x dy/dx = 2
62) If y= sin(x²), z= ₑy² and t= √z, then the value of dt/dx is. 2xyz/t cos(x²)
63) If x + y= eˣ⁻ʸ Prove dy/dx = {2(x+y)}/(x+y+1)³
64) If x= 2 cos t- cos 2t and y= 2 Sint - sin2t, then the value of d²y/dx² at t= π/2 is. -3/2
65) If (sin⁻¹x)² + (cos⁻¹x)², then prove (1+ x²) d²y/dx² - x dy/dx = 4
66) If x²+ y² = t+ 1/t and x⁴ + y⁴ = t² + 1/t², then the value of - x³y dy/dx is.. 1
67) If siny = x sin(a+ y), then prove dy/dx= (sin²(a+y))/(sina)
Mean Theorem
1) let f(x)= eˣ, x ∈ [0, 1], then a number c of Lagrange's mean value theorem is.. log(e -1)
2) If the function f(x) satisfies the conditions of Rolle's theorem in (1,2) and f'(x) is continuous in (1,2), then ²₁∫ f'(x) dx is. 0
3) If f(x)= x(x -1)(x -2), 0≤ x ≤ 4, then the point x= c which satisfies mean value theorem satisfies. 1< c <3
4) The value of c in Rolle's theorem when f(x)= 2x³ - 5x² - 4x +3, x ∈ (1/2,3). 2
5) The mean value theorem f(b) - f(a)= (b - a)f'(c) (a < c < b), if a= 4, b= 9 and f(x)=√x, then the value of c is. 6.25
6) if the function f(x)= 4x³ + ax² + bx - 1 satisfies all the conditions of Rolle's theorem in -1/4 ≤ x≤ 1 and f'(1/2)=0, then the value of a and b. 1, -4
7) in the mean value theorem f(a+ h)= f(a) + f'(a+ ¢h) (0<¢<1), if f(x)= √x, a=1, h= 3, then the value of ¢. 5/12
8) If the conditions of Rolle's theorem are satisfied by the function f(x)= x³ + ax² + bx -5 in 1≤ x≤3 with c= 2 + 1/√3, then the value of a and b. -6, 11
9) In the mean value theorem f(b) - f(a)= (b - a) f'(c) (a< c< b), if a=π/6, b= 5π/6 and f(x)= log(sinx), then the value of c is. π/2
10) if the function f(x)= x⁴ + ax² - bx +4 defined in -2≤ x ≤ 2 satisfies Rolle's theorem when c= 1/3 (1+ √3), then the value of a and b. -1, 4
INTEGRATION (Definite and Indefinite)
1) ∫ eˣ(1- cotx + cot²x) dx. - eˣ cosecx
2) ∫ dx/√(e²ˣ -1). Sec⁻¹(eˣ)
3) ∫ sinx/sin(x -a). (x-a) cosa + sina log|sin(x -a)|
4) ¹₀∫ d/dx [sin⁻¹{2x/(1+ x²)}]. π/2
5) ²₋₂ ∫ |1 - x²|dx. 4
6) ∫ sin 2x log(tanx) dx at (π/2,0). 0
7) ∫ dx/√(2x - x²). sin⁻¹(x -1)
8) ∫ xeˣ/(x+1)² dx. eˣ/(x+1)
9) ∫ cos⁻¹(1/x) dx. x sec⁻¹x - log|x + √(x² -1)|
10) ∫ √x/{√(a-x)+ √x} at ((an-1)/n, 1/n). (an-2)/2n
11) ∫ x/(x² +4x+5). 1/2 log|x²+ 4x+5| - 2 tan⁻¹(x+2)
12) ∫ x|x| dx at (1,-1). 0
13) ∫ {1+ x +√(x+x²)}/{√x + √(1+x)}. 2/3 √(1+x)³
14) ∫ ₑ√x. 2(√x - 1) ₑ√x
15) If ∫ x sinx dx= - x cos x+ m, then the value of m is. Sinx
16) ∫ 1/√x . ₐ√x dx. 2.ₐ√x /(loga)
17) ∫ ₑtan⁻¹x /(1+ x²). ₑtan⁻¹x
18) ∫ dx/(1+ cotx) at (π/2,0). π/4
19) ∫ U= dx/logx at (e², e) and V= ∫ eˣ/x dx at (2,1), then find the value of U - V is. 0
20) ∫ {2x(1+ sinx)}/(1+ cos²x) at (π, - π). π²
21) ∫ dx/(sinx - cosx +√2). -1/√2 cot(x/2 +π/8)
22) ∫ cos³x ₑ sin²x. dx at (π,0). 0
23) ∫ dx/(5+ 3 cosx) at (π,0). π/4
24) ∫ ₑ log(tanx). Log(secx)
25) ∫ (1+ x - 1/x)ₑ x+ 1/x. x. ₑ x+ 1/x
26) ∫ x dx/(a² cos²x + b² sin²x) at (π,0). π²/ab
27) ∫ √tanx/sinx cosx. 2√tanx
28) ∫ x sinx at (π/2,0). 1
29) ∫ log(tanx) at (π/2,0). 0
30) ∫ log(sin²x) at (π,0). 2π log(1/2)
31) ∫ (logx)² at (e,1). e -2
32) ∫ dx/(eˣ+ 1/eˣ). tan⁻¹eˣ
33) ∫ x(1- x)ⁿdx at (1,0). 1/(n+1) - 1/(n+2)
34) ∫ {log (1+ x)}/(1+ x²) at (1,0). π
35) ∫ sinx/(sinx + cosx) at (π/2,0). π/4
36) ∫ x³ sin²x at (π/7, -π/7). 1
37) ∫ cosec⁴x. - cotx -1/3 cot³x
38) ∫ dx/{2√x(x +1)}. tan⁻¹√x
39) ∫ dx/(x² + 2x cos a +1) at (1,0). a/2sina
40) ∫ sin³x cosx. 1/4 sin⁴x
41) ∫ sin[2 tan⁻¹√{(1+x)(1-x)}] at (1,0). π/4
42) ∫ √(1+ sin(x/4)). 8(sin x/8 - cos x/8)
43) ∫ | log x|dx at (e, 1/e). 2(1- 1/e)
44) ∫ | sinx| dx at (10π, π). 18
45) ∫ dx/(x² +4x +13). 1/3 tan⁻¹(x +2)/3
46) ∫ cosx log{(1-x)/(1+x)} at (1/2,-1/2). 0
47) ∫ (3x+1)/(x²+9) at (3,0). Log(2√2)+ π/12
48) ∫ eˣ{(1+ sinx)/(1+ cosx). eˣ tan x/2
49) ∫ tan⁻¹{1/(x²- x+1)} at (1,0). π/2 - log 2
50) ∫ aˣ⁾² /√(a⁻ˣ - a ˣ). 1/log a sin⁻¹ aˣ
51) ∫ √{1-x)/(1+x) at (1,0). π/2 -1
52) ∫ |1- x²|dx at (3,-2). 28/3
53) ∫ x³ log x. 1/16 (4x⁴ log x - x⁴)
54) ∫ x dx/{(1+x)(1+ x²)} at (∞,0). π/4
55) ∫ eˣ/{(eˣ+2)/(eˣ+1)}. Log{(eˣ+2)/(eˣ+1({
56) ∫ dx/(cosx - sinx). 1/√2 log| tan (x/2 - π/8)|
57) ∫ (sinx + cosx)²/√(1+ sin2x) at (π/2,0). 2
58) ∫ ³√(x - x³)/x⁴. -3/8. ³√(1- 1/x²)⁴
59) ∫ eˣ ˡᵒᵍ ᵃ eˣ dx. (ae)ˣ/log(ae)
60) ∫ {(log x -1)/(1+ (lox)²)}². x/(1+ (logx)²)
DIFFERENTIAL EQUATIONS
A) Find the Order and degree of:
1) √dy/dx - 4 dy/dx - 7x = 0. 1, 2
2) (d²y/dx²)⁵ + 4 {d²y/dx²)³/(d³y/dx³)} + d³y/dx³ = x² -1. 3,2
3) [a + (dy/dx)⁶] ⁷⁾⁵= b d²y/dx². 2,5
4) dy/dx - x = (y - x dy/dx)-⁴
5) +1+ 3 dy/dx)⁴⁾³= 4 d³y/dx³. 3,3
6) (d³y/dx³)²⁾³ - 3 d²y/dx² + 5 dy/dx + 4y = 0. 2
B) Write the Integrating factor of:
1) (x+1) dy/dx - ny = eˣ(x +1)ⁿ⁺¹. 1/(x+1)ⁿ
2) x dy/dx +(x -1) y = x². eˣ/x
3) dy/dx + y tanx = secx. secx
4) 3 dy/dx + 3y/x = 2x⁴y⁴. 1/x³
5) cosx dy/dx + y sinx = 1. secx
6) cos²x dy/dx - y tan 2x = cos⁴x. (1- tan²x)
Solve:
1) (x+y) dx + x dy = 0. x²+ 2xy= c
2) dy/dx= xy + 2y at (1, 1). y= ₑ(2x+ x²/2 - 5/2)
3) dy/dx= ʸ ⁻ ˣ. 2 ⁻ ˣ - 2 ⁻ʸ = c
4) dy/dx = eˣ ⁻ʸ + 1. e ʸ⁻ ˣ = x+ c
5) x log x dy/dx + 2y = log x. (Log x)²
6) dy/dx - y tan x = - 2 sinx. y cosx = 1/2 cos 2x + c
7) cosx siny dx + sinx cosy dy= 0. |sinx siny| = c
8) dy/dx = (x-y)/(x+y). x² - y² - 2xy = c
9) y - x dy/dx = a(y² + dy/dx). |(x+ a)(1- ay)|= c|y|
10) dy/dx = y/x + p(y/x)/p'(y/x). |p(y/x|=k|x|
11) (x +2y³) dy/dx = y. x= y(y² + c)
12) dy/dx = (1+ y²)/(1+ x²). y - x = c(1+ xy)
13) dy/dx = √(1- x² - y² + x²y²). sin⁻¹y = x/2 √(1- x²) + 1/2 sin⁻¹x+ c
14) dy/dx= (x log x² + x)/(siny + y cosy). y siny = x² log x + c
15) y dx + (x + x²y)dy = 0. -1/xy + log y + c
16) ₑdy/dx = x +1, when y(0)= 3. y = (x+1) log|x +1| - x +3
17) dy/dx + y = e⁻ ˣ , y(0)= 0. y = xe⁻ ˣ
18) cot y dx = x dy. |x|=|secy|
19) y² dx +(x² - xy + y²) dy= 0. tan⁻¹(x)y) + log y + c= 0
20) cos²x dy/dx + y = tan x. y= tanx - c ₑ- tanx
21) tan y d/dx = sin(x+y)+ sin(x -y). secy + 2 cosx = c
22) (2y-1) dx - (2x +3) dy = 0. |(2x +3)/(2y -1)| = c
23) (2x - y+1) dx + (2y - x +2) dy = 0. x² + y² - xy + x + y = c
24) (x +y)(dx - dy) = dx + dy. Log|x + y|= x - y + c
25) dy= cosx(2 - y cosecx) dx where y= 3/√2 when x= π/4. y = sinx + cosecx
26) dy/dx + y = sinx. y= ce⁻ ˣ
27) sin⁻¹x + sin⁻¹y = c. √(1- x²) dy + √(1- y²) dx = 0
28) tany sec²x dx + tanx sec²y dy = 0. |tan x tan y| = k
29) dy/dx + (1+ cos 2y)/(1- cos 2x) = 0. tany - cot x= c
30) (1+ y²)+ (x - ₑ tan⁻¹y) dy/dx = 0. 2 x ₑ tan⁻¹y = ₑ 2tan⁻¹y + k
31) x dy - y dx + ax (x² + y²) dx = 0. 2 tan⁻¹(y/x) + ax² = c
32) 2(y +3)- xy dy/dx = 0, with y= -2 when x= 1. |x²(y +3)³|= ʸ⁺²
33) (x +y)(dx - dy)= dx+ dy. x+ y = ˣ⁻ʸ
34) dy/dx + y/x = sin x. x(y + cosx) = sinx + c
