CHAPTER-01
RELATIONS AND FUNCTIONS
1. Show that the relation S in the set A={x ∊ Z: 0 ≤ x ≤ 12} given by S={(a, b): a, b in Z,|a-b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1 .
2. Show that the relation R in the set of real numbers, defined as R={(a, b): a ≤ b^2} is neither reflexive nor symmetric, nor transitive.
3. Show that the function f: R → R defined by f(x)=2 x³-7, for x in R is bijective.
4. Show that the relation R: N → N defined by (a, b) R(c, d) rightarrow a+d=b+c forall(a, b),(c, d) in N times N is an equivalence relation.
5. Let N} be the set of all natural numbers and R be the relation in N x N defined by ( a, b) R(c, d) →
a. d= b.c. Show that R is an equivalence relation
6. Show that in the set of all triangles in a plane the relation defined by "is similar to' is an equivalence relation,
7. Show that the function f: R → R defined by f(x)=3-4 x is one one and onto.
8. Let A and B are two given sets. Show that f: A X ~B → B X A such that f(a, b)=(b, a), is a bijective function.
9. Check whether the function f:{N → N given by f(x)=x^3 is one one and onto?
10.
Prove that the function f is surjective, where f: N → N such that
f(n)={{}{l}
{n+1}{2}, { if } n { is odd }
{n}{2}, { if } n { is even }
{}.
Is the function injective? Justify your answer.
CHAPTER 02
Inverse Trigonometric Functions
1. Simplify $tan ^{-1}$$[{√{1+x²}-1}/{x}]$, x ≠ 0.
2. Prove that $cot ^{-1}$$[{√{1-sin x}+√{1-sin x}}/{√{1+sin x}-√{1-sin x}}]$=${x}/{2}$
3. Find the value of $tan ^{-1}$[2 cos {$2 sin ^{-1}$$({1}/{2})$}]
4. Find the value of $tan ^{-1}$ √3 - $sec ^{-1}$(-2).
5. Find the value of $tan ^{-1}$(tan ${5 π}/{6}$)+ $cos ^{-1}$ (cos ${13π}{/6}$)
6. Find the value of $Cos^{-1}$(Cos ${13π}/{6}$)
7. Find the value of $Tan^{-1}$(tan ${7π}/{6}$)
8. Find the value of Sin(${π}/{3}$ - $Sin^{-1}$(-${1}/{2}$)}
9. Express in the simplest form: Prove that $tan ^{-1}$[${√1+cos x+√1-cos x}/{√1+cos x-√1-cos x$]
10. Find the value of $tan ^{-1}$√3 - cot (-√3).
Remarks: Similar questions to be practiced
CH-03_MATRICES
1. If |{}{cc}x & 2 18 & x{}|=|{}{cc}6 & 2 18 & 6{}| then find the value of x}. (Some more similar questions may be given)
2. If A} be a nonsingular matrix of order 3 times 3, then |{adj} A| is equal to :
(a) |A|
(b) |A|^2
(c) |A|^3
(d) 3|A|
3. If A=[{}{ll}3 & -2 4 & -2{}], then find k if A^2=k A-2 I.
4. If A=[{}{cc}-1 & 4 2 & -4{}], find f(~A}) if f(x)=x^2-2 x+3.
5. Let A=[{}{lll}3 & 2 & 5 4 & 1 & 3 0 & 6 & 7{}]. Express A as the sum of two matrices such that one is symmetric and the other is skew-symmetric.
6. Using determinants find the area of the triangle whose Vertices are (1,3),(-2,4) and (5,3).
7. For what value of k} points (1,3),(-2, k}) and (5,3) are collinear?
8. If A=[{}{lll}1 & 2 & 2 2 & 1 & 2 2 & 2 & 1{}], verify that A^2-4 A-5 I=O
9. Find A}^{-1} where A=[{}{ccc}1 & 1 & 1 1 & 2 & -3 2 & -1 & 3{}].
Hence solve the system of equations x+y+2 z=0, x+2 y-z=9, and x-3 y+3 z=-14.
10. Find the product of matrices A=[{}{rrr}-5 & 1 & 3 7 & 1 & -5 1 & -1 & 1{}], B=[{}{lll}1 & 1 & 2 3 & 2 & 1 2 & 1 & 3{}]
and use it for solving the equations:
x+y+2 z=1, 3 x+2 y+z=7, 2 x+y+3 z=2 .
11. Find X and Y if X+Y=[{}{ll}5 & 2 0 & 9{}] and X-Y=[{}{cc}3 & 6 0 & -1{}]
12. If A and B are symmetric matrices of same order then show that: (i) AB}-BA} is skew symmetric
(ii) AB}+BA} is Symmetric
13. Using matrix method, solve the following system of equations : 2 x}-y}+z}=3, -x}+2 y}-z}=-4, x}-2 y}+2 z}=1
14. Using matrix method, solve the following system of equations :
x+2 y-3 z=-4,2 x+3 y+2 z=2, 3 x-3 y-4 z=11
Remark: Questions based on properties of transpose matrix and adjoint matrix to be practiced for MCQ
CH-05 CONTINUITY AND DIFFERENTIABILITY
1. For what value of k, the function f(x)={{}{cl}{1-cos 4 x}{8 x^2} & x ≠ 0 k & x=0{}. is continuous at x=0.
2. Find the value of k} if f(x)={{}{cc}{1-cos k x}{x sin x} & x ≠ 0 1 / 2 & x=0{}. is continuous at x=0.
3. Examine the continuity of the following function: {{}{ll}{x}{2|x|} & x ≠ 0 {1}{2} & x=0{} x=0..
4. Find the value of k} if f}(x}) is continuous at x}=0
f(x)={{}{cc}
k(x^2-2 x), & { if } x ≤ 0
4 x+1, & { if } x>0
{}}
5. Find the relation between a} and b} so that f(x)={{}{ll}a x+1, & x ≤ 3 b x+3, & x>3{}. is continues at x}=3.
6. If x^y=e^{x-y}, then prove that {d y}{d x}={2-log x}{(1-log x)^2}
7. If e^x+e^y=e^{x+y} then find {d y}{d x}.
8. If y=5 sin x-3 cos x, then find {d^2 y}{d x^2}
9. If x}=a sec 𝛳 and y=b tan 𝛳 then find {d^2 y}{d x^2} at 𝛳={π}{4}
Remark: Questions based on same concepts to be practiced for MCQ
CH-06 APPLICATION OF DERIVATIVES
1. Find the intervals in which given function is increasing or decreasing f(x)=4 x^3-6 x^2-72 x+30.
2. Find the intervals in which given function f(x)=x^2-4 x+6, is strictly increasing
3. Find the intervals in (0.2 π) in which f(x)=sin x+cos x is strictly increasing or decreasing.
4. What is the minimum value of f(x)=2 cos x-3 in [0, π / 2].
Note: some similar questions from above concept from NCERT book example to be practiced.
5. From a square sheet of tin with each side 18 ~cm}, a square is cut off from each corner, and an open box is made. Find the length of each side of removed square of the box of maximum volume?
6. Show that the right circular cylinder of given surface area and maximum volume is such that its height is equal to the diameter of the base.
7. Show that the right circular cone of the least curved surface and given volume has an altitude equal to √{2} times the radius of base.
8. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan ^{-1} √{2}.
9. Show that the height of the cylinder of maximum volume that can be inscribed in a radius R} is {2 r}{√{3}}. Also find the maximum volume.
10. An open box, with a square base, is to be made out of a given quantity of metal sheet of area c^2. Show that the maximum volume of the box is {c^3}{6 √{3}}.
Remark: Some important and easy questions from Rate of change of quantities may be given.
CHAPTER 7 INTEGRALS
1. Evaluate ∫ {d x}{sin ^2 x cos ^2 x}
2. Evaluate ∫ {(1-sin x)}{cos ^2 x} d x
3. Evaluate ∫ {cosec} x({cosec} x+cot x) d x
Also give Question No. 18, 19, 20 of exercise 7.1 and 18, 19, 20 0f 7.2 NCERT
4. Find the following integrals:
(a) ∫({1}{1+cot x}) d x
(b) ∫({1}{1+tan x}) d x
5. Evaluate (a) ∫ {(1+log x)^2}{x} d x
(b) ∫ {(x+1)(x+log x)^2}{x} d x
(c) ∫({1}{x+x log x}) d x
6. Evaluate: ∫ {x^3 tan ^{-1} x^4}{1+x^8} dx}
7. Find ∫ {(3 sin emptyset-2) cos emptyset}{.5-cos ^2 emptyset-4 sin emptyset)} d emptyset
8. Special integrals Exercise 7.4, Using Partial fraction Exercise 7.5 and 7.7, specially of the type ∫ √{c-b x-a x^2} dx}
9. Ex. 20 page 325, Ex. 22 Page No. 327, Q11 Exercise7.6,
10.Evaluate ∫ {x e^x}{(x+1)^2} d x
Also questions of the same type 16, 18, 19, 20, 21 and 22 of page No. 328 .
11. Integrals of. sin ^{-1} x and tan ^{-1} x
Definite Integrals
1. ∫_0^{{π}{2}} {√{cot x}}{√{cot x}+√{tan x}} d x
2. ∫_0^{{π}{2}} {√{sin x}}{√{sin x}+√{cos x}} d x
3. Evaluate I=∫_0^π {x sin x}{1+cos ^2 x} d x
4. Evaluate I=∫_{π / 6}^{π / 3} {1}{1+√{tan x}} d x
5. Evaluate ∫_0^{{π}{4}} log (1+tan x) d x
6. Find the integral ∫_0^π log (1+cos x) d x
7. Determine ∫_0^a {√{x}}{√{x}+√{a-x}} d x
8. Evaluate ∫_0^π {x d x}{a^2 cos ^2 x+b^2 sin ^2 x}
9. Find the value of ∫^{{π}{2}}(2 log sin x-log sin 2 x) d x
10. Evaluate ∫_1^3 {√{4-x}}{√{x}+√{4-x}} d x
CHAPTER 8
APPLICATION OF INTEGRALS
1. Find the area of the region in the first quadrant enclosed by the x-axis, the line y=x. and the circle x^2+y^2=32.
2. Using the method of integration, find the area of the region bounded by the lines 2 x+y=4,3 x-2 y=6 and x-3 y+5=0.
3. Find the area bounded between the circle x^2+y^2=16 and the line y}=2 x} above x} -
4. Find the area bounded between the parabola y^2=4 a x and the line y}=mx} using in
5. Find the area bounded between the parabola y²=x and the line y=x+2 and the X-axis.
6. Using integrals find the area of triangle whose vertices are (1,0),(2,2), and (3,1).
7. Find the area between x^2+y^2=4 and (x-2)^2+y^2=4.
8. Using integration, find the area of the triangle ABC} with vertices as A}(-1,0), B}(3,2) and C (1,3).
9. The ellipse 9 x^2+y^2=36 intersects positive X}-axis and y} axis at A} and B} respectively. Find the area bod between the ellipse and hor AB}.
10. Find the are of the region in the first quart enclosed by the x-axis, the liner √{3} x and the circle x^2+y^2=4.
CHAPTER 8
APPLICATION OF INTEGRALS
1. 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.
2. Using the method of integration, find the area of the region bounded by the lines 2 x+y=4,3 x-2 y=6 and x-3 y+5=0.
3. Find the area bounded between the circle x²+y²=16 and the line y=2x above x-axis.
4. Find the area bounded between the parabola y²=4 a x and the line y=mx using integral.
5. Find the area bounded between the parabola y²=x and the line y=x+2 and the X-axis.
6. Using integrals find the area of triangle whose vertices are (1,0),(2,2), and (3,1).
7. Find the area between x^2+y^2=4 and (x-2)^2+y^2=4.
8. Using integration, find the area of the triangle ABC with vertices as A(-1,0), B(3,2) and C(1,3).
9. The ellipse 9 x²+y²=36 intersects positive X-axis and y axis at A and B respectively. Find the area bounded between the ellipse and chord AB.
10. Find the area of the region in the first quadrant enclosed by the x-axis, the line y=√3 x. and the circle x²+y²=4.
11. Find the area of the circle 4x²+4 y²=9 which is interior to the parabola x²=4y.
3-D Geometry
1. Find the shortest distance between the lines
{r}=(6 {i}+2 {j}+2 k)+λ({i}-2 {j}+2 {k}) & {r}=-4 {i}-{k}+μ(3 {i}-2 {j}-2 {k})
2. Find the shortest distance between the lines
{r}=({i}+2 {j}+k)+λ({i}-{j}+{k}) & {r}=(2 {i}-{j}-{k})+μ(2 {i}+{j}+2 {k})
3. Find the shortest distance between the lines
{r}=(1+2 λ) {i}+(1-λ) {j}+λ {k} & {r}=(2 {i}+{j}-{k})+μ(2 {i}+{j}+2 {k})
4. Find the shortest distance between the lines
{r}=(1-t) {i}+(t-2) {j}+(3-2 t) {k} & {r}=(s+1) {i}+(2 s-1) {j}-(2 s+1) {k}
5. Find the shortest distance between the lines {x-1}{1}={y-2}{-1}={z-1}{1} & {x-2}{2}={y+1}{1}={z+1}{2}
6. Find the angle between following pair of lines {r}=2 {i}-5 j+{k}+λ(3 {i}+2 {j}+6 {k}) and
{r}=(7 {i}-6 j-6 {k})+μ({i}+2 j+2 {k}) .
7. Find the angle between following pair of lines {-x+2}/{-2}={y-1}{7}={z+3}{-3} and
{x+2}{-1}={2 y-8}{4}={z-5}{4}
8. Find the value of λ, so that the following lines are perpendicular to each other
{x-5}{5 λ+2}={2-y}{5}={1-z}{-1} { and } {x}{1}={2 y+1}{4 λ}={1-z}{-3}
9. Find the value of λ, so that the following lines are perpendicular to each other
{1-x}{3}={7 y-14}{2 λ}={5 z-10}{11} { and } {7-7 x}{3 λ}={y-5}{1}={6-z}{5}
10. Find the value of λ, so that the following lines are perpendicular to each other
{1-x}{3}={y-2}{2 λ}={z-3}{2} { and } {x+1}{3 λ}={y-1}{1}={6-z}{7}.
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Baye's Theorem
1. Three boxes contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black balls respectively. One of the boxes is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it has come from first box.
2. A factory has two machines A and B. Past record shows that machine A produced 60 % of items of output and machine B produced 40 %of items. Further, 2 % of items, produced by machine A and 1% produced by machine B were defective.One item is chosen at random from total output and this is found to be defective. What is the probability that it was produced by machine B?
3. There are three coins. One is a two headed coin another is a biassed coin that comes up heads 75 % of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is a probability that it was two headed coin?
4. Suppose that 5 % men and 0.25 % women have grey hair. A grey haired person is selected at random. What is the probability of this person being a male? Assume that there are equal number of men and women.
5. Of the students in a college, it is known that 60 % reside in hostel and 40 % do not reside in hostel. Previous year result report that 30% of students residing in hostel attain A grade and 20 % of ones not residing in hostel attain A grade in their annual examination. At the of the year, one student is chosen at random from the college and he has an A grade. What is the probability that selected student is a hosteler?
6. An insurance company insured 2000 scooter drivers, 4000 car drivers, 6000 truck drivers. The probability of their meeting with an accident are 0.01,0.03 and 0.15 , respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver?
7. A man is known to speak 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.
8. A card from a pack of 52 playing cards is lost. From the remaining cards, two cards are drawn and are found to be of diamond. Find the probability that the missing card is also of diamond.
Baye’s Theorem
1. Three boxes contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black balls respectively. One of the boxes is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it has come from first box.
2. A factory has two machines A and B. Past record shows that machine A produced 60% of items of output and machine B produced 40% of items. Further, 2% of items, produced by machine A and 1% produced by machine B were defective.One item is chosen at random from total output and this is found to be defective. What is the probability that it was produced by machine B?
3. There are three coins. One is a two headed coin another is a biased coin that comes up heads 75% of the time and third is unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is a probability that it was two headed coin?
4. Suppose that 5 % men and 0.25 % women have gray hair. A gray haired person is selected at random. What is the probability of this person being a male? Assume that there are equal number of men and women.
5. Of the students in a college, it is known that 60% reside in hostel and 40% do not reside in hostel. Previous year result report that 30% of students residing in hostel attain A grade and 20 % of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade. What is the probability that selected student is a hosteler?
6. An insurance company insured 2000 scooter drivers, 4000 car drivers, 6000 truck drivers. The probability of their meeting with an accident are 0.01, 0.03 and 0.15, respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver?
7. A man is known to speak 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.
8. A card from a pack of 52 playing cards is lost. From the remaining cards, two cards are drawn and are found to be of diamond. Find the probability that the missing card is also of diamond.
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