Master Class 12 Maths: Engaging Questions & Answers for Success

intlimits pi 4pi 4 fracx11 3x9 + 8x3 4x + 1sec 2xdx

The ratio of angles in one regular polygon to that in another is 3: 2.The number of sides in the first is twice the second. How many sides do the second polygon have?

A box contains 10 red marbles 20 blue marbles and 30 green marbles 5 marbles are drawn from the box what is the probability that$\\left( i \\right)$ All are blue?$\\left( {ii} \\right)$ At least one will be green?

Show that the matrix $A + B$ is symmetric or skew symmetric according as $A$and $B$ are symmetric or skew symmetric.

If tan 1y tan 1x + tan 1left frac2x1 x2 right where x frac1sqrt 3 Then the value of y is

Define a function \\[f:R \\to R{\\text{ }}by{\\text{ }}f(x) = \\max \\left\\{ {\\left| x \\right|\\left| {x - 1} \\right|.......\\left| {x - 2n} \\right|} \\right\\}\\]Where n is a fixed natural number. Then \\[\\int\\limits_0^{2n} {f(x)dx} \\] is.\\[(A){\\text{ }}n{\\text{ }}\\]\\[(B){\\text{ }}{n^2}\\]\\[(C){\\text{ }}3n\\]\\[(D){\\text{ }}3{n^2}\\]

How many natural numbers are there from $1$ to $1000$ which have none of their digits repeated?

The integral $\\int {\\left[ {\\dfrac{{2{x^{12}} + 5{x^9}}}{{{{\\left( {{x^5} + {x^3} + 1} \\right)}^3}}}} \\right]dx} $ is equal to\\[{\\text{A}}{\\text{. }}\\dfrac{{{x^{10}}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{B}}{\\text{. }}\\dfrac{{{x^5}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{C}}{\\text{. }}\\dfrac{{ - {x^{10}}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{D}}{\\text{. }}\\dfrac{{ - {x^5}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\ \\]

Prove that: ${1^2}.{C_1} + {2^2}.{C_2} + {3^2}.{C_3} + {\\text{ }} \\ldots {n^2}.{C_n} = n\\left( {n + 1} \\right){2^{n - 2}}$

Write the value of the integral $\\int {{e^{2{x^2} + \\ln x}}dx} $.

A particle starts from rest and its angular displacement (in radians) is given by $\\theta =\\dfrac{{{t}^{2}}}{20}+\\dfrac{t}{5}$. If the angular velocity at the end of $t=4s$ is $'k'$. Then what will be the value of $'5k'$ ?

Find the distance of the point (2 12 5) from the point of intersection of the line \\[\\overrightarrow{r}=2\\widehat{i}-4\\widehat{j}+2\\widehat{k}+\\lambda (3\\widehat{i}+4\\widehat{j}+2\\widehat{k})\\]and the plane\\[\\overrightarrow{r}.\\left( \\widehat{i}-2\\widehat{j}+\\widehat{k} \\right)=0.\\]

Let $a=\\min \\left\\{ {{x}^{2}}+2x+3x\\in R \\right\\}$ and $b=\\underset{\\theta \\to 0}{\\mathop{\\lim }}\\\\dfrac{1-\\cos \\theta }{{{\\theta }^{2}}}$. The value of $\\sum\\limits_{r=0}^{n}{{{a}^{r}}.{{b}^{n-r}}}$ is(a) $\\dfrac{{{2}^{n+1}}-1}{{{3.2}^{n}}}$ (b) $\\dfrac{{{2}^{n+1}}+1}{{{3.2}^{n}}}$(c) $\\dfrac{{{4}^{n+1}}-1}{{{3.2}^{n}}}$ (d) none of these

Find \\[{{f}^{-1}}\\] if it exists: \\[f:A\\to B\\] where (i) \\[A=\\left\\{ 0-1-32 \\right\\};B=\\left\\{ -9-306 \\right\\}\\] and \\[f\\left( x \\right)=3x\\](ii) \\[A=\\left\\{ 13579 \\right\\};B=\\left\\{ 019254981 \\right\\}\\] and \\[f\\left( x \\right)={{x}^{2}}\\]

The geometric means of the abscissa of the points of intersection of the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$ and the line $y=1$ is(a)$\\sqrt{7}$(b)$\\sqrt{2}$(c)$\\sqrt{14}$(d) 1

An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn then a ball is drawn from the later. Find the probability that it is a white ball.

An insurance company insured 2000 scooter drivers 4000 car drivers and 6000 truck drivers. The probability of accidents are 0.010.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Find the equation of the plane passing through the points whose coordinates are (-1 1 1) and (1 -1 1) and perpendicular to the plane x + 2y + 2z = 5

A curve parametrically given by \\[x=t+{{t}^{3}}\\]and \\[y={{t}^{2}}\\] where \\[t\\in R\\]. For what value(s) of \\[t\\] is \\[\\dfrac{dy}{dx}=\\dfrac{1}{2}\\]?(A) \\[\\dfrac{1}{3}\\] (B) \\[2\\](C) \\[3\\](D) \\[1\\]

If $\\omega $ is a cube root of unity then the value of determinant $\\left| {\\begin{array}{*{20}{c}}{1 + \\omega }{{\\omega ^2}}\\omega \\\\ {{\\omega ^2} + \\omega }{ - \\omega }{{\\omega ^2}} \\\\ {1 + {\\omega ^2}}\\omega {{\\omega ^2}} \\end{array}} \\right|$ is equal toA. $ - 1 + \\omega $B. $1 - \\omega $C. 0D. ${\\omega ^2}$

Find the value of \\[\\int {\\dfrac{{{e^x}}}{x}(x\\log x + 1)dx} \\].(a). \\[\\dfrac{{{e^x}}}{x} + C\\](b). \\[x{e^x}\\log \\left| x \\right| + C\\](c). \\[{e^x}\\log \\left| x \\right| + C\\](d). \\[x\\left( {{e^x} + \\log \\left| x \\right|} \\right) + C\\](e). \\[x{e^x} + \\log \\left| x \\right| + C\\]

Find the square root of $ - 8 + 6i$ ?${\\text{A}}{\\text{.}} \\pm {\\text{(1 + 3i)}} \\\\{\\text{B}}{\\text{.}} \\pm {\\text{(1 - 3i)}} \\\\{\\text{C}}{\\text{.}} \\pm {\\text{(3 + i)}} \\\\{\\text{D}}{\\text{.}} \\pm {\\text{(3 - i)}} \\\\ $

Find the value of a and b such that$\\int{\\dfrac{dx}{1\\ +\\ \\sin x}}\\ =\\ \\tan (x+\\ a)\\ +\\ b$.

Integrate $\\int {\\cot x dx} $

If the coordinates of the point A B C D be $(123)(457)( - 43 - 6)$and $\\left( {292} \\right)$respectively. Then find the angle between the line AB and CD

What is the value of $\\sin {105^0} + \\cos {105^0}$?${\\text{A}}{\\text{. }}\\sin {50^0} \\\\{\\text{B}}{\\text{. cos}}{50^0} \\\\{\\text{C}}{\\text{. }}\\dfrac{1}{{\\sqrt 2 }} \\\\{\\text{D}}{\\text{. 0}} \\\\ $

If the three vectors $\\overrightarrow{a}+\\overrightarrow{b}$ $\\overrightarrow{b}+\\overrightarrow{c}$ and $\\overrightarrow{c}+\\overrightarrow{a}$ are coterminous edges of a parallelepiped then its volume is _.A) $3\\left[ \\overrightarrow{a}\\overrightarrow{c}\\overrightarrow{b} \\right]$ B) $0$ C) $2\\left[ \\overrightarrow{a}\\overrightarrow{b}\\overrightarrow{c} \\right]$D) $4\\left[ \\overrightarrow{b}\\overrightarrow{a}\\overrightarrow{c} \\right]$

Given the probability that A can solve a problem is $\\dfrac{2}{3}$ and the probability that B can solve the same problem is $\\dfrac{3}{5}$. Find the probability that none of two will be able to solve the problem.

How many ways to get a 6-digit telephone number which has at least one of their digits repeated ?

Evaluate the integral $\\int\\limits_{0}^{\\pi }{\\dfrac{x\\sin x}{1+\\sin x}dx}$.

Integrate the following trigonometric term:$\\int {\\cos 3x\\cos 4x.dx} $

Find the value of x if \\[\\left| {\\begin{array}{*{20}{c}}aax \\\\ mmm \\\\ bxb \\end{array}} \\right| = 0\\]A) aB) bC) a-bD) mThis question has multiple correct answers.

Find a vector parallel to the given vector $5\\hat i - \\hat j + 2\\hat k$ which has magnitude 8 units?

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\\dfrac{11}{25}$ and the probability of none of them occurring is $\\dfrac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$ thenA. $P(E)=\\dfrac{4}{5}P(F)=\\dfrac{3}{5}$B. $P(E)=\\dfrac{1}{5}P(F)=\\dfrac{2}{5}$C. $P(E)=\\dfrac{2}{5}P(F)=\\dfrac{1}{5}$D. $P(E)=\\dfrac{3}{5}P(F)=\\dfrac{4}{5}$

A child has a die whose six faces show the letters as given belowA B C D E AThe die is thrown once. What is the probability of getting(a) A?(b) D?

Two fair dice are rolled. What is the probability of getting a sum of 7?${\\text{A}}{\\text{. }}\\dfrac{1}{{36}} \\\\{\\text{B}}{\\text{. }}\\dfrac{1}{6} \\\\{\\text{C}}{\\text{. }}\\dfrac{7}{{12}} \\\\{\\text{D}}{\\text{. }}\\dfrac{5}{{12}} \\\\ $

The focal distance of a point on the parabola is ${y^2} = 8x{\\text{ is 4}}$; Find the coordinates of the point?

If A (-5 7) B (-4 -5) C (-1 -6) and D (4 5) are vertices quadrilateral find the area of the quadrilateral ABCD.

An Apache helicopter of an enemy is flying along the curve given by $y={{x}^{2}}+7$. A soldier placed at (3 7) wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

The plane passing through the point (4 – 1 2) and parallel to the lines \\[\\dfrac{x+2}{3}=\\dfrac{y-2}{-1}=\\dfrac{z+1}{2}\\] and \\[\\dfrac{x-2}{1}=\\dfrac{y-3}{2}=\\dfrac{z-4}{3}\\] also passes through the point. (a) (– 1 – 1 – 1)(b) (– 1 – 1 1)(c) (1 1 – 1)(d) (1 1 1)

Find the integral of ${\\text{sinx}}{\\text{.sin2x}}{\\text{.sin3x}}$ denoted by $\\int {{\\text{sinx sin2x sin3x}}} {\\text{ dx}}$.

Find the value of the given trigonometric ratio $\\tan 15{}^\\circ $ .

The sum of the binomial coefficient of ${\\left[ {2x + \\dfrac{1}{x}} \\right]^n}$ is equal to 256. The constant term in the expansion isA.1120B.2110C.1210D.None of the above

Find the equation of the plane passing through the following points: (234) (−351) and (4−12).

A coin is tossed three times where(i) E: head on third toss F: heads on first two tosses(ii) E: at least two heads F: at most two heads (iii) E: at most two tails F: at least one tailDetermine P(E|F).A. 0.42 0.50 0.85B. 0.50 0.42 0.85C. 0.85 0.42 0.30D. 0.42 0.46 0.47

If f(x) = x.cos(x) then find the value of $f'(\\pi )$.Formula: $\\cos (A + B) = \\cos A\\cos B - \\sin A\\sin B$

Show that $\\cos hx + \\sin hx = {e^x}$ and simplify $\\cos hx - \\sin hx = ?$ . By considering ${(\\cos hx + \\sin hx)^2} + {(\\cos hx - \\sin hx)^2}$ show that ${\\cos ^2}hx - {\\sin ^2}hx = \\cos h2x.$

What is the probability that a leap year has 53 Sundays?A. \\[\\dfrac{1}{7}\\] B. \\[\\dfrac{12}{52}\\]C. \\[\\dfrac{2}{7}\\]D. \\[\\dfrac{1}{52}\\]

If \\[\\left( {{{\\cos }^{ - 1}}x} \\right) + \\left( {{{\\cos }^{ - 1}}y} \\right) = 2\\pi \\] then find the value of \\[\\left( {{{\\sin }^{ - 1}}x} \\right) + \\left( {{{\\sin }^{ - 1}}y} \\right)\\].${\\text{A}}{\\text{. }}\\pi \\\\{\\text{B}}{\\text{. }} - \\pi \\\\{\\text{C}}{\\text{. }}\\dfrac{\\pi }{2} \\\\ $ ${\\text{D}}{\\text{.}}$ None of these

The value of \\[\\dfrac{d}{{dx}}\\left( {{x^x}} \\right)\\] is equal to:A. \\[{x^x}\\log \\left( {\\dfrac{e}{x}} \\right)\\]B. \\[{x^x}\\log ex\\]C. \\[{x^x}\\left( {1 + \\log x} \\right)\\]D. \\[{x^x}\\log x\\]

Evaluate the following integral: $\\int{{{\\tan }^{\\dfrac{3}{2}}}x{{\\sec }^{2}}xdx}$

Out of the number of four different digits formed from the digits 1 2 3 4 5 6 7 8 9 how many of them are greater than 4000 (Repetition is not allowed)?(a). 2010(b). \\[{}^9{P_4}\\](c). 2016(d). None of these

A tank with rectangular box and rectangular sidesopen at the top is to be constructed so that its depth is 2m and volume is $8{m^3}$. If the building of a tank costs Rs.70 per sq metres for the base and RS.45 per square metre for sides what is the cost of the least expensive tank?

Evaluate $\\int {\\sin x \\cdot \\sin 2x \\cdot \\sin 3xdx} $

The number of real solutions of the equation $\\sin \\left( {{e^x}} \\right) = {5^x} + {5^{ - x}}$$a.0 \\\\b.1 \\\\c.2 \\\\d.{\\text{None of these}}{\\text{.}} \\\\ $

Solve the following differential equation$\\dfrac{{dy}}{{dx}} = 1 + x + y + xy$

Prove that $\\sum\\limits_{k=0}^{n}{{{\\left( -1 \\right)}^{k}}\\left( \\begin{matrix}n \\\\k \\\\\\end{matrix} \\right)}\\dfrac{1}{k+m+1}=\\sum\\limits_{k=0}^{m}{{{\\left( -1 \\right)}^{k}}\\left( \\begin{matrix}m \\\\k \\\\\\end{matrix} \\right)\\dfrac{1}{k+n+1}}$

Area of a rectangle having vertices A B C D with position vectors $ - \\hat i + \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }}\\hat i + \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }}\\hat i - \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }} - \\hat i - \\dfrac{1}{2}\\hat j + 4\\hat k$ respectively is$a.{\\text{ }}\\dfrac{1}{2} \\\\b.{\\text{ }}1 \\\\c.{\\text{ }}2 \\\\d.{\\text{ }}4 \\\\ $

Find the the value given determinant $\\left| {\\begin{array}{*{20}{c}}{\\cos {{15}^ \\circ }}{\\sin {{15}^ \\circ }} \\\\ {\\sin {{75}^ \\circ }}{\\cos {{75}^ \\circ }} \\end{array}} \\right|$ A.$1$ B. $0$ C. $2$ D. $3$

Solve \\[2x - y = 7;{\\text{ }}3x - 2y = 11\\] by using the matrix inversion method.

A piggy bank contains a hundred 50 paise coins fifty Rs.1 coins twenty Rs.2 coins and ten Rs.5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down find the probability that the coin which fell(i) will be a 50 paise coin (ii) will be of value more than Rs.1(iii) will be of value less than Rs.5 (iv) will be Rs.1 or Rs.2 coin

If ${\\text{A}}\\left( {{\\text{adjA}}} \\right) = 5{\\text{I}}$ where ${\\text{I}}$ is the identity matrix of order 3 then $|adjA|$ is equal toA. 125B. 25C. 5D. 10

The total revenue in Rupees received from the sale of $x$ units of a product is given by \\[R(x) = 3{x^2} + 36x + 5\\]. The marginal revenue when $x = 15$ is(a) $116$(b) $96$(c) $90$(d) $126$

The points \\[(119){\\text{ }}(21)\\] and \\[(2 - 1)\\] are the midpoints of the sides of the triangle.Then the centroid is.\\[(A){\\text{ }}( - 5 - 3){\\text{ }}(B){\\text{ }}(5 - 3) \\\\(C){\\text{ }}(35){\\text{ }}(D){\\text{ }}(53) \\\\ \\]

If $A + B + C = \\pi $then $\\left| {\\begin{array}{*{20}{c}}{\\sin (A + B + C)}{\\sin B}{\\cos C} \\\\ {\\sin B}0{\\tan A} \\\\ {\\cos (A + B)}{ - \\tan A}0 \\end{array}} \\right| = .......................$

The number of arbitrary constants in the particular solution of a differential equation of third order is${\\text{a}}{\\text{. 0}} \\\\{\\text{b}}{\\text{. 1}} \\\\{\\text{c}}{\\text{. 2}} \\\\{\\text{d}}{\\text{. 3}} \\\\ $

Prove the following:$\\left| {\\begin{array}{*{20}{c}}{ax}{by}{cz} \\\\ {{x^2}}{{y^2}}{{z^2}} \\\\ 111 \\end{array}} \\right| = \\left| {\\begin{array}{*{20}{c}}abc \\\\ xyz \\\\ {yz}{zx}{xy} \\end{array}} \\right|$

Five cards are drawn successively with replacement from a well shuffled pack of$52$ cards. The probability that none is spade is $\\dfrac{243}{{{4}^{k}}}$. What is the valueof k.

A number $x$ is chosen at random from the set $\\{ 1234...........100\\} .$ Define the event:$A = $ the chosen number $x$ satisfies $\\dfrac{{(x - 10)(x - 50)}}{{(x - 30)}} \\geqslant 0.$ Then $P(A)$is :

Two tangents to a parabola intercept on a fixed tangent segment whose product is a constant. Prove that the locus of their point of intersection is a straight line.

If four coins are tossed find the chance that there should be two heads and two tails.

Two dice are thrown together. What is the probability that the sum of the numbers on two dice is \\[5\\] or number on the second die is greater than or equal to the number on the first die?

A card from a pack of 52 cards is lost. From the remaining cards of the pack two cards are drawn and are found to be diamonds .Find the probability of the lost card being a diamond.

If \\[A\\subset Bn\\left( A \\right)=5\\] and \\[n\\left( B \\right)=7\\] then \\[n\\left( A\\cup B \\right)=\\]_ _ _ _ _ _ _ _ _.(a) 5(b) 7(c) 2(d) 12

The value of ${{\\cos }^{-1}}\\left\\{ \\cos 2{{\\cot }^{-1}}\\left( \\sqrt{2}-1 \\right) \\right\\}$ is equal to(a) $\\sqrt{2}-1$ (b) $\\dfrac{\\pi }{4}$ (c) $\\dfrac{3\\pi }{4}$ (d) $0$

The function \\[f\\left( x \\right)=\\left\\{ \\begin{align} \\left| x-3 \\right|\\text{ }x\\ge 1 \\\\ \\dfrac{{{x}^{2}}}{4}-\\dfrac{3x}{2}+\\dfrac{13}{4}\\text{ }x1 \\\\ \\end{align} \\right.\\] is(a) continuous at \\[x=1\\](b) differentiable at \\[x=1\\](c) continuous at \\[x=3\\](d) differentiable at \\[x=3\\]

Solve the integral $I = \\int\\limits_0^\\pi {{{\\sin }^2}x{\\text{ }}dx} $.

A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Find the real roots of the equation \\[{{\\text{x}}^{\\text{2}}}{\\text{ - f '(x) = 0}}\\] if ${\\text{f(x) = }}\\int\\limits_{\\text{1}}^{\\text{x}} {\\sqrt {{\\text{2 - }}{{\\text{t}}^{\\text{2}}}} {\\text{dt}}} $ A. $ \\pm 1$B. $ \\pm \\dfrac{1}{2}$C. $ \\pm \\dfrac{1}{2}$D. 0 and 1

The differential equation obtained by eliminating A and B from $y = A\\cos \\omega t + B\\sin \\omega t$ is$(a){\\text{ y'' + y' = 0}} \\\\(b){\\text{ y'' + }}{\\omega ^2}{\\text{y = 0}} \\\\(c){\\text{ y'' = }}{\\omega ^2}{\\text{y}} \\\\(d){\\text{ y'' + y = 0}} \\\\ $

The straight line $\\dfrac{x-3}{3}=\\dfrac{y-2}{1}=\\dfrac{z-1}{0}$ is:(a) parallel to x axis (b) parallel to y axis (c) parallel to z axis (d) perpendicular to z axis

For what values of p are the points \\[\\left( {21} \\right)\\left( {p - 1} \\right){\\text{ }}and{\\text{ }}\\left( {1 - 3} \\right)\\] collinear.

For any vector \\[\\overrightarrow r \\] prove that \\[\\overrightarrow r = \\left( {\\overrightarrow r .\\mathop i\\limits^ \\wedge } \\right)\\mathop i\\limits^ \\wedge + \\left( {\\overrightarrow r .\\mathop j\\limits^ \\wedge } \\right)\\mathop j\\limits^ \\wedge + \\left( {\\overrightarrow {r.} \\mathop k\\limits^ \\wedge } \\right)\\mathop k\\limits^ \\wedge \\].

If ${\\cos ^{ - 1}}x = \\alpha (0 x 1)$ and ${\\sin ^{ - 1}}(2x\\sqrt {1 - {x^2}} ) + {\\sec ^{ - 1}}\\left( {\\dfrac{1}{{2{x^2} - 1}}} \\right) = \\dfrac{{2\\pi }}{3}$ then ${\\tan ^{ - 1}}(2x)$ is equal to \\[A.{\\text{ }}\\dfrac{\\pi }{6} \\\\B.{\\text{ }}\\dfrac{\\pi }{4} \\\\C.{\\text{ }}\\dfrac{\\pi }{3} \\\\D.{\\text{ }}\\dfrac{\\pi }{2} \\\\ \\]

(A) From a pack of 52 playing cards jacks queens kings and aces of red colour are removed . From the remaining a card is drawn at random .Find the probability that the card drawn is(i) a black queen (ii) a red card (iii) a ten(iv) a picture card ( jacks queen and kings are picture cards ) .(B) All cards of ace jack and queen are removed from a deck of playing cards. One card is drawn at random from the remaining cards. Find the probability that the card drawn is(i) A face card (ii) Not a face card

Bag 1 contains 3 red and 4 black balls and bag 2 contains 4 red and 5 black balls. Two balls are transferred at random from bag 1 to bag 2 then a ball is drawn from bag 2. The ball so drawn is found to be red in color. Find the probability that the transferred balls were both black.

Find the equation of the hyperbola whose conjugate axis is along the y-axis and having length 3 units and distance between foci is 5.$(a){\\text{ }}\\dfrac{{4{x^2}}}{9} - \\dfrac{{{y^2}}}{4} = 1 \\\\(b){\\text{ }}\\dfrac{{{x^2}}}{4} - \\dfrac{{4{y^2}}}{9} = 1 \\\\(c){\\text{ }}\\dfrac{{{x^2}}}{4} - \\dfrac{{{y^2}}}{9} = 1 \\\\(d){\\text{ 4}}{{\\text{x}}^2}{\\text{ - 36}}{{\\text{y}}^2}{\\text{ = 9 }} \\\\ $

Find the value of the following expression\\[\\operatorname{cosec}\\left\\{ {{\\cot }^{-1}}\\left( -\\dfrac{12}{5} \\right) \\right\\}\\]

A man borrows Rs. 20000 at 12% per annum compounded semi – annually and agrees to pay it in 10 equal semi-annual instalments. Find the value of each instalment if the first payment is due at the end of two years.

A radio station surveyed 190 students to determine the types of music they liked. The survey revealed that 114 liked rock music 50 liked folk music and 41 liked classical music 14 liked rock music and folk music 15 liked rock music and classical music 11 liked classical music and folk music 5 liked all three of music.Find:(i) How many did not like any of the 3 types?(ii) How many liked any two types only?(iii) How many liked folk music but not rock music?

Using trapezoidal rule by dividing the interval [0 4] into 4 equal parts the approximate value of \\[\\int\\limits_{0}^{4}{{{x}^{2}}+1}\\] is equal to(a) 25(b) 26(c) 27(d) 28

If x and y are positive and xy 1 then what is ${{\\tan }^{-1}}x+{{\\tan }^{-1}}y$ equal to?a.${{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$b.$\\pi +{{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$c.$\\pi -{{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$d.${{\\tan }^{-1}}\\left( \\dfrac{x-y}{1+xy} \\right)$

What is the distance between the parallel lines 3x + 4y + 7 = 0 and 3x + 4y – 5 = 0?(a). \\[\\dfrac{2}{5}\\](b). \\[\\dfrac{{12}}{5}\\](c). \\[\\dfrac{5}{{12}}\\](d). \\[\\dfrac{3}{5}\\]

A coin is tossed twice what is the probability that at least one tail occurs?

In any $\\Delta ABC$ prove that$a\\cos A+b\\cos B+c\\cos C=2a\\sin B\\sin C$

Solve the following equations:$\\left( {x + y} \\right)\\left( {x + z} \\right) = 30 \\\\\\left( {y + z} \\right)\\left( {y + x} \\right) = 15 \\\\\\left( {z + x} \\right)\\left( {z + y} \\right) = 18 \\\\ $This question has multiple correct answers.$A.\\left( {241} \\right) \\\\B.\\left( { - 2 - 4 - 1} \\right) \\\\C.\\left( {312} \\right) \\\\D.\\left( { - 3 - 1 - 2} \\right) \\\\ $

If \\[B\\] is a nonsingular matrix and \\[A\\] is a square matrix then \\[\\det \\left( {{B^{ - 1}}AB} \\right)\\] is equal to A. \\[\\det \\left( A \\right)\\]B. \\[\\det \\left( B \\right)\\]C. \\[\\det \\left( {{B^{ - 1}}} \\right)\\]D. \\[\\det \\left( {{A^{ - 1}}} \\right)\\]

Give an example of an upper triangular matrix.

Find the length of the arc in terms of \\[\\pi \\] that subtends an angle of \\[{{30}^{o}}\\] at the center of a circle of radius 4 cm.

Integrate the function $\\dfrac{1}{x{{\\left( \\log x \\right)}^{m}}}x0m\\ne 1$

intlimits pi 4pi 4 fracx11 3x9 + 8x3 4x + 1sec 2xdx

The ratio of angles in one regular polygon to that in another is 3: 2.The number of sides in the first is twice the second. How many sides do the second polygon have?

A box contains 10 red marbles 20 blue marbles and 30 green marbles 5 marbles are drawn from the box what is the probability that$\\left( i \\right)$ All are blue?$\\left( {ii} \\right)$ At least one will be green?

Show that the matrix $A + B$ is symmetric or skew symmetric according as $A$and $B$ are symmetric or skew symmetric.

If tan 1y tan 1x + tan 1left frac2x1 x2 right where x frac1sqrt 3 Then the value of y is

Define a function \\[f:R \\to R{\\text{ }}by{\\text{ }}f(x) = \\max \\left\\{ {\\left| x \\right|\\left| {x - 1} \\right|.......\\left| {x - 2n} \\right|} \\right\\}\\]Where n is a fixed natural number. Then \\[\\int\\limits_0^{2n} {f(x)dx} \\] is.\\[(A){\\text{ }}n{\\text{ }}\\]\\[(B){\\text{ }}{n^2}\\]\\[(C){\\text{ }}3n\\]\\[(D){\\text{ }}3{n^2}\\]

How many natural numbers are there from $1$ to $1000$ which have none of their digits repeated?

The integral $\\int {\\left[ {\\dfrac{{2{x^{12}} + 5{x^9}}}{{{{\\left( {{x^5} + {x^3} + 1} \\right)}^3}}}} \\right]dx} $ is equal to\\[{\\text{A}}{\\text{. }}\\dfrac{{{x^{10}}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{B}}{\\text{. }}\\dfrac{{{x^5}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{C}}{\\text{. }}\\dfrac{{ - {x^{10}}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\{\\text{D}}{\\text{. }}\\dfrac{{ - {x^5}}}{{2{{\\left( {{x^5} + {x^3} + 1} \\right)}^2}}} + C \\\\ \\]

Prove that: ${1^2}.{C_1} + {2^2}.{C_2} + {3^2}.{C_3} + {\\text{ }} \\ldots {n^2}.{C_n} = n\\left( {n + 1} \\right){2^{n - 2}}$

Write the value of the integral $\\int {{e^{2{x^2} + \\ln x}}dx} $.

A particle starts from rest and its angular displacement (in radians) is given by $\\theta =\\dfrac{{{t}^{2}}}{20}+\\dfrac{t}{5}$. If the angular velocity at the end of $t=4s$ is $'k'$. Then what will be the value of $'5k'$ ?

Find the distance of the point (2 12 5) from the point of intersection of the line \\[\\overrightarrow{r}=2\\widehat{i}-4\\widehat{j}+2\\widehat{k}+\\lambda (3\\widehat{i}+4\\widehat{j}+2\\widehat{k})\\]and the plane\\[\\overrightarrow{r}.\\left( \\widehat{i}-2\\widehat{j}+\\widehat{k} \\right)=0.\\]

Let $a=\\min \\left\\{ {{x}^{2}}+2x+3x\\in R \\right\\}$ and $b=\\underset{\\theta \\to 0}{\\mathop{\\lim }}\\\\dfrac{1-\\cos \\theta }{{{\\theta }^{2}}}$. The value of $\\sum\\limits_{r=0}^{n}{{{a}^{r}}.{{b}^{n-r}}}$ is(a) $\\dfrac{{{2}^{n+1}}-1}{{{3.2}^{n}}}$ (b) $\\dfrac{{{2}^{n+1}}+1}{{{3.2}^{n}}}$(c) $\\dfrac{{{4}^{n+1}}-1}{{{3.2}^{n}}}$ (d) none of these

Find \\[{{f}^{-1}}\\] if it exists: \\[f:A\\to B\\] where (i) \\[A=\\left\\{ 0-1-32 \\right\\};B=\\left\\{ -9-306 \\right\\}\\] and \\[f\\left( x \\right)=3x\\](ii) \\[A=\\left\\{ 13579 \\right\\};B=\\left\\{ 019254981 \\right\\}\\] and \\[f\\left( x \\right)={{x}^{2}}\\]

The geometric means of the abscissa of the points of intersection of the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$ and the line $y=1$ is(a)$\\sqrt{7}$(b)$\\sqrt{2}$(c)$\\sqrt{14}$(d) 1

An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn then a ball is drawn from the later. Find the probability that it is a white ball.

An insurance company insured 2000 scooter drivers 4000 car drivers and 6000 truck drivers. The probability of accidents are 0.010.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Find the equation of the plane passing through the points whose coordinates are (-1 1 1) and (1 -1 1) and perpendicular to the plane x + 2y + 2z = 5

A curve parametrically given by \\[x=t+{{t}^{3}}\\]and \\[y={{t}^{2}}\\] where \\[t\\in R\\]. For what value(s) of \\[t\\] is \\[\\dfrac{dy}{dx}=\\dfrac{1}{2}\\]?(A) \\[\\dfrac{1}{3}\\] (B) \\[2\\](C) \\[3\\](D) \\[1\\]

If $\\omega $ is a cube root of unity then the value of determinant $\\left| {\\begin{array}{*{20}{c}}{1 + \\omega }{{\\omega ^2}}\\omega \\\\ {{\\omega ^2} + \\omega }{ - \\omega }{{\\omega ^2}} \\\\ {1 + {\\omega ^2}}\\omega {{\\omega ^2}} \\end{array}} \\right|$ is equal toA. $ - 1 + \\omega $B. $1 - \\omega $C. 0D. ${\\omega ^2}$

Find the value of \\[\\int {\\dfrac{{{e^x}}}{x}(x\\log x + 1)dx} \\].(a). \\[\\dfrac{{{e^x}}}{x} + C\\](b). \\[x{e^x}\\log \\left| x \\right| + C\\](c). \\[{e^x}\\log \\left| x \\right| + C\\](d). \\[x\\left( {{e^x} + \\log \\left| x \\right|} \\right) + C\\](e). \\[x{e^x} + \\log \\left| x \\right| + C\\]

Find the square root of $ - 8 + 6i$ ?${\\text{A}}{\\text{.}} \\pm {\\text{(1 + 3i)}} \\\\{\\text{B}}{\\text{.}} \\pm {\\text{(1 - 3i)}} \\\\{\\text{C}}{\\text{.}} \\pm {\\text{(3 + i)}} \\\\{\\text{D}}{\\text{.}} \\pm {\\text{(3 - i)}} \\\\ $

Find the value of a and b such that$\\int{\\dfrac{dx}{1\\ +\\ \\sin x}}\\ =\\ \\tan (x+\\ a)\\ +\\ b$.

Integrate $\\int {\\cot x dx} $

If the coordinates of the point A B C D be $(123)(457)( - 43 - 6)$and $\\left( {292} \\right)$respectively. Then find the angle between the line AB and CD

What is the value of $\\sin {105^0} + \\cos {105^0}$?${\\text{A}}{\\text{. }}\\sin {50^0} \\\\{\\text{B}}{\\text{. cos}}{50^0} \\\\{\\text{C}}{\\text{. }}\\dfrac{1}{{\\sqrt 2 }} \\\\{\\text{D}}{\\text{. 0}} \\\\ $

If the three vectors $\\overrightarrow{a}+\\overrightarrow{b}$ $\\overrightarrow{b}+\\overrightarrow{c}$ and $\\overrightarrow{c}+\\overrightarrow{a}$ are coterminous edges of a parallelepiped then its volume is _.A) $3\\left[ \\overrightarrow{a}\\overrightarrow{c}\\overrightarrow{b} \\right]$ B) $0$ C) $2\\left[ \\overrightarrow{a}\\overrightarrow{b}\\overrightarrow{c} \\right]$D) $4\\left[ \\overrightarrow{b}\\overrightarrow{a}\\overrightarrow{c} \\right]$

Given the probability that A can solve a problem is $\\dfrac{2}{3}$ and the probability that B can solve the same problem is $\\dfrac{3}{5}$. Find the probability that none of two will be able to solve the problem.

How many ways to get a 6-digit telephone number which has at least one of their digits repeated ?

Evaluate the integral $\\int\\limits_{0}^{\\pi }{\\dfrac{x\\sin x}{1+\\sin x}dx}$.

Integrate the following trigonometric term:$\\int {\\cos 3x\\cos 4x.dx} $

Find the value of x if \\[\\left| {\\begin{array}{*{20}{c}}aax \\\\ mmm \\\\ bxb \\end{array}} \\right| = 0\\]A) aB) bC) a-bD) mThis question has multiple correct answers.

Find a vector parallel to the given vector $5\\hat i - \\hat j + 2\\hat k$ which has magnitude 8 units?

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\\dfrac{11}{25}$ and the probability of none of them occurring is $\\dfrac{2}{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$ thenA. $P(E)=\\dfrac{4}{5}P(F)=\\dfrac{3}{5}$B. $P(E)=\\dfrac{1}{5}P(F)=\\dfrac{2}{5}$C. $P(E)=\\dfrac{2}{5}P(F)=\\dfrac{1}{5}$D. $P(E)=\\dfrac{3}{5}P(F)=\\dfrac{4}{5}$

A child has a die whose six faces show the letters as given belowA B C D E AThe die is thrown once. What is the probability of getting(a) A?(b) D?

Two fair dice are rolled. What is the probability of getting a sum of 7?${\\text{A}}{\\text{. }}\\dfrac{1}{{36}} \\\\{\\text{B}}{\\text{. }}\\dfrac{1}{6} \\\\{\\text{C}}{\\text{. }}\\dfrac{7}{{12}} \\\\{\\text{D}}{\\text{. }}\\dfrac{5}{{12}} \\\\ $

The focal distance of a point on the parabola is ${y^2} = 8x{\\text{ is 4}}$; Find the coordinates of the point?

If A (-5 7) B (-4 -5) C (-1 -6) and D (4 5) are vertices quadrilateral find the area of the quadrilateral ABCD.

An Apache helicopter of an enemy is flying along the curve given by $y={{x}^{2}}+7$. A soldier placed at (3 7) wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

The plane passing through the point (4 – 1 2) and parallel to the lines \\[\\dfrac{x+2}{3}=\\dfrac{y-2}{-1}=\\dfrac{z+1}{2}\\] and \\[\\dfrac{x-2}{1}=\\dfrac{y-3}{2}=\\dfrac{z-4}{3}\\] also passes through the point. (a) (– 1 – 1 – 1)(b) (– 1 – 1 1)(c) (1 1 – 1)(d) (1 1 1)

Find the integral of ${\\text{sinx}}{\\text{.sin2x}}{\\text{.sin3x}}$ denoted by $\\int {{\\text{sinx sin2x sin3x}}} {\\text{ dx}}$.

Find the value of the given trigonometric ratio $\\tan 15{}^\\circ $ .

The sum of the binomial coefficient of ${\\left[ {2x + \\dfrac{1}{x}} \\right]^n}$ is equal to 256. The constant term in the expansion isA.1120B.2110C.1210D.None of the above

Find the equation of the plane passing through the following points: (234) (−351) and (4−12).

A coin is tossed three times where(i) E: head on third toss F: heads on first two tosses(ii) E: at least two heads F: at most two heads (iii) E: at most two tails F: at least one tailDetermine P(E|F).A. 0.42 0.50 0.85B. 0.50 0.42 0.85C. 0.85 0.42 0.30D. 0.42 0.46 0.47

If f(x) = x.cos(x) then find the value of $f'(\\pi )$.Formula: $\\cos (A + B) = \\cos A\\cos B - \\sin A\\sin B$

Show that $\\cos hx + \\sin hx = {e^x}$ and simplify $\\cos hx - \\sin hx = ?$ . By considering ${(\\cos hx + \\sin hx)^2} + {(\\cos hx - \\sin hx)^2}$ show that ${\\cos ^2}hx - {\\sin ^2}hx = \\cos h2x.$

What is the probability that a leap year has 53 Sundays?A. \\[\\dfrac{1}{7}\\] B. \\[\\dfrac{12}{52}\\]C. \\[\\dfrac{2}{7}\\]D. \\[\\dfrac{1}{52}\\]

If \\[\\left( {{{\\cos }^{ - 1}}x} \\right) + \\left( {{{\\cos }^{ - 1}}y} \\right) = 2\\pi \\] then find the value of \\[\\left( {{{\\sin }^{ - 1}}x} \\right) + \\left( {{{\\sin }^{ - 1}}y} \\right)\\].${\\text{A}}{\\text{. }}\\pi \\\\{\\text{B}}{\\text{. }} - \\pi \\\\{\\text{C}}{\\text{. }}\\dfrac{\\pi }{2} \\\\ $ ${\\text{D}}{\\text{.}}$ None of these

The value of \\[\\dfrac{d}{{dx}}\\left( {{x^x}} \\right)\\] is equal to:A. \\[{x^x}\\log \\left( {\\dfrac{e}{x}} \\right)\\]B. \\[{x^x}\\log ex\\]C. \\[{x^x}\\left( {1 + \\log x} \\right)\\]D. \\[{x^x}\\log x\\]

Evaluate the following integral: $\\int{{{\\tan }^{\\dfrac{3}{2}}}x{{\\sec }^{2}}xdx}$

Out of the number of four different digits formed from the digits 1 2 3 4 5 6 7 8 9 how many of them are greater than 4000 (Repetition is not allowed)?(a). 2010(b). \\[{}^9{P_4}\\](c). 2016(d). None of these

A tank with rectangular box and rectangular sidesopen at the top is to be constructed so that its depth is 2m and volume is $8{m^3}$. If the building of a tank costs Rs.70 per sq metres for the base and RS.45 per square metre for sides what is the cost of the least expensive tank?

Evaluate $\\int {\\sin x \\cdot \\sin 2x \\cdot \\sin 3xdx} $

The number of real solutions of the equation $\\sin \\left( {{e^x}} \\right) = {5^x} + {5^{ - x}}$$a.0 \\\\b.1 \\\\c.2 \\\\d.{\\text{None of these}}{\\text{.}} \\\\ $

Solve the following differential equation$\\dfrac{{dy}}{{dx}} = 1 + x + y + xy$

Prove that $\\sum\\limits_{k=0}^{n}{{{\\left( -1 \\right)}^{k}}\\left( \\begin{matrix}n \\\\k \\\\\\end{matrix} \\right)}\\dfrac{1}{k+m+1}=\\sum\\limits_{k=0}^{m}{{{\\left( -1 \\right)}^{k}}\\left( \\begin{matrix}m \\\\k \\\\\\end{matrix} \\right)\\dfrac{1}{k+n+1}}$

Area of a rectangle having vertices A B C D with position vectors $ - \\hat i + \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }}\\hat i + \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }}\\hat i - \\dfrac{1}{2}\\hat j + 4\\hat k{\\text{ }} - \\hat i - \\dfrac{1}{2}\\hat j + 4\\hat k$ respectively is$a.{\\text{ }}\\dfrac{1}{2} \\\\b.{\\text{ }}1 \\\\c.{\\text{ }}2 \\\\d.{\\text{ }}4 \\\\ $

Find the the value given determinant $\\left| {\\begin{array}{*{20}{c}}{\\cos {{15}^ \\circ }}{\\sin {{15}^ \\circ }} \\\\ {\\sin {{75}^ \\circ }}{\\cos {{75}^ \\circ }} \\end{array}} \\right|$ A.$1$ B. $0$ C. $2$ D. $3$

Solve \\[2x - y = 7;{\\text{ }}3x - 2y = 11\\] by using the matrix inversion method.

A piggy bank contains a hundred 50 paise coins fifty Rs.1 coins twenty Rs.2 coins and ten Rs.5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down find the probability that the coin which fell(i) will be a 50 paise coin (ii) will be of value more than Rs.1(iii) will be of value less than Rs.5 (iv) will be Rs.1 or Rs.2 coin

If ${\\text{A}}\\left( {{\\text{adjA}}} \\right) = 5{\\text{I}}$ where ${\\text{I}}$ is the identity matrix of order 3 then $|adjA|$ is equal toA. 125B. 25C. 5D. 10

The total revenue in Rupees received from the sale of $x$ units of a product is given by \\[R(x) = 3{x^2} + 36x + 5\\]. The marginal revenue when $x = 15$ is(a) $116$(b) $96$(c) $90$(d) $126$

The points \\[(119){\\text{ }}(21)\\] and \\[(2 - 1)\\] are the midpoints of the sides of the triangle.Then the centroid is.\\[(A){\\text{ }}( - 5 - 3){\\text{ }}(B){\\text{ }}(5 - 3) \\\\(C){\\text{ }}(35){\\text{ }}(D){\\text{ }}(53) \\\\ \\]

If $A + B + C = \\pi $then $\\left| {\\begin{array}{*{20}{c}}{\\sin (A + B + C)}{\\sin B}{\\cos C} \\\\ {\\sin B}0{\\tan A} \\\\ {\\cos (A + B)}{ - \\tan A}0 \\end{array}} \\right| = .......................$

The number of arbitrary constants in the particular solution of a differential equation of third order is${\\text{a}}{\\text{. 0}} \\\\{\\text{b}}{\\text{. 1}} \\\\{\\text{c}}{\\text{. 2}} \\\\{\\text{d}}{\\text{. 3}} \\\\ $

Prove the following:$\\left| {\\begin{array}{*{20}{c}}{ax}{by}{cz} \\\\ {{x^2}}{{y^2}}{{z^2}} \\\\ 111 \\end{array}} \\right| = \\left| {\\begin{array}{*{20}{c}}abc \\\\ xyz \\\\ {yz}{zx}{xy} \\end{array}} \\right|$

Five cards are drawn successively with replacement from a well shuffled pack of$52$ cards. The probability that none is spade is $\\dfrac{243}{{{4}^{k}}}$. What is the valueof k.

A number $x$ is chosen at random from the set $\\{ 1234...........100\\} .$ Define the event:$A = $ the chosen number $x$ satisfies $\\dfrac{{(x - 10)(x - 50)}}{{(x - 30)}} \\geqslant 0.$ Then $P(A)$is :

Two tangents to a parabola intercept on a fixed tangent segment whose product is a constant. Prove that the locus of their point of intersection is a straight line.

If four coins are tossed find the chance that there should be two heads and two tails.

Two dice are thrown together. What is the probability that the sum of the numbers on two dice is \\[5\\] or number on the second die is greater than or equal to the number on the first die?

A card from a pack of 52 cards is lost. From the remaining cards of the pack two cards are drawn and are found to be diamonds .Find the probability of the lost card being a diamond.

If \\[A\\subset Bn\\left( A \\right)=5\\] and \\[n\\left( B \\right)=7\\] then \\[n\\left( A\\cup B \\right)=\\]_ _ _ _ _ _ _ _ _.(a) 5(b) 7(c) 2(d) 12

The value of ${{\\cos }^{-1}}\\left\\{ \\cos 2{{\\cot }^{-1}}\\left( \\sqrt{2}-1 \\right) \\right\\}$ is equal to(a) $\\sqrt{2}-1$ (b) $\\dfrac{\\pi }{4}$ (c) $\\dfrac{3\\pi }{4}$ (d) $0$

The function \\[f\\left( x \\right)=\\left\\{ \\begin{align} \\left| x-3 \\right|\\text{ }x\\ge 1 \\\\ \\dfrac{{{x}^{2}}}{4}-\\dfrac{3x}{2}+\\dfrac{13}{4}\\text{ }x1 \\\\ \\end{align} \\right.\\] is(a) continuous at \\[x=1\\](b) differentiable at \\[x=1\\](c) continuous at \\[x=3\\](d) differentiable at \\[x=3\\]

Solve the integral $I = \\int\\limits_0^\\pi {{{\\sin }^2}x{\\text{ }}dx} $.

A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Find the real roots of the equation \\[{{\\text{x}}^{\\text{2}}}{\\text{ - f '(x) = 0}}\\] if ${\\text{f(x) = }}\\int\\limits_{\\text{1}}^{\\text{x}} {\\sqrt {{\\text{2 - }}{{\\text{t}}^{\\text{2}}}} {\\text{dt}}} $ A. $ \\pm 1$B. $ \\pm \\dfrac{1}{2}$C. $ \\pm \\dfrac{1}{2}$D. 0 and 1

The differential equation obtained by eliminating A and B from $y = A\\cos \\omega t + B\\sin \\omega t$ is$(a){\\text{ y'' + y' = 0}} \\\\(b){\\text{ y'' + }}{\\omega ^2}{\\text{y = 0}} \\\\(c){\\text{ y'' = }}{\\omega ^2}{\\text{y}} \\\\(d){\\text{ y'' + y = 0}} \\\\ $

The straight line $\\dfrac{x-3}{3}=\\dfrac{y-2}{1}=\\dfrac{z-1}{0}$ is:(a) parallel to x axis (b) parallel to y axis (c) parallel to z axis (d) perpendicular to z axis

For what values of p are the points \\[\\left( {21} \\right)\\left( {p - 1} \\right){\\text{ }}and{\\text{ }}\\left( {1 - 3} \\right)\\] collinear.

For any vector \\[\\overrightarrow r \\] prove that \\[\\overrightarrow r = \\left( {\\overrightarrow r .\\mathop i\\limits^ \\wedge } \\right)\\mathop i\\limits^ \\wedge + \\left( {\\overrightarrow r .\\mathop j\\limits^ \\wedge } \\right)\\mathop j\\limits^ \\wedge + \\left( {\\overrightarrow {r.} \\mathop k\\limits^ \\wedge } \\right)\\mathop k\\limits^ \\wedge \\].

If ${\\cos ^{ - 1}}x = \\alpha (0 x 1)$ and ${\\sin ^{ - 1}}(2x\\sqrt {1 - {x^2}} ) + {\\sec ^{ - 1}}\\left( {\\dfrac{1}{{2{x^2} - 1}}} \\right) = \\dfrac{{2\\pi }}{3}$ then ${\\tan ^{ - 1}}(2x)$ is equal to \\[A.{\\text{ }}\\dfrac{\\pi }{6} \\\\B.{\\text{ }}\\dfrac{\\pi }{4} \\\\C.{\\text{ }}\\dfrac{\\pi }{3} \\\\D.{\\text{ }}\\dfrac{\\pi }{2} \\\\ \\]

(A) From a pack of 52 playing cards jacks queens kings and aces of red colour are removed . From the remaining a card is drawn at random .Find the probability that the card drawn is(i) a black queen (ii) a red card (iii) a ten(iv) a picture card ( jacks queen and kings are picture cards ) .(B) All cards of ace jack and queen are removed from a deck of playing cards. One card is drawn at random from the remaining cards. Find the probability that the card drawn is(i) A face card (ii) Not a face card

Bag 1 contains 3 red and 4 black balls and bag 2 contains 4 red and 5 black balls. Two balls are transferred at random from bag 1 to bag 2 then a ball is drawn from bag 2. The ball so drawn is found to be red in color. Find the probability that the transferred balls were both black.

Find the equation of the hyperbola whose conjugate axis is along the y-axis and having length 3 units and distance between foci is 5.$(a){\\text{ }}\\dfrac{{4{x^2}}}{9} - \\dfrac{{{y^2}}}{4} = 1 \\\\(b){\\text{ }}\\dfrac{{{x^2}}}{4} - \\dfrac{{4{y^2}}}{9} = 1 \\\\(c){\\text{ }}\\dfrac{{{x^2}}}{4} - \\dfrac{{{y^2}}}{9} = 1 \\\\(d){\\text{ 4}}{{\\text{x}}^2}{\\text{ - 36}}{{\\text{y}}^2}{\\text{ = 9 }} \\\\ $

Find the value of the following expression\\[\\operatorname{cosec}\\left\\{ {{\\cot }^{-1}}\\left( -\\dfrac{12}{5} \\right) \\right\\}\\]

A man borrows Rs. 20000 at 12% per annum compounded semi – annually and agrees to pay it in 10 equal semi-annual instalments. Find the value of each instalment if the first payment is due at the end of two years.

A radio station surveyed 190 students to determine the types of music they liked. The survey revealed that 114 liked rock music 50 liked folk music and 41 liked classical music 14 liked rock music and folk music 15 liked rock music and classical music 11 liked classical music and folk music 5 liked all three of music.Find:(i) How many did not like any of the 3 types?(ii) How many liked any two types only?(iii) How many liked folk music but not rock music?

Using trapezoidal rule by dividing the interval [0 4] into 4 equal parts the approximate value of \\[\\int\\limits_{0}^{4}{{{x}^{2}}+1}\\] is equal to(a) 25(b) 26(c) 27(d) 28

If x and y are positive and xy 1 then what is ${{\\tan }^{-1}}x+{{\\tan }^{-1}}y$ equal to?a.${{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$b.$\\pi +{{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$c.$\\pi -{{\\tan }^{-1}}\\left( \\dfrac{x+y}{1-xy} \\right)$d.${{\\tan }^{-1}}\\left( \\dfrac{x-y}{1+xy} \\right)$

What is the distance between the parallel lines 3x + 4y + 7 = 0 and 3x + 4y – 5 = 0?(a). \\[\\dfrac{2}{5}\\](b). \\[\\dfrac{{12}}{5}\\](c). \\[\\dfrac{5}{{12}}\\](d). \\[\\dfrac{3}{5}\\]

A coin is tossed twice what is the probability that at least one tail occurs?

In any $\\Delta ABC$ prove that$a\\cos A+b\\cos B+c\\cos C=2a\\sin B\\sin C$

Solve the following equations:$\\left( {x + y} \\right)\\left( {x + z} \\right) = 30 \\\\\\left( {y + z} \\right)\\left( {y + x} \\right) = 15 \\\\\\left( {z + x} \\right)\\left( {z + y} \\right) = 18 \\\\ $This question has multiple correct answers.$A.\\left( {241} \\right) \\\\B.\\left( { - 2 - 4 - 1} \\right) \\\\C.\\left( {312} \\right) \\\\D.\\left( { - 3 - 1 - 2} \\right) \\\\ $

If \\[B\\] is a nonsingular matrix and \\[A\\] is a square matrix then \\[\\det \\left( {{B^{ - 1}}AB} \\right)\\] is equal to A. \\[\\det \\left( A \\right)\\]B. \\[\\det \\left( B \\right)\\]C. \\[\\det \\left( {{B^{ - 1}}} \\right)\\]D. \\[\\det \\left( {{A^{ - 1}}} \\right)\\]

Give an example of an upper triangular matrix.

Find the length of the arc in terms of \\[\\pi \\] that subtends an angle of \\[{{30}^{o}}\\] at the center of a circle of radius 4 cm.

Integrate the function $\\dfrac{1}{x{{\\left( \\log x \\right)}^{m}}}x0m\\ne 1$

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