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Class 11 Maths Question Answers

Question Answer

Class 11

Maths

Master Class 11 Maths: Engagin...

Class 11 Maths Question Answers

Let z1=23–√+i67–√67–√+i23–√ and z2=11−−√+i313−−√313−−√+i11−−√ then,∣∣∣1z1+1z2∣∣∣ is equal to(a) 47(b) 264(c) |z1−z2|(d) |z1+z2|(e) |z1z2|

Consider the following statements in respect of the function $f(x) = {x^3} - 1$$x \\in [ - 1 1] $I. f(x) is increasing in [ - 1 1]II. f'(x) has no root in ( - 1 1].Which of the statements given above is\/are correct?A. only IB. only IIC. Both I and IID. Neither I nor II

Expand the following determinant:\\[\\left| {\\begin{array}{*{20}{c}}1{ - 3}4 \\\\35{ - 3} \\\\2{ - 5}0\\end{array}} \\right| \\\\\\]

If aN = {ax: x $\\in$ N} and bN $\\cap$ cN = dN where b c $\\in$ N are relatively prime then A. d = bcB .c = bdC. b = cdD. None of these.

The equation of the straight line joining the point $\\left( a b \\right)$ to the point ofintersection of the lines $\\dfrac{x}{a}+\\dfrac{y}{b}=1$ and $\\dfrac{x}{b}+\\dfrac{y}{a}=1$ is(a) ${{a}^{2}}y-{{b}^{2}}x=ab\\left( a-b \\right)$(b) ${{a}^{2}}x+{{b}^{2}}y=ab\\left( a+b \\right)$(c) ${{a}^{2}}y+{{b}^{2}}x=ab$(d) ${{a}^{2}}x+{{b}^{2}}y=ab\\left( a-b \\right)$

IIf the parametric equation of a line is given by $x=4+\\dfrac{t}{\\sqrt{2}}$ and $y=-1+\\sqrt{2}t$ where $t$ is the parameter then(a) Slope of the line is ${{\\tan }^{-1}}\\left( 2 \\right)$(b) Slope of the line is ${{\\tan }^{-1}}\\left( \\dfrac{1}{2} \\right)$(c) Intercept made by the line on the x-axis $=\\dfrac{9}{2}$(d) Intercept made by the line on the y-axis $=-9$

Prove that : $\\sin 20^\\circ \\sin 40^\\circ \\sin 60^\\circ \\sin 80^\\circ = \\dfrac{3}{{16}}$

The number of selections of n objects from 2n objects of which n are identical and the rest are different is:\\[(A){\\text{ }}{2^n} \\\\(B){\\text{ }}{2^{n - 1}} \\\\(C){\\text{ }}{2^n} - 1 \\\\(D){\\text{ }}{2^n} + 1 \\\\ \\]

The points \\[\\left( {0 \\dfrac{8}{3}} \\right) \\left( {1 3} \\right)\\] and \\[\\left( {82 30} \\right)\\] are the vertices of \\[\\left( {\\text{A}} \\right)\\]An obtuse angled triangle\\[\\left( {\\text{B}} \\right)\\] An acute angled triangle\\[\\left( {\\text{C}} \\right)\\] A right-angled triangle\\[\\left( {\\text{D}} \\right)\\]None of these

The line joining the points \\[{\\text{(}} - {\\text{6}} 8{\\text{)}}\\] and \\[{\\text{(8 }} - {\\text{6)}}\\] is divided into four equal parts; find the coordinates of the points of section.

Let alpha and beta be the roots of equation x2 6x 2 0 If an alpha n beta n for n geqslant 1 then the value of fraca10 2a82a9 is

A function $f$ is defined by$f\\left( x \\right) = 2x - 5$ . Write down the values of$\\left( i \\right)f\\left( 0 \\right) \\\\\\left( {ii} \\right)f\\left( 7 \\right) \\\\\\left( {iii} \\right)f\\left( { - 3} \\right) \\\\ $

Find the logarithms of 125 to base$5\\sqrt 5 $ and 0.25 to base 4.

State whether the following statement is true or false. Justify your answer.The set of all integers is contained in the set of all rational numbers.\\[{\\text{(A)}}\\]True\\[{\\text{(B)}}\\]False

Find the derivative of the following function (n is an integer).${\\sin ^n}x$

If \\[{\\text{A}} = \\{ 10 15 20 25 30 35 40 45 50\\} \\] \\[{\\text{B}} = \\{ 1 5 10 15 20 30\\} \\] and \\[{\\text{C}} = \\{ 7 8 15 20 35 45 48\\} \\] find \\[{\\text{A}} - ({\\text{B}} \\cap {\\text{C}})\\].

The value of ${}^{10}{{\\text{C}}_1} + {}^{10}{C_2} + {}^{10}{C_3} + \\ldots + {}^{10}{C_9}$is${\\text{A}}{\\text{. }}{{\\text{2}}^{10}} \\\\{\\text{B}}{\\text{. }}{2^{11}} \\\\{\\text{C}}.{\\text{ }}{2^{10}} - 2 \\\\{\\text{D}}.{\\text{ }}{2^{10}} - 1 \\\\ $

Obtain all the zeroes of \\[f(x) = {x^3} + 13{x^2} + 32x + 20\\] if one of the zeroes is \\[{\\text{ - 2}}\\] .

An example for a function which is a relation (Domain-R Codomain-R) is:This question has multiple correct optionsA. $y = x$ B. $y = x - 1$ C. $y = {x^2}$ D. None of these\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

A ladder has rungs 25cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are ${\\text{2}}\\dfrac{1}{2}{\\text{m}}$apart then what is the length of the wood required for the rungs?\n \n \n \n \n

If \\[\\underset{x\\to a}{\\mathop{Lt}}\\ \\left[ f(x)g(x) \\right]\\] exists then both \\[\\underset{x\\to a}{\\mathop{Lt}}\\ f(x)\\]and $\\underset{x\\to a}{\\mathop{Lt}}\\ g(x)$ exist. State if True \/ False.

The value of $\\sin {12^ \\circ }\\sin {48^ \\circ }\\sin {54^ \\circ }$is equal to:A.$\\frac{2}{3}$B.$\\frac{1}{2}$C.$\\frac{1}{8}$D.\\[\\frac{1}{3}\\]

If ${\\sin ^{ - 1}}x + {\\sin ^{ - 1}}y + {\\sin ^{ - 1}}z = \\frac{{3\\pi }}{2} $then write the value of $x + y + z.$

If $\\sec a + \\tan a = p$ then show that $\\sec a - \\tan a = \\dfrac{1}{p}$ . Hence find the value of $\\cos a$ and $\\sin a$.

Find the zeroes of the following quadratic polynomials and verify the relationship Between the zeroes and the coefficients.$\\left( {\\text{i}} \\right){x^2} - 2x - 8{\\text{ }}\\left( {{\\text{ii}}} \\right)4{s^2} - 4s + 1\\left( {{\\text{iii}}} \\right)6{x^2} - 3 - 7x \\\\\\left( {{\\text{iv}}} \\right)4{u^2} + 8u\\left( {\\text{v}} \\right){t^2} - 15\\left( {{\\text{vi}}} \\right)3{x^2} - x - 4 \\\\ $

Find the equation of the plane that bisects the line joining $\\left( {1 2 3} \\right)$ and $\\left( {3 4 5} \\right)$ and is at the right angle to the line.

The total number of natural number of 6 digits that can be made with digits 1 2 3 4 where all the digits are to appear in the same number at least once isA. $1560$B. $840$C. $1080$D. $480$

Prove that the equation of circle in the$z$plane can be written in the form$\\alpha z\\overline z + \\overline \\beta z + \\beta \\overline z + c = 0$. Deduce the equation of the line.A. $\\overline \\beta z + \\beta \\overline z + c = 0$B. $\\overline \\beta z - \\beta \\overline z + c = 0$C. $\\overline \\beta z + \\beta \\overline z - c = 0$D. None of these

Cards numbered from $11$ to $60$ are kept in a box if a card is drawn at random from the box find the probability that the number on the card drawn is$\\left( i \\right)$ an odd number $\\left( ii \\right)$ a perfect square$\\left( iii \\right)$ divisible by $5$ $\\left( iv \\right)$ a prime number less than $20$

Find the volume of the largest right circular cone that can be cut of a cube whose edge is \\[9cm\\].

Find the value of k for which the simultaneous equation x + y + z = 3; x + 2y + 3z = 4 and x + 4y + kz = 6 will not have a unique solution.A.0B.5C.6D.7

The arithmetic mean of the first n natural numbers is$A)\\dfrac{{\\left( {n + 1} \\right)}}{2} \\\\B)\\dfrac{{\\left( {n - 1} \\right)}}{2} \\\\C)\\dfrac{n}{2} \\\\D){\\text{none of the above}} \\\\ $

The centroid and a vertex of an equilateral triangle are $\\left( 1 1 \\right)$and $\\left( 1 2 \\right)$ respectively. Another vertex of the triangle can beA) $\\left( \\dfrac{2-\\sqrt{3}}{2} \\dfrac{1}{2} \\right)$ B) $\\left( \\dfrac{2+3\\sqrt{3}}{2} \\dfrac{1}{2} \\right)$C) $\\left( \\dfrac{2+\\sqrt{3}}{2} \\dfrac{1}{2} \\right)$D) None of these

Tangents are drawn to the circle ${x^2} + {y^2} = 12$ at the points where it is met by the circle ${x^2} + {y^2} - 5x + 3y - 2 = 0;$ Find the point of intersection of these tangents.

Find the set E of the value of X for which the binomial expansion \\[{{\\left( 2+5x \\right)}^{\\dfrac{-1}{2}}}\\]is valid.

If \\[\\sin \\theta +{{\\sin }^{2}}\\theta =1\\] find the value of \\[{{\\cos }^{12}}\\theta +3{{\\cos }^{10}}\\theta +3{{\\cos }^{8}}\\theta +{{\\cos }^{6}}\\theta +2{{\\cos }^{4}}\\theta +2{{\\cos }^{2}}\\theta -2\\]

In a group of 500 people 200 can speak Hindi alone while only 125 speak English alone. The number of people who can speak both Hindi and English is${\\text{A}}{\\text{. 175}} \\\\{\\text{B}}{\\text{. 325}} \\\\{\\text{C}}{\\text{. 300}} \\\\{\\text{D}}{\\text{. 375}} \\\\ $

Find the logarithms of \\[\\dfrac{1}{256}\\text{ to base }2\\] \\[\\text{and 0}\\text{.3 to base 9}\\].

Prove the following expression$\\sin x+\\sin 3x+\\sin 5x+\\sin 7x=4\\cos x\\cos 2x\\sin 4x$

Find a point on x-axis which is equidistant from points (5 4) and (-2 3).\\[{\\text{A}}{\\text{. }}\\left( {2 0} \\right) \\\\{\\text{B}}{\\text{. }}\\left( { - 2 0} \\right) \\\\{\\text{C}}{\\text{. }}\\left( {3 0} \\right) \\\\{\\text{D}}{\\text{. }}\\left( { - 3 0} \\right) \\\\ \\]

Four 100 rupees notes and five notes each of rupees five rupees twenty rupees and fifty rupees are distributed among 3 children such that a child gets at least one 100 rupee note. The number of ways of distributing is..A. $3 \\times {5^3}$ B. ${3^3} \\times 5$ C. ${3^6}$ D. None

The income of a person is Rs.300 000 in the first year and he receives an increase of Rs.10000 to his income per year for the next 19 years. Find the total amount he received in 20 years.

A milk container is made of metal sheet in the shape of frustum of a cone whose volume is $10459\\dfrac{3}{7}c{{m}^{3}}$. The radii of its lower and upper circular ends are 8cm and 20cm respectively. Find the cost of the metal sheet used in making the container at the rate of $Rs.1.40\\ per\\ c{{m}^{3}}.\\left( Use\\ \\pi =22.7 \\right)$.

Find the next number in the series: 499 622 868 1237 1729 2344.(a) 3205(b) 3082(c) 2959(d) 3462(e) 2876

Prove the following trigonometric equation :${{\\text{sin}}^2}{\\text{A = co}}{{\\text{s}}^2}({\\text{A - B}}) + {\\cos ^2}{\\text{B}} - 2\\cos ({\\text{A - B}})\\cos {\\text{AcosB}}$

A flashlight with a parabolic reflector has the light bulb placed at the focus. The distance from the light bulb to the side of the flashlight called the focal diameter is 3cm. Determine the equation for the Parabola.\n \n \n \n \n

What is the value of $({\\cos ^2}{67^ \\circ }{\\text{ - }}{\\sin ^2}{23^ \\circ })?$\n\n

\\[ {\\text{Consider the following statements in respect of the function }}f(x) = {x^3} - 1 \\\\x \\in [ - 1 1] \\\\{\\text{I}}{\\text{. }}f(x){\\text{ is increasing in }}[ - 1 1] \\\\{\\text{II}}f'(x){\\text{ has no root in (}} - 1 1]. \\\\{\\text{Which of the statements given above is\/are correct?}} \\\\{\\text{A}}{\\text{. only I}} \\\\{\\text{B}}{\\text{. only II}} \\\\{\\text{C}}{\\text{. Both I and II}} \\\\{\\text{D}}{\\text{. Neither I nor II}} \\\\ \\]

${\\text{If sin}}\\theta {\\text{ cos}}\\theta {\\text{ and tan}}\\theta {\\text{ are in G}}{\\text{.P}}{\\text{. then co}}{{\\text{t}}^6}\\theta - {\\cot ^2}\\theta {\\text{ is}} \\\\{\\text{A}}{\\text{. 1}} \\\\{\\text{B}}{\\text{. }}\\dfrac{1}{2} \\\\{\\text{C}}{\\text{. 2}} \\\\{\\text{D}}{\\text{. 3}} \\\\ $

What are represented by the equation ${x^3} + {y^3} - xy(x + y) + {a^2}(y - x) = 0$\n\n

If ${\\text{U}} = \\left\\{ {1 2 3 4 5 6 7 8 9} \\right\\}$ and ${\\text{A}} = \\left\\{ {1 2 3 4} \\right\\}$ ${\\text{B}} = \\left\\{ {2 4 6 8} \\right\\}$ ${\\text{C}} = \\left\\{ {3 4 5 6} \\right\\}$ then prove the following: ${\\text{A’}} = \\left\\{ {5 6 7 8 9} \\right\\}$.

Make the greatest and the smallest 4-digit number using any four different digits with the condition that the digit 9 is always at hundreds place.

Find the modulus and the argument of the complex number$z = - 1 - i\\sqrt 3 $.

The sum of first m terms of an A.P. is $4{m^2} - m$. if it's ${n^{th{\\text{ }}}}$term is $107$ find the value of n.

If $10$ times the ${10^{th}}$term of an A.P. is equal to $15$ times the ${15^{th}}$ term show that the ${25^{th}}$term of A.P. is zero.

If AM and GM of two positive numbers a and b are 10 and 8 respectively find the numbers

Let a 0b 0 and c 0 Then both the roots of the equation ax2 + bx + c 0

If the 9th term of an AP is zero then prove that the 29th term is double of the 19th term.

A dice is rolled twice. Find the probability that:$(i)$$5$ will not come up either time $(ii)$ $5$ will come up exactly one time.

The harmonic conjugate of $(4 1)$ with respect to the points $(3 2)$ and$( - 1 6)$ is

In a set of four numbers the first three are in G.P and the last three in A.P with common difference 6. If the first number is the same as the fourth find the four numbers.

The given sum $1 \\times 1! + 2 \\times 2! + ............. + 50 \\times 50!$ is equal to(a) $51!$(b) $51! - 1$(c) $51! + 1$(d) $2 \\times 51!$

If\\[y = \\sin (x + y)\\] find \\[\\dfrac{{{d^2}y}}{{d{x^2}}}\\]

The equation of a line through the intersection of lines $x=0$ and $y=0$ and through thepoint $\\left( 2 ~2 \\right)$ is(a) $y=x-1$(a) $y=-x$(b) $y=x$(c) $y=-x+2$

Find the area bounded by the ellipse \\[\\dfrac{{{x^2}}}{{{a^2}}} + \\dfrac{{{y^2}}}{{{b^2}}} = 1\\] and the ordinates $x = 0$and $x = ae$ where ${b^2} = {a^2}\\left( {1 - {e^2}} \\right)$ and $e 1$.

Find the general solution of \\[sinx + sin3x + sin5x = 0\\].

The first term of an infinite G.P is 1 and any term is equal to the sum of all the succeeding terms. Find the series.

Prove that the points whose coordinates are respectively (5 1) (1 -1) and (11 4) lie on a straight line and find its intercepts on the axis.

Find out the sum of the series up to n terms.$\\left( {a + b} \\right) + \\left( {{a^2} + 2b} \\right) + \\left( {{a^3} + 3b} \\right) + ...........$ to n terms.

Find the number of divisors of 8064.

An iron pillar consists of a cylindrical portion $2.8m$ high and $20cm$ in diameter and a cone $42cm$ high is surmounted on it. Find the weight of the pillar given that $1c{m^3}$ of iron weights $7.5gm$.

How many changes can be rung with a peal of 7 bells the tenor always being last?

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Find the value of $\\sin {2^ \\circ }\\sin {4^ \\circ }\\sin {6^ \\circ }........\\sin {180^ \\circ }$ .

The value of $\\tan {81^0} - \\tan {63^0} - \\tan {27^0} + \\tan {9^0}$is$\\left( A \\right).{\\text{ 0}} \\\\\\left( B \\right).{\\text{ 2}} \\\\\\left( C \\right).{\\text{ 3}} \\\\\\left( D \\right).{\\text{ 4}} \\\\ $

Find the common ratio of the geometric sequence: $ - 1 1 - 1 1...........$${\\text{a}}{\\text{. 1}} \\\\{\\text{b}}{\\text{. }} - 1 \\\\{\\text{c}}{\\text{. 2}} \\\\{\\text{d}}{\\text{. }} - 2 \\\\ $

Evaluate the expression $\\underset{x\\to \\sqrt{10}}{\\mathop{\\lim }}\\ \\dfrac{\\sqrt{7+2x}-\\left(\\sqrt{5}+\\sqrt{2} \\right)}{{{x}^{2}}-10}$

Which of the following limit is not in the indeterminate form?(a) $\\underset{x\\to a}{\\mathop{\\lim }}\\ \\dfrac{{{x}^{3}}-{{a}^{3}}}{x-a}$(b) $\\underset{x\\to 0}{\\mathop{\\lim }}\\ \\sin x\\text{ }\\cos \\text{ec }x$(c) $\\underset{x\\to 0}{\\mathop{\\lim }}\\ {{x}^{x}}$(d) $\\underset{x\\to 0}{\\mathop{\\lim }}\\ \\dfrac{0}{x}$

The AM GM and HM in any series are equal thenA). The distribution is symmetricB). All the values are sameC). The distribution is unimodalD). None of these

Let $U$ be the set of all people and $M$ = {Males} $S$ = {college students} $T$= {teenagers} $W$ ={people having heights more than five feet}. Express each of the following in the notation of set theory.(i) College people having heights more than five feet.(ii) People who are not teenagers and have their heights less than five feet.(iii) All people who are neither males nor teenagers nor college students.

If $f(x) = \\dfrac{{{e^{\\dfrac{1}{x}}} - 1}}{{{e^{\\dfrac{1}{x}}} + 1}} x \\ne 0{\\text{ and f}}\\left( 0 \\right) = 0$ then $f(x)$ is(a) Continuous at $0$(b) Right continuous at $0$(c) Discontinuous at $0$(d) Left continuous at $0$

The number of integer’s n with $100 \\leqslant n \\leqslant 999$and containing at most two distinct digits is:${\\text{a}}{\\text{. }}252 \\\\{\\text{b}}{\\text{. }}280 \\\\{\\text{c}}{\\text{. }}324 \\\\{\\text{d}}{\\text{. }}360 \\\\ $

Integrate the following function:\\[\\sqrt{\\sin 2x}\\cos 2x\\]

A fair coin is tossed four times and a person wins \\[\\operatorname{Re}1\\] for each head and loses \\[Rs.1.5\\] for each tail that turns up. Let \\[p\\] be the probability of a person losing \\[Rs3.50\\] after \\[4\\] tosses. Find \\[4p\\]?

Find the Arithmetic mean of the following sequence a-2 a a+2

Solve the given matrix equation for the value of y.\\[y:\\left( {\\begin{array}{*{20}{c}}{x + y}xx \\\\ x{x + y}x \\\\ xx{x + y} \\end{array}} \\right) = 16(3x + 4)\\]

A letter is known to have come down either from \\[LONDON\\] or \\[CLIFTON\\]. On the envelope just two consecutive letters \\[ON\\] are visible. What is the probability that the letter has come from(i) \\[LONDON\\](ii) \\[CLIFTON\\]

Complete the following statement.Two lines are congruent if…….

There are n points in a plane of which p points are collinear. How many lines can be formed from these points?A. $^n{C_2}{ - ^P}{C_2}$ B. $^n{C_2}{ - ^p}{C_2} + 1$ C. $^n{C_2}{ - ^p}{C_2} - 1$ D.$^{n - p}{C_2}$

An unbiased die is thrown. What is the probability of getting a number greater than 3?

To find the probability that a card drawn at random from a pack of 52 cards is neither a heart nor a king?(a)$\\dfrac{4}{13}$ (b)$\\dfrac{9}{13}$(c) $\\dfrac{2}{13}$(d) $\\dfrac{5}{13}$

State whether the following statement is true or false.All even numbers are composite numbers.A. TrueB. False

The number of positive integral solutions ${x^2} + 9 {\\left( {x + 3} \\right)^2} 8x + 25$ is$A.{\\text{ 2}} \\\\{\\text{B}}{\\text{. 3}} \\\\{\\text{C}}{\\text{. 4}} \\\\{\\text{D}}{\\text{. 5}} \\\\ $

What is the probability of getting a number less than 5 when an unbiased die is thrown?

Show that the function defined by g(x)=x-[x] is discontinuous at all integral points.Here [x] denotes the greatest integer less than or equal to x.

Let R be a relation on N defined by x+2y=8.The domain of R isA. {2 4 6}B. {2 4 6 8}C. {2 4 6}D. {1 2 3 4}

Find the sum to \\[n\\] terms of the sequence \\[8 88 888 8888 ...............\\]

In a parabola Prove that the length of a focal chord which is inclined at ${30^0}$ to the axis is four times the length of the latus-rectum.

Find the value r from the following combination ${}^8{C_r} - {}^7{C_3} = {}^7{C_2}$ .