General Instructions:
(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory.
(ii) Section A: Q. No.1 to 17 comprises of 17 questions of one mark each.
(iii) Section B: Q. No. 18 to 26 comprises of 9 questions of two mark each.
(iv) Section C: Q. No. 27 to 34 comprises of 8 questions of three mark each.
(v) Section D: Q. No. 35 to 40 comprises of 6 questions of four mark each.
(vi) Use of Calculators are not permitted.
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Find the coordinates of a point A, where AB is diameter of a circle whose centre is (2, 3) and B is the point (1, 4).
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Let the coordinates of point A be (x, y) and point O (2, 3) be point the centre, then
By midpoint formula,
and
or and
The coordinates of point A are (3, 10)
For what values of \(k \), the roots of the equation \(x^2+4x+k=0 \) are real?
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The given equation is
On comparing the given equation with , we get
and
For real roots,
or
or
or
For , equation will have real roots.
Find the value of k for which the roots of the equation \(3x^210x+k=0 \) are reciprocal of each other.
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The given equation is
On comparing it with , we get
Let the roots of the equation are and
Product of the roots
or
Find \(A \) if \(\tan2A=\cot(A24^o) \)
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Given,
or
or
or
or
Find the value of \((\sin^2\ 33^o+\sin^2\ 57^o) \)
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How many two digits numbers are divisible by 3?
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The twodigit numbers divisible by are
This is an A.P. in which
or
or
So, there are twodigit numbers divisible by .
In Figure below, \(DE  BC, AD = 1\ cm \) and \(BD = 2\ cm \). What is the ratio of the ar \((\triangle ABC) \) to the ar \((\triangle ADE) \)?s
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Given,
Also,
(Given)Â (corresponding angles)
In and
(common)
[by equation (i)]
(by AA rule)
Now,
Find a rational number between \(\sqrt2 \) and \(\sqrt3 \).
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As
So, a rational number between and is
Find the coordinates of a point A, where AB is a diameter of the circle with centre (2, 2) and B is the point with coordinates (3, 4).
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By midpoint formula
and
Coordinates of point A are .
Find the value of \(k \) for which the following pair of linear equations have infinitely many solutions.
\(2x + 3y = 7, (k + 1)x + (2k – 1)y = 4k + 1 \)
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Given, and
On comparing above equations with
and , we get
For infinitely many solutions
or
Hence, .
Two positive integers a and b can be written as \(a=x^3y^2 \) and \(b=xy^3. \ x,y \) are prime numbers. Find LCM \((a,b) \).
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Given, and
L.C.M = Product of the greatest power of each prime factors =
Find, how many twodigit natural numbers are divisible by 7.
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Two digit numbers which are divisible by 7 are 14, 21, 28,â€¦. 98
Hence, there are 13 two digit numbers, divisible by 7.
If the sum of first n terms of an A.P. is \(n^2 \), then find its 10th term.
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Given,
Put
Hence 10th term
Find the HCF of 1260 and 7344 using Euclidâ€™s algorithm.
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Two numbers are and
Since , we apply the Euclid division lemma to and , we get
H.C.F. of and is .
Which term of the A.P. 3, 15, 27, 39, â€¦â€¦ will be 120 more than its 21st term?
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The given A.P. is
Here
Now,
Hence, the term which is 120 more than its 21st term will be its 31st term.
A game consists of tossing a coin 3 times and noting the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game.
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When a coin is tossed three times, the set of all possible outcomes is given by,
Same result on all tosses
A die is thrown once. Find the probability of getting a number which is a prime number
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In throwing a die Total possible outcomes
i.e.,
Prime numbers
If the common difference of an AP is 3, find \(a_{20}a_{15} \)
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Find the value of \(a \), So that the point \((3,a) \) lies on the line represented by \(2x3y=5 \)
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For what value of \(k \) will \(k+9,2k1 \) and \(2k+7 \) are the consecutive terms of an AP?
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Find \( c\) if the system of equations \(cx + 3y + (3 – c) = 0,\ 12x + cy – c = 0 \) has infinitely many solutions?
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The given equations are
and
On comparing with equation and equation , we get
and
For infinitely many solutions
or
or
So, from both the above cases
Prove that \(\sqrt2 \) is an irrational number.
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Let is a rational number.
So, where and are coprime integers and
or
Squaring on both sides, we get
Therefore, divides
or divides a (from theorem)
Let , for some integer
From equation (i)
or
or
It means that divides and so divides
Therefore and have at least as a common factor.
But this contradicts the fact that and are coprime.
This contradiction is due to our wrong assumption that is rational.
So, we conclude that is irrational.
Hence Proved.
Find the value of \(k \) such that the polynomial \(x^2(k+6)x+2(2k1) \) has sum of its zeros equal to half to their product.
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The given quadratic polynomial is
Comparing with , we get and
Let the zeroes of the polynomial be and
we know that
or
Also,
or
According to question
Sum of zeroes of their product
or [using equations (i) & (ii)]
or
A fatherâ€™s age is three times the sum of the ages of his two children. After 5 years his age will be two times the sum of their ages. Find the present age of the father.
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Let the present age of father be years and sum of ages of his two children be years
According to question
After 5 years
Fatherâ€™s age years
Sum of ages of two children years years
According to question
or
or
or (Using equations (i))
Now from equation (i)
(Put )
or
So, Present age of father years.
Find the point on yaxis which is equidistant from the points (5, 2) and (3, 2)
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We know that a point on the axis is of the form .
So, let the point be equidistant from and
Then
or
or
or
So, the required point is
The line segment joining the points \(A(2, 1) \) and \(B(5, 8) \) is trisected at the points \(P \) and \(Q \) such that \( P\) is nearer to \(A \). If \(P \) also lies on the line given by \(2x – y + k = 0 \), find the value of \(k \).
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The line segment is trisected at the points and and is nearest to
So, divides in the ratio
Then coordinates of P, by section formula
lies on the line
On putting and in the given equation, we get
Hence,
Prove that \((\sin\theta+\text{cosec}\ \theta)^2+(\cos\theta+\sec\theta)^2=7+\tan^2\theta+\cot^2\theta \).
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L.H.S.
.
Prove that \((1+\cot A\text{cosec}\ A)\ (1+\tan A+\sec A)=2 \).
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L.H.S.
In Fig. \(PQ \) is a chord of length 8 cm of a circle of radius 5 cm and centre O. The tangents at \(P \) and \(Q \) intersect at point \(T \). Find the length of \(TP \).
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Join , let it intersect at the point
Now, is an isosceles triangle and is the angle bisector of .
So, and therefore, bisects
Also,
Now,
Hence, the length of
In Fig. \(\angle ACB=90^o \) and \(CD\bot AB \), prove that \(CD^2=BD\times AD \).
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Given, in which and
To prove:
Proof: In and
(common)
( each)
(By AA rule)â€¦(i)
Similarly,
(By AA rule)â€¦(ii)
From equation (i) and (ii)
(by the definition of similarity of triangles)
or
or
Hence Proved.
If \(P \) and \(Q \) are the points on side \(CA \) and \(CB \) respectively of \(\triangle ABC \), rightangled at \( C\), prove that \((AQ^2+BP^2)=(AB^2+PQ^2) \).
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Given, is a rightangled triangle in which
To prove:
Construction: Join and
Proof: In
(Using Pythagoras theorem)
In
(Using Pythagoras theorem)
Adding equation (i) and (ii)
or
Hence Proved.
Find the area of the shaded region in Fig. if ABCD is a rectangle with sides 8 cm and 6 cm and D is the centre of the circle.
[Take \( \pi=3.14\)]
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Given, is a rectangle with sides and
In
(By Pythagoras Theorem)
The diagonal of the rectangle will be the diameter of the circle radius of the circle
Area of shaded portion = Area of circle â€“ Area of Rectangle
Hence, Area of shaded portion
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/hour. How much area will it irrigate in 30 minutes, if 8 cm standing water is needed?
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Let be the width and be the depth of the canal
and
Water is flowing with a speed
Length of water flowing in
Length (l) of water flowing in
Volume of water flowing in 30 min.
Let the area irrigated in 30 min be
Volume of water required for irrigation = Volume of water flowing in 30 min.
hectares ( hectares )
Hence, the canal will irrigate hectares in min.
Find the mode of the following frequency distribution.
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The given frequency distribution table is
Here, the maximum class frequency is
Modal class
lower limit (l) of modal class
Class size (h)
Frequency of the modal class
Frequency of preceding class
Frequency of succeeding class
Hence, Mode .
Two water taps together can fill a tank in \(1\dfrac{7}{8} \) hours. The tap with longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.
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Portion of tank filled by A in 1 hour
Portion of tank filled by B in 1 hour
Portion of tank filled by both taps in 1 hour
Time taken with long diameter tap hours
Time taken with smaller diameter tap hours
If the sum of first four terms of an A.P. is \(40 \) and that of first \(14 \) terms is \(280 \). Find the sum of its first \(n \) terms.
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Given, and
If a be the first term and d be the common difference of an A.P.
Then, Sum of term
or
or
or
or ….(i)
Also, Sum of first terms
or ….(ii)
On solving equation (i) and (ii), we get
Now, sum of terms
On putting
Hence, the sum of first terms is
A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from \(60^o \) to \(30^o \). Find the speed of the boat in metres per minute. [Use latex]\sqrt3=1.732 [/latex]]
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Let be the lighthouse and be the two positions of the boat, such that,
and
Now, In
….(i)
In
or ….(ii)
From equation (i) and (ii)
metres
Time taken to cover
Speed of boat
Hence, speed of boat
Construct a \(\triangle ABC \) in which \(CA=6\ cm,\ AB=5\ cm \) and \(\angle BAC=45^o \). Then construct a triangle whose sides are \(35 \) of the corresponding sides of \(\triangle ABC \).
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Steps of Construction are as follows:
1. Draw
2. At the point, draw .
3. From cut off
4. Join is formed with given data.
5. Draw making an acute angle with as shown in the figure.
6. Draw arcs and with equal intervals.
7. Join .
8. Draw meeting at .
9. From , draw meeting at
Hence is the required triangle.
A bucket open at the top is in the form of a frustum of a cone with a capacity of \(12308.8\ cm^3 \). The radii of the top and bottom of circular ends of the bucket are \( 20\ cm\) and \(12\ cm \) respectively. Find the height of the bucket and also the area of the metal sheet used in making it. (Use \(\pi=3.14 \))
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Let and be the radii of the top and the bottom circular ends of the bucket respectively.
Let be the height of the bucket.
and
Capacity of the bucket
Volume of bucket (frustum)
or
or
or
Thus, the height of the bucket is .
The area of the metal sheet used in making the bucket = CSA of bucket + area of the circular bottom
Area of metal sheet used
Prove that in a rightangle triangle, the square of the hypotenuse is equal the sum of squares of the other two sides.
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Given, right angled at .
To prove:
Construction :
Proof: In and
(common)
( each)
(By AA rule)
So, (sides are proportional)
or
Also, In and
(common)
( each)
So,
or
Adding equation (i) and (ii), we get
or
Hence Proved.