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 20 comprises of 20 questions of one mark each.
(iii) Section B: Q. No. 21 to 26 comprises of 6 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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If \(x = 3 \) is one root of the quadratic equation \(x^2 – 2kx – 6 = 0 \), then find the value of \(k \).
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Given quadratic equation is,
is a root of above equation, then
What is the HCF of the smallest prime number and the smallest composite number?
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Smallest prime number
Smallest composite number
Prime factorisation of is
Prime factorisation of is
HCF
Find the distance of a point \(P(x,y) \) from the origin.
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The given point is
The origin is
The distance of point from the origin,
unit
In an AP if the common difference \((d)=4 \) and the seventh term \((a_7) \) is \(4 \), then find the first term.
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Given,
What is the value of \((\cos^267^o\sin^2\ 23^o) \)?
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We have,
Given \(\triangle ABC\sim\triangle PQR \), if \(ABPQ=13 \), then find \(\dfrac{\text{ar}\triangle ABC}{\text{ar}\triangle PQR} \)
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Find the sum of the first 8 multiples of 3.
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First 8 multiples of 3 are 3, 6, 9,….. up to 8 terms
Sum of n terms of an A.P is given by,
Two different dice are tossed together. Find the probability of getting a doublet.
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Total outcomes on tossing two different dice = 36
Number of favourable outcomes of
A five digit number is the multiple of 11 and 91. If the second digit from left is 8, then find the fourth digit from the left of the number
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LCM
Find the number of solutions to the pair of equations \(x=0 \) and \(x=3 \)
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is the axis and is the line parallel to axis at a distance of 3 units. These lines do not meet any where any where so no solution exists
If the \(5^{th} \) term of an AP is O, find the relation between \(12^{th} \) and \(26^{th} \) term
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If the sum of \(n \) terms of an AP is \(3n^2+5n \), then which of its terms is \(164 \)?
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Let be term
If \(\text{cosec}\ A=2 \) then find the value of \(\dfrac{1}{\tan A}+\dfrac{\sin A}{1+\cos A} \)
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The minute hand of a clock is 10 cm long. Find the area swept by the minute hand between 8:00 am to 8:25 am
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Angle swept by minute hand in 25 min
Area of sector
In the given figure, \(\angle ABC=90^o \) and \(BD\bot AC \). If \(BD=8\ cm \) and \(AD=4\ cm \) then the value of \(CD \)
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In and
(each )
(each )
(AA)
Find the distance between two parallel tangents of a circle having radius 3 cm
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When we rotate a right triangle about the perpendicular line then write the shape of the three dimensional formed figure
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While rotating the right triangle about the perpendicular line, a cone is formed
Find the value of \(\dfrac{1\tan^245^o}{1+\tan^245^o} \)
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The simplified value of \((1\cos^2A)\ \text{cosec}^2A \)
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If a cylinder is filled with water and spherical ball of r is dropped into the cylinder, find the quantity of water spread out
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When we dropped the ball into full cylinder of water, the quantity of spreading water is equal to the volume of sphere
Given that \(\sqrt2 \) is irrational, prove that \((5+3\sqrt2) \) is an irrational number.
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Given is an irrational number.
Let
Suppose, is a rational number.
So,
or
So,
But is rational number, so is rational number which contradicts the fact that is irrational number.
So, our supposition is wrong.
Hence, is also irrational.
Hence Proved.
In fig. 1, \(ABCD \) is a rectangle. Find the values of \(x \) and \(y \).
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Given, is a rectangle.
or
Similarly,
or
On adding eq. (i) and (ii), we get
Putting the value of in eq. (i), we get
So,
Find the ratio in which \(P(4, m) \) divides the line segment joining the points \( A(2, 3) \) and \(B(6, 3) \). Hence find \(m \).
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Let divides line segment in the ratio
Coordinates of
On comparing, we get
Hence, divides in the ratio .
From (i),
An integer is chosen at random between 1 and 100. Find the probability that it is:
(i) divisible by 8.
(ii) not divisible by 8.
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The total number are
(i) Let be the event of getting a number divisible by .
(ii) Let be the event of getting a number not divisible by .
Then,
Find HCF and LCM of 404 and 96 and verify that HCF \(\times \) LCM = Product of the two given numbers.
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Prime factorization of
Prime factorization of
HCF
And LCM
HCF , LCM
Verification:
HCF LCM = Product of the two given numbers
Hence Verified.
Find all zeroes of the polynomial \((2x^4 – 9x^3 + 5x^2 + 3x – 1) \) if two of its zeroes are \((2+\sqrt3) \) and \( (2\sqrt3)\).
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Here,
And two of its zeroes are and .
Quadratic polynomial with zeroes is given by,
(say)
Now, will be a factor of so will be divisible by
For other zeroes,
Zeroes of are
and .
A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.
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Let the usual speed of plane be km/h.
Increased speed = km/h.
Distance to cover km.
Time taken by plane with usual speed hr
Time taken by plane with increased speed
According to the question,
Either (Rejected)
or
Usual speed of plane km/hr.
Prove that the lengths of tangents drawn from an external point of a circle are equal.
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A circle with centre on which two tangents and are drawn from an external point .
To Prove:
Construction: Join and
Proof: Since tangent and radius are perpendicular at point of contact,
In and ,
(Radii)
(Common)
(RHS cong.)
(C.P.C.T)
Hence Proved.
If \(4\tan\theta=3 \), evaluate \((4\sin\theta\cos\theta+14\sin\theta+\cos\theta1) \)
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Given,
Now,
If \(\tan2A=\cot(A18^o) \), where \(2A \) is an acute angle, find the value of \(A \).
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Given,
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Find the area of the shaded region in Fig. where arcs are drawn with centres A, B, C and D intersect in pairs at midpoints P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use \(\pi=3.14 \)]
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P, Q, R and S are the midpoints of sides AB, BC, CD and AD respectively.
Area of shaded region = Area of square – 4 Area of quadrant
A wooden article was made by scooping out a hemisphere form each end of a solid cylinder, as shown in Fig. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm. Find the total surface area of the article.
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Given, Radius (r) of cylinder = Radius of hemisphere = 3.5 cm.
Total SA of article = CSA of cylinder + 2 CSA of hemisphere
Height of cylinder,
A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?
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Base diameter of cone
Radius
Height of cone,
Volume of rice in conical heap
Now, slant height,
Canvas cloth required to just cover the heap =
CSA of conical heap
.
The table below shows the salaries of 280 persons:
Calculate the median salary of the data.
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The cumulative frequency just greater than is .
Median class is .
, c.f. and
A motorboat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
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Given, speed of motorboat instil
water = km/hr.
Let speed of stream = km/hr.
Speed of boat downstream = km/hr.
And speed of boat upstream km/hr.
Time of the upstream journey =
Time of the downstream journey =
According to the question,
Either
Rejected, as speed cannot be negative
or
Thus, the speed of the stream is km/hr.
The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7:15. Find the numbers.
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Let the first term of AP be and be a common difference.
Let your consecutive term of an AP be and
According to the question,
Also,
For , four terms of AP are,
For , four term are
Thus, the four terms of AP series are or .
In an equilateral \(\triangle ABC,D\) is a point on side \(BC \) such that \(BD=\dfrac{1}{3}\ BC \). Prove that \(9(AD)^2=7(AB)^2 \).
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Given, is an equilateral triangle and is a point on such that .
To prove:
Construction : Draw
Proof: (Given)
We know that perpendicular from a vertex of equilateral triangle to the base divides base in two equal parts.
In ,
(Pythagoras theorem)
or
Similarly, In ,
[from equation (ii) and (iii)]
Hence Proved.
Draw a triangle \(ABC \) with \(BC = 6\ cm, AB = 5\ cm \) and \(\angle ABC=60^o \). Then construct a triangle whose sides are 34 of the corresponding sides of the \(\triangle ABC \).
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1. Draw a line segment
2. Construct
3. With as centre and radius equal to , draw an arc intersecting at .
4. Join . Thus, is obtained.
5. Draw an acute angle below of .
Mark equal parts on as and
7. Join to .
8. From By draw a line parallel to intersecting at .
9. Draw another line parallel to from , intersecting at .
10. is required triangle which is similar to such that .
The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:
(i) The area of the metal sheet used to make the bucket.
(ii) Why we should avoid the bucket made by ordinary plastic? [Use latex]\pi=3.14 [/latex]]
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Given, Height of frustum,
Diameter of lower end
Radius of lower end, .
Diameter of upper end
Radius of upper end,
Slant height,
(i) Area of metal sheet used to make the bucket = CSA of frustum + Area of base
(ii) We should avoid the bucket made by ordinary plastic because plastic is harmful to the environment and to protect the environment its use should be avoided.
As observed from the top of a 100 m high lighthouse from the sealevel, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. [Use \(\sqrt3=1.732 \)]
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Let be the lighthouse and two ships are at and .
In
Similarly, in ,
Distance between two ships
[from equation (i) and (ii)]