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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The ratio of the height of a tower and the length of its shadow on the ground is \(\sqrt3:1 \). What is the angle of elevation of the sun?
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Given,
In ,
Hence, the angle of elevation is .
Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of the hemisphere?
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Let radius of hemisphere be units
Volume of hemisphere = S.A. of hemisphere
or diameter units
A number is chosen at random from the numbers 3, 2, 1,0, 1, 2, 3. What will be the probability that square of this number is less than or equal to 1?
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Possible outcomes
and only three numbers fall under given condition so, Required probability
If the distance between the points \((4,k) \) and \((1,0) \) is \(5 \), then what can be the possible values of \(k \)?
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Distance between and
On squaring both sides,
Find the roots of the quadratic equation \(\sqrt2\ x^2+7x+5\sqrt2=0 \).
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Given the quadratic equation is,
[Splitting middle term]
Hence roots are and .
Find how many integers between 200 and 500 are divisible by 8.
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Smallest divisible no. (by ) in given range
Last divisible no. (by ) in range
So,
So number of terms between 200 and 500 divisible by 8 are 37.
What is the common difference of an A.P. in which \(a_{21} – a_7 = 84 \)?
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Given,
Hence common difference
If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60Â°, then find the length of OP.
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Given,
In right angle ,
If a tower \( 30\ m\) high, casts a shadow \(10\sqrt3\ m \) long on the ground, then what is the angle of elevation of the sun?
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In ,
Hence angle of elevation is .
The probability of selecting a rotten apple randomly from a heap of 900 apples is 018. What is the number of rotten apples in the heap?
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Total apples
No. of rotten apples
Which term of the A.P. 8, 14, 20, 26,â€¦ will be 72 more than its 41st term?
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For what value of \(n \), are the terms of two A.Ps 63, 65, 67,â€¦. and 3, 10, 17,â€¦. equal ?
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AP. Is
According to question,
Hence, 13th term of both A.P. is equal.
If \(\sqrt3\tan\theta=1 \), the value of \(\sin^2\theta\cos^2\theta \) is
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Is this statement correct?
The tangents drawn at the extremities of the diameter of a circle are parallel
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But they form alternate angles So,
Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m. Find the distance between their tops
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A top is in the shape of cone over a hemisplane and the radius of the hemisplane is 3.5 cm. If the height of the top is 15.5 cm. Find the total area of the top.
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Height of the cone
CSA of the cone
CSA of the hemisphere
Total area of the top
If \(n^{th} \) term of an AP is \(3n8 \). Find its \(16^{th} \) term
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If the distance between the points \((4,P) \) and \((1,0) \) is \(5 \). Find the value of \(P \)
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If \(\tan^245^o\cos^230^o=x\sin30^o\cos60^o \) then, find the value of \(x \)
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The simplified form of \( \cos^4\theta\sin^4\theta\) is
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Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other.
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Given, is a diameter of a circle with centre .
The lines and are tangents at and respectively.
To Prove:
Proof: is a tangent to the circle at and is the radius through the point of contact
Similarly, is a tangent to circle at and is radius through the point of contact
But both form pair of alternate angles
Hence Proved.
Find the value of \(k \) for which the equation \(x^2 + k(2x + k – 1) + 2 = 0 \) has real and equal roots.
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Given equation is,
Here and
For real and equal roots
Draw a line segment of length 8 cm and divide it internally in the ratio 4 : 5.
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Steps of construction:
1. Draw
2. Draw any ray making an acute angle with .
3. Draw points on ray namely at equal distance.
4. Join .
5. Through point , draw a line parallel to intersecting at the point .
Then
In the given figure, \(PA \) and \(PB \) are tangents to the circle from an external point \(P.\ CD \) is another tangent touching the circle at \(Q \). If \(PA = 12\ cm, QC = QD = 3\ cm, \) then find \(PC + PD. \)
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Given, [Tangent from external point]
[Tangent from external point]
So,
If mth term of an A.P. is 1n and n term is 1m, then find the sum of its first mn terms.
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Let and be the first term and common difference respectively of the given A.P.
Then, term â€¦(i)
term …(ii)
By subtracting eq. (ii) from eq. (i),
Putting in eq. (i),
We get,
Sum of first terms
If the equation \((1 + m^2)x^2 + 2mcx + c^2 – a^2 = 0 \) has equal roots then show that \(c^2 = a^2( 1 + m^2) \).
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The given equation has equal roots
Here,
For equal roots,
The \(\dfrac{3}{4}^{th} \) part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with an internal radius of 10 cm. Find the height of water in a cylindrical vessel.
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According to the question,
Volume of water in conical vessel = Volume of cylindrical vessel
Hence the height of water in a cylindrical vessel is 1.5 cm.
In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.
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Area of shaded region = Area of quadrant OACB â€“ Area of DOB
Hence, area of the shaded region is
Two tangents \(TP \) and \(TQ \) are drawn to a circle with centre \(O \) from an external point \(T \). Prove that \(\angle PTQ=2\angle OPQ \).
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Given, a circle with centre , an external point and two tangents and .
Let
To Prove:
Proof: [Tangent from an external point]
So, is an isosceles triangle
[Angle opposite to equal sides of a latex]\triangle [/latex]]
So,
But, [Angle between tangent and radius]
or
Hence Proved.
Show that \(\triangle ABC \), where \(A(2, 0), B(2, 0), C(0, 2) \) and \(\triangle PQR \) where \(P(4, 0), Q(4, 0), R(0, 4) \) are similar triangles.
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Coordinates of vertices are
We see that sides of are twice the sides of .
Hence, both triangles are similar.
Hence Proved.
The area of a triangle is \(5 \) sq units. Two of its vertices are \((2,1) \) and \((3,2) \). If the third vertex is \((72, y)\), find the value of \(y\).
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Given, and
Now, Area
Two different dice are thrown together. Find the probability that the numbers obtained
(i) have a sum less than 7
(ii) have a product less than 16
(iii) is a doublet of odd numbers.
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Total possible outcomes in each case
(i) Have a sum less than 7, Possible outcomes are,
So, probability
(ii) Have a product less than 16, Possible outcomes are,
So, probability
(iii) Is a doublet of odd no.,
Possible outcomes are
P(doublet of odd no.)
A moving boat is observed from the top of a \(150\ m \) high cliff moving away from the cliff. The angle of depression of the boat changes from \(60^o \) to \(45^o \) in \(2 \) minutes. Find the speed of the boat in \(\dfrac{\text m}{\text h} \).
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From ,
or
From
Distance covered in 2 min.
Distance covered in 1 hour
Speed
Hence, the speed of boat is 1902 m/hr.
Construct an isosceles triangle with base \(8\ cm \) and altitude \(4\ cm \). Construct another triangle whose sides are \(\dfrac{2}{3} \) times the corresponding sides of the isosceles triangle.
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Steps of construction:
1. Draw .
2. Construct , the perpendicular bisector of line segment , meeting at .
3. Cut on . Join is obtained.
4. At , draw an acute angle in a downward direction. Draw arcs and on it.
5. Join and at draw line parallel to , cutting at .
6. At , draw parallel to .
Thus, is the required triangle.
The ratio of the sums of the first \(m \) and first \(n \) terms of an A. P. is \(m^2:n^2 \).
Show that the ratio of its mth and nth and terms is \((2m – 1): (2n – 1). \)
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Let be first term and is a common difference.
Hence Proved.
Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.
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Let speed of the stream be .
According to question,
If \(a\ne b\ne c \), prove that the points \((a, a^2), (b, b^2) (c, c^2) \) will not be collinear.
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This can never be zero as
Hence, these points can never be collinear.
Hence Proved.
The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.
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Let
Since is midpoint of
is midpoint of or
Also
Now,
Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25.
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Total possible events in case of peter is 36 favourable outcome is (5, 5)
So, P(getting as product) =
While total possible event in case of Rina is 6
Favourable outcome is 5
So, P(square is ) =
As , so Rina has better chance.
A chord \(PQ \) of a circle of radius \(10\ cm \) subtends an angle of \(60^o \) at the centre of circle. Find the area of major and minor segments of the circle.
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Area of minor segment
Area of major segment
Area of circle – Area of minor segment