Q1
mcq
1 mark
The pair of linear equations $2x = 5y + 6$ and $15y = 6x – 18$ represents two lines which are :
- A. intersecting
- B. parallel
- C. coincident
- D. intersecting or parallel
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Q2
mcq
1 mark
In the given figure, TA is a tangent to the circle with centre O such that OT = 4 cm, $\angle OTA = 30^o$, then length of TA is :
- A. $2\sqrt{3}$ cm
- B. 2 cm
- C. $2\sqrt{2}$ cm
- D. 3 cm
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Q3
mcq
1 mark
The ratio of HCF to LCM of the least composite number and the least prime number is :
- A. 1:2
- B. 2:1
- C. 1:1
- D. 1:3
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Q4
mcq
1 mark
If a pole 6 m high casts a shadow $2\sqrt{3}$ m long on the ground, then sun’s elevation is :
- A. $60^o$
- B. $45^o$
- C. $30^o$
- D. $90^o$
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Q5
mcq
1 mark
In the given figure, $\triangle ABC \sim \triangle QPR$. If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; then the value of x is :
- A. 3.6 cm
- B. 2.5 cm
- C. 10 cm
- D. 3.2 cm
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Q6
mcq
1 mark
The distance of the point (– 6, 8) from origin is :
- A. 6
- B. – 6
- C. 8
- D. 10
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Q7
mcq
1 mark
The next term of the A.P. : 7, 28, 63 is :
- A. 70
- B. 80
- C. 97
- D. 112
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Q8
mcq
1 mark
($\sec^2$ θ – 1) ($\csc^2$ θ – 1) is equal to :
- A. – 1
- B. 1
- D. 2
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Q9
mcq
1 mark
Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equals to 3 is :
- A. 1/9
- B. 2/9
- C. 1/6
- D. 1/12
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Q10
mcq
1 mark
A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not an ace is :
- A. 1/13
- B. 9/13
- C. 4/13
- D. 12/13
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Q11
mcq
1 mark
The roots of the equation $x^2 + 3x – 10 = 0$ are :
- A. 2, –5
- B. –2, 5
- C. 2, 5
- D. –2, –5
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Q12
mcq
1 mark
If α, β are zeroes of the polynomial $x^2 – 1$, then value of (α + β) is :
- A. 2
- B. 1
- C. –1
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Q13
mcq
1 mark
If α, β are the zeroes of the polynomial $p(x) = 4x^2 – 3x – 7$, then $(1/α + 1/β)$ is equal to :
- A. 7/3
- B. -7/3
- C. 3/7
- D. -3/7
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Q13
mcq
1 mark
If $\alpha, \beta$ are the zeroes of the polynomial $p(x) = 4x^2 - 3x - 7$, then $\frac{1}{\alpha} + \frac{1}{\beta}$ is equal to :
- A. $\frac{7}{3}$
- B. $-\frac{7}{3}$
- C. $\frac{3}{7}$
- D. $-\frac{3}{7}$
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Q14
mcq
1 mark
What is the area of a semi-circle of diameter ‘d’ ?
- A. $1/16 πd^2$
- B. $1/4 πd^2$
- C. $1/8 πd^2$
- D. $1/2 πd^2$
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Q15
mcq
1 mark
If $\alpha, \beta$ are zeroes of the polynomial $x^2 - 1$, then value of $(\alpha + \beta)$ is :
- A. 2
- B. 1
- C. –1
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Q15
mcq
1 mark
For the following distribution : Marks Below 10 20 30 40 50 60 Number of Students 3 12 27 57 75 80 The modal class is :
- A. 10-20
- B. 20-30
- C. 30-40
- D. 50-60
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Q16
mcq
1 mark
In the given figure, PT is a tangent at T to the circle with centre O. If ∠TPO = 25°, then x is equal to :
- A. 25°
- B. 65°
- C. 90°
- D. 115°
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Q17
mcq
1 mark
In the given figure, PQ || AC. If BP = 4 cm, AP = 2.4 cm and BQ = 5 cm, then length of BC is :
- A. 8 cm
- B. 3 cm
- C. 0.3 cm
- D. 25/3 cm
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Q18
mcq
1 mark
The points (– 4, 0), (4, 0) and (0, 3) are the vertices of a :
- A. right triangle
- B. isosceles triangle
- C. equilateral triangle
- D. scalene triangle
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Q19
mcq
1 mark
Assertion (A) : The probability that a leap year has 53 Sundays is 2/7. Reason (R) : The probability that a non-leap year has 53 Sundays is 5/7.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- C. Assertion (A) is true but Reason (R) is false.
- D. Assertion (A) is false but Reason (R) is true.
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Q20
mcq
1 mark
Assertion (A) : a, b, c are in A.P. if and only if 2b = a + c. Reason (R) : The sum of first n odd natural numbers is n^2.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- C. Assertion (A) is true but Reason (R) is false.
- D. Assertion (A) is false but Reason (R) is true.
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Q22
short answer
2 marks
(A) _mZ kmV H$s{OE … 2sec^2 θ + 3cosec^2 θ – 2sin θcos θ, if θ = 45°. (B) `{X sin θ – cos θ = 0 h¡, Vmo sin^4 θ + cos^4 θ H$m _mZ kmV H$s{OEŸ&
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Q23
short answer
2 marks
If a fair coin is tossed twice, find the probability of getting ‘atmost one head’.
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Q24
short answer
2 marks
Find the sum and product of the roots of the quadratic equation $2x^2 - 9x + 4 = 0$.
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Q24
short answer
2 marks
Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers ?
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Q25
short answer
2 marks
Find the discriminant of the quadratic equation $4x^2 - 5 = 0$ and hence comment on the nature of roots of the equation.
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Q25
short answer
2 marks
If one zero of the polynomial $p(x) = 6x^2 + 37x - (k - 2)$ is reciprocal of the other, then find the value of k.
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Q26
short answer
2 marks
Evaluate $2\sec^2 \theta + 3\csc^2 \theta - 2\sin \theta \cos \theta$ if $\theta = 45^\circ$.
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Q26
short answer
3 marks
Find the value of ‘p’ for which one root of the quadratic equation $px^2 - 14x + 8 = 0$ is 6 times the other.
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Q27
short answer
2 marks
If $\sin \theta - \cos \theta = 0$, then find the value of $\sin^4 \theta + \cos^4 \theta$.
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Q27
short answer
3 marks
From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the centre of the circle bisects the angle between the two tangents.
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Q28
short answer
3 marks
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
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Q31
short answer
3 marks
Prove that $\sqrt{5}$ is an irrational number.
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Q33
long answer
5 marks
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the figure. 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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Q34
short answer
3 marks
How many terms are there in A.P. whose first and fifth term are -14 and 2, respectively and the last term is 62.
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Q34
long answer
5 marks
The monthly expenditure on milk in 200 families of a Housing Society is given below. Find the value of $x$ and also, find the median and mean expenditure on milk.
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Q35
short answer
3 marks
Which term of the A.P. : 65, 61, 57, 53, .................. is the first negative term ?
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Q36
short answer
3 marks
Prove that $\frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} = \tan A$
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Q37
short answer
3 marks
Prove that $\sec A (1 - \sin A) (\sec A + \tan A) = 1$.
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Q39
long answer
5 marks
In a $\triangle PQR$, N is a point on PR, such that $QN \perp PR$. If $PN \times NR = QN^2$, prove that $\angle PQR = 90^\circ$.
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Q40
long answer
5 marks
In the given figure, $\triangle ABC$ and $\triangle DBC$ are on the same base BC. If AD intersects BC at O, prove that $\frac{ar(ABC)}{ar(DBC)} = \frac{AO}{DO}$.
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Q43
long answer
5 marks
A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30° and 60°, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use $\sqrt{3} = 1.73$)
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Q44
long answer
5 marks
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.
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Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.
Q45
short answer
What is the perimeter of the parking area?
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Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.
Q46
short answer
2 marks
What is the total area of parking and the two quadrants ?
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Q47
short answer
2 marks
What is the ratio of area of playground to the area of parking area ?
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Q48
short answer
1 mark
Find the cost of fencing the playground and parking area at the rate of ` 2 per unit.
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Two schools ‘P’ and ‘Q’ decided to award prizes to their students for two games of Hockey ` x per student and Cricket ` y per student. School ‘P’ decided to award a total of ` 9,500 for the two games to 5 and 4 students respectively; while school ‘Q’ decided to award ` 7,370 for the two games to 4 and 3 students respectively.
Q49
short answer
1 mark
Represent the following information algebraically (in terms of x and y).
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Q50
short answer
2 marks
What is the prize amount for hockey ?
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Q51
short answer
2 marks
Prize amount on which game is more and by how much ?
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Q52
short answer
1 mark
What will be the total prize amount if there are 2 students each from two games ?
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Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.
Q53
short answer
1 mark
Taking O as origin, coordinates of P are (-200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S ?
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Q54
short answer
2 marks
What is the area of square PQRS ?
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Q55
short answer
2 marks
What is the length of diagonal PR in square PQRS ?
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Q56
short answer
1 mark
If S divides CA in the ratio K:1, what is the value of K, where point A is (200, 800) ?
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