Q1
mcq
1 mark
Which of the following is true for all values of $\theta$ ($0^{\circ} \le \theta \le 90^{\circ}$) ?
- A. $\cos^2 \theta – \sin^2 \theta = 1$
- B. $\operatorname{cosec}^2 \theta – \sec^2 \theta = 1$
- C. $\sec^2 \theta – \tan^2 \theta = 1$
- D. $\cot^2 \theta – \tan^2 \theta = 1$
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Q2
mcq
1 mark
If $k + 2, 4k – 6$ and $3k – 2$ are three consecutive terms of an A.P., then the value of $k$ is :
- A. 3
- B. – 3
- C. 4
- D. – 4
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Q3
mcq
1 mark
In $\triangle ABC$, $PQ \parallel BC$. If $PB = 6$ cm, $AP = 4$ cm, $AQ = 8$ cm, find the length of $AC$.
- A. 12 cm
- B. 20 cm
- C. 6 cm
- D. 14 cm
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Q4
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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Q5
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. $\frac{1}{13}$
- B. $\frac{9}{13}$
- C. $\frac{4}{13}$
- D. $\frac{12}{13}$
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Q6
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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Q7
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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Q8
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$^{\circ}$
- B. 45$^{\circ}$
- C. 30$^{\circ}$
- D. 90$^{\circ}$
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Q9
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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Q10
mcq
1 mark
What is the area of a semi-circle of diameter ‘d’ ?
- A. $\frac{1}{16} \pi d^2$
- B. $\frac{1}{4} \pi d^2$
- C. $\frac{1}{8} \pi d^2$
- D. $\frac{1}{2} \pi d^2$
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Q11
mcq
1 mark
For the following distribution: The sum of lower limits of median class and modal class is :
- A. 15
- B. 25
- C. 30
- D. 35
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Q12
mcq
1 mark
The length of tangent drawn to a circle of radius 9 cm from a point 41 cm from the centre is :
- A. 40 cm
- B. 9 cm
- C. 41 cm
- D. 50 cm
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Q13
mcq
1 mark
In the given figure, O is the centre of the circle and PQ is the chord. If the tangent PR at P makes an angle of 50$^{\circ}$ with PQ, then the measure of $\angle POQ$ is :
- A. 50$^{\circ}$
- B. 40$^{\circ}$
- C. 100$^{\circ}$
- D. 130$^{\circ}$
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Q14
mcq
1 mark
The sum of lower limits of median class and modal class is :
- A. 15
- B. 25
- C. 30
- D. 35
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Q14
mcq
1 mark
A bag contains 5 red balls and n green balls. If the probability of drawing a green ball is three times that of a red ball, then the value of n is :
- A. 18
- B. 15
- C. 10
- D. 20
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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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Q16
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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Q17
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. either intersecting or parallel
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Q19
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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Q20
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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Q21
short answer
2 marks
Evaluate : $\frac{2\cot^2 30^\circ + \sin^2 60^\circ - \cot^2 45^\circ + 2\sin^2 90^\circ}{...}$
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Q22
short answer
2 marks
If $\theta$ is an acute angle and $\sin \theta = \cos \theta$, find the value of $\tan^2 \theta + \cot^2 \theta - 2$.
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Q22
short answer
2 marks
If a fair coin is tossed twice, find the probability of getting ‘atmost one head’.
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Q23
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 sum and product of the roots of the quadratic equation $2x^2 - 9x + 4 = 0$.
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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
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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Q26
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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Q27
short answer
3 marks
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
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Q28
short answer
3 marks
Find the value of ‘p’ for which the quadratic equation $px(x - 2) + 6 = 0$ has two equal real roots.
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Q30
short answer
3 marks
Prove that $3$ is an irrational number.
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Q31
short answer
3 marks
The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20th term.
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Q32
short answer
3 marks
Rohan repays his total loan of ₹ 1,18,000 by paying every month starting with the first instalment of ₹ 1,000. If he increases the instalment by ₹ 100 every month, what amount will be paid by him in the 30th instalment ? What amount of loan has he paid after 30th instalment ?
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Q32
long answer
5 marks
From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hallowed out. Find the total surface area of the remaining solid.
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Q33
short answer
3 marks
Prove that $\sec A (1 - \sin A) (\sec A + \tan A) = 1$
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Q33
long answer
5 marks
The monthly expenditure on milk in 200 families of a Housing Society is given below: Monthly Expenditure (in `): 1000-1500, 1500-2000, 2000-2500, 2500-3000, 3000-3500, 3500-4000, 4000-4500, 4500-5000; Number of families: 24, 40, 33, x, 30, 22, 16, 7. Find the value of x and also, find the median and mean expenditure on milk.
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Q35
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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Q38
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 $30^\circ$ and $60^\circ$, 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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Q39
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 $60^\circ$ and the angle of depression of its foot is $30^\circ$. Determine the height of the tower.
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Q40
long answer
5 marks
In the given figure, $\angle ADC = \angle BCA$; prove that $\triangle ACB \sim \triangle ADC$. Hence find BD if AC = 8 cm and AD = 3 cm.
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Q41
long answer
5 marks
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
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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.
Q42
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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Q44
short answer
2 marks
What is the length of diagonal PR in square PQRS ?
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Q45
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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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.
Q46
short answer
1 mark
What is the total perimeter of the parking area ?
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Q47
short answer
2 marks
What is the total area of parking and the two quadrants ?
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Q48
short answer
2 marks
What is the ratio of area of playground to the area of parking area ?
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Q49
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.
Q50
short answer
1 mark
Represent the following information algebraically (in terms of x and y).
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Q51
short answer
2 marks
What is the prize amount for hockey ?
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Q52
short answer
2 marks
Prize amount on which game is more and by how much ?
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Q53
short answer
1 mark
What will be the total prize amount if there are 2 students each from two games ?
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