Q1
mcq
If one zero of the polynomial $x^2 - 3kx + 4k$ be twice the other, then the value of k is :
- A. 2
- B. -2
- C. 1/2
- D. -1/2
Solution hidden
Log in to view solution →
Q2
mcq
The ratio in which the x-axis divides the line segment joining the points (2, 3) and (6, 7) is :
- A. 1 : 3
- B. 3 : 7
- C. 7 : 3
- D. 1 : 2
Solution hidden
Log in to view solution →
Q3
mcq
What is the total surface area of a solid hemisphere of diameter d?
- A. 3 $d^2$
- B. 2 $d^2$
- C. 1/2 $d^2$
- D. 3/4 $d^2$
Solution hidden
Log in to view solution →
Q4
mcq
What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90°?
- A. 22 cm
- B. 44 cm
- C. 88 cm
- D. 11 cm
Solution hidden
Log in to view solution →
Q5
mcq
If x = 0·3 is a root of the equation $x^2 - 0·9k = 0$, then k is equal to :
- A. 1
- B. 10
- C. 0·1
- D. 100
Solution hidden
Log in to view solution →
Q6
mcq
In the given figure, AB || PQ. If AB = 6 cm, PQ = 2 cm and OB = 3 cm, then the length of OP is :
- A. 9 cm
- B. 3 cm
- C. 4 cm
- D. 1 cm
Solution hidden
Log in to view solution →
Q7
mcq
If p and q are prime numbers and a = p^3q^2 and b = pq^2, then HCF of a and b is:
- A. pq^2
- B. p^2q
- C. q
- D. p + q
Solution hidden
Log in to view solution →
Q8
mcq
If ΔABC ~ ΔPQR, ∠A = 32° and ∠R = 65°, then the measure of ∠B is:
- A. 32°
- B. 65°
- C. 83°
- D. 97°
Solution hidden
Log in to view solution →
Q9
mcq
The pair of equations x = a and y = b graphically represents lines which are:
- A. parallel
- B. intersecting at (b, a)
- C. coincident
- D. intersecting at (a, b)
Solution hidden
Log in to view solution →
Q10
mcq
The area of the triangle formed by the line x/a + y/b = 1 with the coordinate axes is:
- A. ab
- B. 1/2 ab
- C. 1/4 ab
- D. 2ab
Solution hidden
Log in to view solution →
Q11
mcq
In a single throw of two dice, the probability of getting 12 as a product of two numbers obtained is:
- A. 1/9
- B. 2/9
- C. 4/9
- D. 5/9
Solution hidden
Log in to view solution →
Q12
mcq
If α and β are the zeroes of the quadratic polynomial ax^2 - 5x + c and α + β = αβ = 10, then:
- A. a = 5, c = 1/2
- B. a = 1, c = 5/2
- C. a = 5/2, c = 1
- D. a = 1/2, c = 5
Solution hidden
Log in to view solution →
Q13
mcq
A bag contains 100 cards numbered 1 to 100. A card is drawn at random from the bag. What is the probability that the number on the card is a perfect cube?
- A. 1/20
- B. 3/50
- C. 1/25
- D. 7/100
Solution hidden
Log in to view solution →
Q14
mcq
In the given figure, DE || BC. If AD = 2 units, DB = 3 units, AE = 3 units and EC = x units, then the value of x is:
- A. 2
- B. 3
- C. 5
- D. 9/2
Solution hidden
Log in to view solution →
Q15
mcq
If the pair of equations 3x - y + 8 = 0 and 6x - ry + 16 = 0 represent coincident lines, then the value of r is:
- A. 1/2
- B. -1/2
- C. 2
- D. -2
Solution hidden
Log in to view solution →
Q16
mcq
The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is:
- A. 35/4°
- B. 35/2°
- C. 35°
- D. 70°
Solution hidden
Log in to view solution →
Q17
mcq
In the given figure, the quadrilateral PQRS circumscribes a circle. Here PA + CS is equal to:
- A. QR
- B. PR
- C. PS
- D. PQ
Solution hidden
Log in to view solution →
Q18
mcq
If α and β are the zeroes of the quadratic polynomial p(x) = x^2 - ax - b, then the value of α^2 + β^2 is:
- A. a^2 - 2b
- B. a^2 + 2b
- C. b^2 - 2a
- D. b^2 + 2a
Solution hidden
Log in to view solution →
Q18
mcq
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - ax + b$, then the value of $\alpha^2 + \beta^2$ is :
- A. $a^2 - 2b$
- B. $a^2 + 2b$
- C. $b^2 - 2a$
- D. $b^2 + 2a$
Solution hidden
Log in to view solution →
Q19
mcq
1 mark
Assertion (A): The polynomial p(x) = x^2 + 3x + 3 has two real zeroes.
Reason (R): A quadratic polynomial can have at most two real zeroes.
- A. Both (A) and (R) are true and (R) is the correct explanation of (A).
- B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C. (A) is true but (R) is false.
- D. (A) is false but (R) is true.
Solution hidden
Log in to view solution →
Q20
mcq
1 mark
Assertion (A): If two tangents PA and PB are drawn from an external point P to a circle with centre O, then the quadrilateral AOBP is cyclic.
Reason (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
- A. Both (A) and (R) are true and (R) is the correct explanation of (A).
- B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C. (A) is true but (R) is false.
- D. (A) is false but (R) is true.
Solution hidden
Log in to view solution →
Q22
short answer
2 marks
The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. Find the angle of elevation of the sun.
Solution hidden
Log in to view solution →
Q22
short answer
2 marks
In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If $\angle BAC = 65^\circ$, then find the measure of $\angle BOC$.
Solution hidden
Log in to view solution →
Q23
short answer
2 marks
The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of the tower, is $30^\circ$. Find the height of the tower.
Solution hidden
Log in to view solution →
Q24
short answer
2 marks
Show that the points (2, 3), (8, 3) and (6, 7) are the vertices of a right-angled triangle.
Solution hidden
Log in to view solution →
Q25
short answer
2 marks
If $4 \cot^2 45^\circ - \sec^2 60^\circ + \sin^2 60^\circ + p = \frac{4}{3}$, then find the value of p.
Solution hidden
Log in to view solution →
Q25
short answer
2 marks
Prove that $4^n$ can never end with digit 0, where n is a natural number.
Solution hidden
Log in to view solution →
Q26
short answer
2 marks
If $\cos A + \cos^2 A = 1$, then find the value of $\sin^2 A + \sin^4 A$.
Solution hidden
Log in to view solution →
Q26
short answer
3 marks
Prove that: $\frac{\cos \theta}{1 - \sin \theta} + \frac{\sin \theta}{1 - \cos \theta} = 2 \text{ cosec } \theta$
Solution hidden
Log in to view solution →
Q29
short answer
3 marks
In the given figure, $O$ is the centre of the circle and $QPR$ is a tangent to it at $P$. Prove that $\angle QAP + \angle APR = 90^\circ$.
Solution hidden
Log in to view solution →
Q30
short answer
If the system of linear equations $2x + 3y = 7$ and $2ax + (a + b)y = 28$ has infinitely many solutions, find the values of $a$ and $b$.
Solution hidden
Log in to view solution →
Q30
short answer
3 marks
If $Q(0, 1)$ is equidistant from $P(5, 3)$ and $R(x, 6)$, find the values of $x$.
Solution hidden
Log in to view solution →
Q31
short answer
If $217x + 131y = 913$ and $131x + 217y = 827$, then solve the equations for the values of $x$ and $y$.
Solution hidden
Log in to view solution →
Q31
short answer
3 marks
Reeti prepares a Rakhi for her brother Ronit. The Rakhi consists of a rectangle of length $8\text{ cm}$ and breadth $6\text{ cm}$ inscribed in a circle as shown in the figure. Find the area of the shaded region. (Use $\pi = 3.14$)
Solution hidden
Log in to view solution →
Q32
short answer
Find by prime factorisation the LCM of the numbers $18180$ and $7575$. Also, find the HCF of the two numbers.
Solution hidden
Log in to view solution →
Q32
long answer
5 marks
A student noted the number of cars passing through a spot on a road for $100$ periods each of $3$ minutes and summarised it in the table given below. Find the mean and median of the following data. (Table: Number of cars $0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80$ with frequency $7, 14, 13, 12, 20, 11, 15, 8$ respectively)
Solution hidden
Log in to view solution →
Q33
short answer
Three bells ring at intervals of $6, 12$ and $18$ minutes. If all the three bells rang at $6$ a.m., when will they ring together again ?
Solution hidden
Log in to view solution →
Q34
long answer
5 marks
Solve the equation for $x$ : $1 + 4 + 7 + 10 + .... + x = 287$
Solution hidden
Log in to view solution →
Q38
long answer
Sides $AB$ and $BC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $QR$ and median $PM$ of $\Delta PQR$. Show that $\Delta ABC \sim \Delta PQR$.
Solution hidden
Log in to view solution →
Q39
long answer
Through the mid-point $M$ of the side $CD$ of a parallelogram $ABCD$, the line $BM$ is drawn intersecting $AC$ in $L$ and $AD$ (produced) in $E$. Prove that $EL = 2BL$.
Solution hidden
Log in to view solution →
Q41
long answer
The angle of elevation of the top of a tower $30\text{ m}$ high from the foot of another tower in the same plane is $60^\circ$ and the angle of elevation of the top of the second tower from the foot of the first tower is $30^\circ$. Find the distance between both towers and the height of the first tower.
Solution hidden
Log in to view solution →
Q42
short answer
From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30 and 45 respectively. Find the distance between the two cars. (Use $\sqrt{3} = 1.73$)
Solution hidden
Log in to view solution →Reading Passage
Computer-based learning (CBL) refers to any teaching methodology that makes use of computers for information transmission. At an elementary school level, computer applications can be used to display multimedia lesson plans. A survey was done on 1000 elementary and secondary schools of Assam and they were classified by the number of computers they had. Number of Computers: 1-10, 11-20, 21-50, 51-100, 101 and more Number of Schools: 250, 200, 290, 180, 80 One school is chosen at random.
Q43
short answer
1 mark
Find the probability that the school chosen at random has more than 100 computers.
Solution hidden
Log in to view solution →Reading Passage
In an annual day function of a school, the organizers wanted to give a cash prize along with a memento to their best students. Each memento is made as shown in the figure and its base ABCD is shown from the front side. The rate of silver plating is 20 per cm^2.
Q44
short answer
1 mark
What is the area of the quadrant ODCO?
Solution hidden
Log in to view solution →
Q45
short answer
2 marks
What is the total cost of silver plating the shaded part ABCD?
Solution hidden
Log in to view solution →Reading Passage
In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter 7 cm and height 14 cm and the other is hemispherical with diameter 21 cm.
Q47
short answer
1 mark
Find the area of the base of the cylindrical cup.
Solution hidden
Log in to view solution →
Q48
short answer
2 marks
Find the capacity of the cylindrical cup.
Solution hidden
Log in to view solution →
Q49
short answer
1 mark
Find the curved surface area of the cylindrical cup.
Solution hidden
Log in to view solution →Reading Passage
In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter 7 cm and height 14 cm and the other is hemispherical with diameter 21 cm. Based on the above, answer the following questions :
Q50
short answer
2 marks
What is the capacity of the hemispherical cup ?
Solution hidden
Log in to view solution →