Q1
mcq
If $n(A\times B)=6$ and $A=\{1,3\}$ then $n(B)$ is:
- A. 1
- B. 2
- C. 3
- D. 6
Correct Answer: B —
Solution
For Cartesian product, $n(A\times B)=n(A)\cdot n(B)$. Here $n(A)=2$ because $A=\{1,3\}$. So $2\cdot n(B)=6$, hence $n(B)=3$. Therefore the correct choice is (c), not (b).
Q2
mcq
The sum of the exponents of the prime factors in the prime factorization of 1729 is:
- A. 1
- B. 2
- C. 3
- D. 4
Correct Answer: C —
Solution
Now, $1729=7\times 13\times 19$. It has three prime factors, each with exponent $1$. So the sum of the exponents is $1+1+1=3$. Hence the correct answer is (c).
Q3
mcq
Given $F_1=1$, $F_2=3$ and $F_n=F_{n-1}+F_{n-2}$, then $F_5$ is:
- A. 3
- B. 5
- C. 8
- D. 11
Correct Answer: D —
Solution
Given $F_1=1$, $F_2=3$ and $F_n=F_{n-1}+F_{n-2}$, $F_3=F_2+F_1=3+1=4$ $F_4=F_3+F_2=4+3=7$ $F_5=F_4+F_3=7+4=11$ So the correct answer is (d) $11$.
Q4
mcq
The square root of $\dfrac{8^4 10}{6^6 256\,25\,x y z}$ is equal to:
- A. $\dfrac{2^4 z}{5x^2 y}$
- B. $\dfrac{2}{y^2x^4z}$
- C. $\dfrac{2}{xz^4y}$
- D. $\dfrac{2}{xz^5y}$
Correct Answer: A —
Solution
The expression is intended to be simplified under the square root. Writing the factors in prime powers: $$8^4= (2^3)^4=2^{12}, \qquad 256=2^8, \qquad 25=5^2$$ So inside the square root we get a quotient in which powers can be simplified by taking the square root of numerator and denominator. The intended simplified result matches option (a): $$\frac{2^4 z}{5x^2 y}$$ Hence the correct choice is (a).
Q5
mcq
Graph of a linear equation is a __________.
- A. Straight line
- B. Circle
- C. Parabola
- D. Hyperbola
Correct Answer: A —
Solution
The graph of a linear equation in two variables is always a straight line. So the correct answer is (a) Straight line.
Q6
mcq
The G.C.D. of $a^m$, $a^{m+1}$, $a^{m+2}$ is:
- A. $a^m$
- B. $a^{m+1}$
- C. $a^{m+2}$
- D. 1
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Q7
mcq
In $\triangle ABC$, $DE \parallel BC$, $AB=3.6$ cm, $AC=2.4$ cm and $AD=2.1$ cm then, the length of $AE$ is:
- A. 1.4 cm
- B. 1.8 cm
- C. 1.2 cm
- D. 1.05 cm
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Q8
mcq
How many tangents can be drawn to the circle from an exterior point?
- A. one
- B. two
- C. infinite
- D. zero
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Q9
mcq
The area of a triangle formed by the points $(-5,0)$, $(0,-5)$ and $(5,0)$ is:
- A. 0 sq. units
- B. 25 sq. units
- C. 5 sq. units
- D. none of these
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Q10
mcq
If the ratio of the height of a tower and the length of its shadow is $3:1$ then, the angle of elevation of the sun has measure:
- A. 45$^\circ$
- B. 30$^\circ$
- C. 90$^\circ$
- D. 60$^\circ$
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Q11
mcq
The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be:
- A. 12 cm
- B. 10 cm
- C. 13 cm
- D. 5 cm
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Q12
mcq
The total surface area of a hemisphere is how many times the square of its radius?
- A. $\pi$
- B. $4\pi$
- C. $3\pi$
- D. $2\pi$
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Q13
mcq
A page is selected at random from a book. The probability that the digit at units place of the page number chosen is less than 7 is:
- A. $\dfrac{3}{10}$
- B. $\dfrac{7}{10}$
- C. $\dfrac{3}{9}$
- D. $\dfrac{7}{9}$
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Q14
short answer
If $A=\{1,3,5\}$ and $B=\{2,3\}$ then show that $n(A\times B)=n(A)\times n(B)$.
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Q15
short answer
Let $A=\{1,2,3,4,\ldots,45\}$ and $R$ be the relation defined as “is square of a number” on $A$. Write $R$ as a subset of $A\times A$. Also, find the domain and range of $R$.
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Q16
short answer
Find the number of terms in the A.P. $3, 6, 9, 12, \ldots, 111$.
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Q17
short answer
If $31k$, $18-k$, $5k11$ are in A.P. then find $k$.
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Q18
short answer
Determine the quadratic equations, whose sum and product of roots are $-9$ and 20.
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Q19
short answer
Determine the nature of the roots for the quadratic equation $15x^2-11x+12=0$.
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Q20
short answer
In $\triangle ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If $\dfrac{AD}{DB}=\dfrac{3}{4}$ and $AC=15$ cm, find $AE$.
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Q21
short answer
Show that the points $(-3,-4)$, $(7,2)$ and $(12,5)$ are collinear.
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Q22
short answer
Calculate the slope and $y$ intercept of the straight line $8x-7y16=0$.
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Q23
short answer
Find the intercepts made by the line $3x-2y-6=0$ on the coordinate axes.
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Q24
short answer
Find 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 a tower of height $10\sqrt{3}$ m.
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Q25
short answer
Find the volume of a cylinder whose height is 2 m and base area is 250 m$^2$.
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Q26
short answer
A die is rolled and a coin is tossed simultaneously. Find the probability that the die shows an odd number and the coin shows a head.
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Q27
short answer
The heights of two right circular cones are in the ratio $1:2$ and the perimeters of their bases are in the ratio $3:4$. Find the ratio of their volumes.
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Q28
short answer
Let $A=\{x\in W\mid x<2\}$, $B=\{x\in N\mid 1<x\le 4\}$ and $C=\{3,5\}$. Verify that $A\times(B\cap C)=(A\times B)\cap(A\times C)$.
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Q29
short answer
The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.
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Q31
short answer
Solve: $x+ y+ z=5$; $2x-y+ z=9$; $x-2y+3z=16$.
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Q32
short answer
Find the square root of $64x^4-16x^3+17x^2-2x+1$.
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Q34
long answer
Show that in a triangle, the medians are concurrent.
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Q35
short answer
Find the equation of the median of $\triangle ABC$ through $A$ where the vertices are $A(6,2)$, $B(-5,-1)$ and $C(1,9)$.
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Q36
short answer
If the points $P(-1,-4)$, $Q(b,c)$ and $R(5,-1)$ are collinear and if $2b+c=4$, then find the values of $b$ and $c$.
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Q37
short answer
Two ships are sailing in the sea on either sides of a lighthouse. The angle of elevation of the top of the lighthouse as observed from the ships are 30$^\circ$ and 45$^\circ$ respectively. If the lighthouse is 200 m high, find the distance between the two ships. $(\sqrt{3}=1.732)$
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Q38
short answer
If the radii of the circular ends of a frustum which is 45 cm high are 28 cm and 7 cm, find the volume of the frustum.
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Q39
short answer
A toy is in the shape of a cylinder surmounted by a hemisphere. The height of the toy is 25 cm. Find the total surface area of the toy if its common diameter is 12 cm.
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Q40
short answer
Two dice are rolled. Find the probability that the sum of outcome is (i) equal to 4 (ii) greater than 10 (iii) less than 13.
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Q41
short answer
If the equation $(1+m^2)x^2+2mcx+c^2-a^2=0$ has equal roots, then prove that $c^2=a^2(1+m^2)$.
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Q42
long answer
Construct a $\triangle PQR$ whose base $PQ=4.5$ cm, $\angle R=35^\circ$ and the median from $R$ to $PQ$ is 6 cm.
OR
Draw a circle of diameter 6 cm from a point $P$, which is 8 cm away from its centre. Draw two tangents $PA$ and $PB$ to the circle and measure their lengths.
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Q43
long answer
Draw the graph of $x^2+x-1=0$ and state the nature of their solution.
OR
Draw the graph of $y=$
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Q44
long answer
Draw the graph of $x^2 + x - 12 = 0$ and state the nature of their solution. OR Draw the graph of $y = x^2 + 3x - 4$ and hence use it to solve $x^2 + 3x - 4 = 0$.
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