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given that \( 4(y-2)^{2}=k(y-1) \), where k is a constant. (i) Show hat the sum of the roots equals their product. (ii) Find the value for which the difference between the roots is \( 33 / 4 \)
Algebra Jan 22, 2025
If The matrix \( x=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right) \) and \( y=\left(\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right. \) find \( 4 x-3 y \)
Algebra Jan 22, 2025
If \( \alpha, \beta \) are the roots of the equation \( 2 x^{2}+6 x-10=0 \). Find \( \alpha^{4}+\beta^{4} \) (b) \( \alpha^{3}+\beta^{3} \) (c) \( \alpha^{6}+\beta^{6} \) (d) \( \alpha^{6}-\beta^{6} \)
Algebra Jan 22, 2025
Question 1: The general expansion of \( (a+b)^{\wedge} n \) is given by: - A) \( a^{\wedge} n+b^{\wedge} n \) - B) \( \Sigma k=0^{\wedge} n n k a^{\wedge} n-k b^{\wedge} k \) - C) \( (a+b) \) - D) \( a+b \) Question 2 : In the binomial expansion of \( (x+2)^{\wedge} 3 \), what is the coefficient of \( x^{\wedge} 2 \) ? - A) 6 - B) 12 - C) 4 - D) 3 Question 3 : In the expansion
Algebra Jan 22, 2025
Avaluate \( \left(\begin{array}{cc}3 & -2 \\ -7 & 5\end{array}\right)+2\left(\begin{array}{cc}-2 & 4 \\ 3 & -1\end{array}\right) \)
Algebra Jan 22, 2025
2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
2. A particle moves along the curve \[ \begin{array}{l}x=2 t^{2}, \quad y=t^{2}-4 t \quad \text { and } \quad z=3 t-5 \\ \text { where } t \text { is the time. Find the components of its velocity and acceleration at time } t=1 \text {, in the } \\ \text { direction } \hat{i}-3 \hat{j}+2 \hat{k} \text {. } \\ \text { (Nagpur, Summer 200l) Ans. } \frac{8 \sqrt{14}}{7},-\frac{\sqrt{14}}{7}\end{array} \]
Calculus Jan 22, 2025
1 The coovdinate of the end point of a diancia of 1 circle zre \( \langle-5,3\rangle \) and \( \langle 3,-4\rangle \) Find 2. Equation of the curcle. 1) Lengtis of the diameter c Equation of the diameter
Geometry Jan 22, 2025
33. Solve \( \sin \theta=\tan \theta \). (A) \( 200^{\circ} \) (B) \( 90^{\circ} \) (C) \( 60^{\circ} \) (D.) \( 0^{\circ} \) 34. The sum of the first three terms of an Arithmetic progression (A.P.) is 18 . If the first term is 4 . Find their product. (A) 260 (B) 210 (C) 192 (D) 130 35. Evaluate \( \lim _{x \rightarrow 3}\left(\frac{x^{2}-2 x-3}{x-3}\right) \). (A) 4 (B) 3 (C) 2 (D) 0 36. Given that \( \overrightarrow{A B}=5 i+3 j \) and \( \overrightarrow{A C}=2 i+5 j \), find \( \overrightarrow{B C} \). (A) \( 3 i-2 j \) (B) \( -7 i-8 j \) (C) \( -3 i+2 j \) (D) \( 3 i+8 j \) 37. Find the derivative of \( 3 x^{2}+\frac{1}{x^{2}} \). (A) \( 6 x-\frac{1}{2 x^{2}} \) (B) \( 6 x-\frac{2}{x^{3}} \) (C) \( 6 x-\frac{1}{2 x} \) (D) \( 6 x+2 x^{2} \) 38. A body starts from rest and moves in a straight line with a uniform acceleration of \( 5 \mathrm{mls}{ }^{2} \). How far, in \( m \), does it go in 10 seconds? (A) 50 m (B) 250 m (C) 350 m (D) 500 m 39. If \( n \) items are arranged two at a time, the number obtained is 20 . Find the value of \( n \). (A) 40 (B) 15 (C) 10 (D) 5 40. The gradient of point \( P \) on the curve \( y=3 x^{2}-x+3 \) is 5 . Find the coordinates of \( P \). (A) \( (1,5) \) (B) \( (1,7) \) (C) \( (1,13) \) (D) \( (1,17) \) 41 Evaluate \( \frac{1}{1-\sin 60^{\circ}} \) leaving your answer in surd form. (A) \( 4-2 \sqrt{3} \) (B) \( 2-\sqrt{3} \) (C) \( 1-\sqrt{3} \) (D) \( 4+2 \sqrt{3} \) 42. Calculate, correct to one decimal place, the standard deviation of the numbers \( 85,84,83,86 \) and 87 . (A) 1.8 (B) 1.6 (C) \( \cdot 1.4 \) (D) 2.0 43. Find the value of \( x \) at the point of intersection of the curve \( y=x^{2}+2 x-3 \) and the line \( y+x=1 \). (A) \( (1,-2) \) (B) \( \left(0,,^{4}\right) \) (C) \( (1,-4) \) (D) \( (2,3) \) 44. Given that \( n=3 \), evaluate \( \frac{1}{(n-1)!}-\frac{1}{(n+1)!} \) (A) 12 (B) \( 2 \frac{1}{2} \) (C) \( \frac{11}{24} \) (D) 2
Calculus Jan 22, 2025
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